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<title>Rogers-Ramanujan-Identitäten</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading">Rogers-Ramanujan-Identitäten</h1>

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr">Die Rogers-Ramanujan-Identitäten sind ursprünglich zwei Identitäten zwischen unendlichen Reihen und Produkten, die zuerst <a href="Leonard_James_Rogers" title="Leonard James Rogers">Leonard James Rogers</a> 1894<a href="#cite_note-1">[1]</a> bewies. <a href="S._Ramanujan" class="mw-redirect" title="S. Ramanujan">S. Ramanujan</a> fand sie unabhängig vor 1913 (ohne Beweis).<a href="#cite_note-2">[2]</a> Ramanujan stieß danach durch Zufall auf den Aufsatz von Rogers, der bis dahin kaum beachtet worden war, und veröffentlichte mit Rogers 1919 einen neuen Beweis.<a href="#cite_note-3">[3]</a> Unabhängig fand <a href="Issai_Schur" title="Issai Schur">Issai Schur</a> 1917 die Identitäten und einen Beweis.<a href="#cite_note-4">[4]</a> Es gibt auch Verallgemeinerungen der Identitäten.

<div class="mw-heading mw-heading2"><h2 id="Hauptteil">Hauptteil</h2></div>

Die Identitäten lauten (mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |q|<1}">

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<annotation encoding="application/x-tex">{\displaystyle |q|<1}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ef46eeba8cb2497a797ec6652c0f76d69e2175bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.624ex; height:2.843ex;" alt="{\displaystyle |q|<1}" loading="lazy">):

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<annotation encoding="application/x-tex">{\displaystyle G(q)=\sum _{n=0}^{\infty }{\frac {q^{n^{2}}}{(q;q)_{n}}}={\frac {1}{(q;q^{5})_{\infty }(q^{4};q^{5})_{\infty }}}=1+q+q^{2}+q^{3}+2q^{4}+2q^{5}+3q^{6}+\cdots \,}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ceb271c2772fffa63fc25449fda66e9a04f20fec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:81.905ex; height:7.176ex;" alt="{\displaystyle G(q)=\sum _{n=0}^{\infty }{\frac {q^{n^{2}}}{(q;q)_{n}}}={\frac {1}{(q;q^{5})_{\infty }(q^{4};q^{5})_{\infty }}}=1+q+q^{2}+q^{3}+2q^{4}+2q^{5}+3q^{6}+\cdots \,}" loading="lazy"></dd></dl>

und

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<annotation encoding="application/x-tex">{\displaystyle H(q)=\sum _{n=0}^{\infty }{\frac {q^{n(n+1)}}{(q;q)_{n}}}={\frac {1}{(q^{2};q^{5})_{\infty }(q^{3};q^{5})_{\infty }}}=1+q^{2}+q^{3}+q^{4}+q^{5}+2q^{6}+\cdots \,}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bf4ccfc6dcebd645be70e57757dc0963d19ced76.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:77.434ex; height:7.009ex;" alt="{\displaystyle H(q)=\sum _{n=0}^{\infty }{\frac {q^{n(n+1)}}{(q;q)_{n}}}={\frac {1}{(q^{2};q^{5})_{\infty }(q^{3};q^{5})_{\infty }}}=1+q^{2}+q^{3}+q^{4}+q^{5}+2q^{6}+\cdots \,}" loading="lazy"></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle G(q)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c09834d81f560d5fab4385a3e3102e7c7182c29e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.706ex; height:2.843ex;" alt="{\displaystyle G(q)}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(q)}">

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<annotation encoding="application/x-tex">{\displaystyle H(q)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bb7258e7274db86188844940fad02593b9ec5666.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.942ex; height:2.843ex;" alt="{\displaystyle H(q)}" loading="lazy"> definiert über den jeweils linken Teil der Identitäten (als unendliche Reihe) heißen Rogers-Ramanujan-Funktionen.

Dabei sind <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\cdot ;\cdot )_{n}}">

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<annotation encoding="application/x-tex">{\displaystyle (\cdot ;\cdot )_{n}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d0df47a5a50bac57b33642e810673d2fd0ad5ecc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.355ex; height:2.843ex;" alt="{\displaystyle (\cdot ;\cdot )_{n}}" loading="lazy"> die <a href="Pochhammer-Symbol" title="Pochhammer-Symbol">q-Pochhammer-Symbole</a>:

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<annotation encoding="application/x-tex">{\displaystyle (q;q)_{n}=\prod _{k=1}^{n}(1-q^{k})=(1-q)(1-q^{2})\cdots (1-q^{n})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/10b612fcad244a6cc03cc0b2bd25e4d08380cd65.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:49.782ex; height:6.843ex;" alt="{\displaystyle (q;q)_{n}=\prod _{k=1}^{n}(1-q^{k})=(1-q)(1-q^{2})\cdots (1-q^{n})}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle (q;q^{5})_{\infty }=\prod _{k=0}^{\infty }(1-q^{5k+1})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bb37535618c3582bcbe4ae4cf8f7b61dc4ac6f17.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:24.893ex; height:7.009ex;" alt="{\displaystyle (q;q^{5})_{\infty }=\prod _{k=0}^{\infty }(1-q^{5k+1})}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle (q^{4};q^{5})_{\infty }=\prod _{k=0}^{\infty }(1-q^{5k+4})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0a6963bd17e778b210170661e3e75f1de1435cfb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:25.957ex; height:7.009ex;" alt="{\displaystyle (q^{4};q^{5})_{\infty }=\prod _{k=0}^{\infty }(1-q^{5k+4})}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle (q^{2};q^{5})_{\infty }=\prod _{k=0}^{\infty }(1-q^{5k+2})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f8d693c835c00fa87d4407e355f3d4847b137363.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:25.957ex; height:7.009ex;" alt="{\displaystyle (q^{2};q^{5})_{\infty }=\prod _{k=0}^{\infty }(1-q^{5k+2})}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle (q^{3};q^{5})_{\infty }=\prod _{k=0}^{\infty }(1-q^{5k+3})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/eab5e00c10e6a9fdddfb3b8bc3cd4f02552261cc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:25.957ex; height:7.009ex;" alt="{\displaystyle (q^{3};q^{5})_{\infty }=\prod _{k=0}^{\infty }(1-q^{5k+3})}" loading="lazy"></dd></dl>

So dass die Identitäten sich auch schreiben lassen:

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<annotation encoding="application/x-tex">{\displaystyle G(q)=\sum _{n=0}^{\infty }{\frac {q^{n^{2}}}{(1-q)(1-q^{2})\cdots (1-q^{n})}}={\frac {1}{\prod _{k=0}^{\infty }(1-q^{5k+1})(1-q^{5k+4})}}={\frac {1}{\prod _{k=1}^{\infty }(1-q^{5k-1})(1-q^{5k-4})}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c900b45fa94f339a5a61c34341f8dcbf2843d8df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:101.064ex; height:7.176ex;" alt="{\displaystyle G(q)=\sum _{n=0}^{\infty }{\frac {q^{n^{2}}}{(1-q)(1-q^{2})\cdots (1-q^{n})}}={\frac {1}{\prod _{k=0}^{\infty }(1-q^{5k+1})(1-q^{5k+4})}}={\frac {1}{\prod _{k=1}^{\infty }(1-q^{5k-1})(1-q^{5k-4})}}}" loading="lazy"></dd></dl>

und

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<annotation encoding="application/x-tex">{\displaystyle H(q)=\sum _{n=0}^{\infty }{\frac {q^{n(n+1)}}{(1-q)(1-q^{2})\cdots (1-q^{n})}}={\frac {1}{\prod _{k=0}^{\infty }(1-q^{5k+2})(1-q^{5k+3})}}={\frac {1}{\prod _{k=1}^{\infty }(1-q^{5k-2})(1-q^{5k-3})}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/981d3b35cdbd2003ed68aac57a13c4f5afb1a611.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:101.3ex; height:7.009ex;" alt="{\displaystyle H(q)=\sum _{n=0}^{\infty }{\frac {q^{n(n+1)}}{(1-q)(1-q^{2})\cdots (1-q^{n})}}={\frac {1}{\prod _{k=0}^{\infty }(1-q^{5k+2})(1-q^{5k+3})}}={\frac {1}{\prod _{k=1}^{\infty }(1-q^{5k-2})(1-q^{5k-3})}}}" loading="lazy"></dd></dl>

Es gibt auch verallgemeinerte Identitäten vom Rogers-Ramanujan-Typ, die insbesondere in Arbeiten von <a href="Wilfrid_Norman_Bailey" title="Wilfrid Norman Bailey">Wilfrid Norman Bailey</a>,<a href="#cite_note-5">[5]</a> <a href="Freeman_Dyson" class="mw-redirect" title="Freeman Dyson">Freeman Dyson</a>, <a href="Atle_Selberg" title="Atle Selberg">Atle Selberg</a> und <a href="Lucy_Joan_Slater" title="Lucy Joan Slater">Lucy Joan Slater</a><a href="#cite_note-6">[6]</a> aufgestellt wurden (Slater listet in ihrem Aufsatz von 1952 130 solche Identitäten). Weitere fand z. B. <a href="George_E._Andrews" class="mw-redirect" title="George E. Andrews">George E. Andrews</a> (Andrews-Gordon-Identität, mit <a href="Basil_Gordon" title="Basil Gordon">Basil Gordon</a>),<a href="#cite_note-7">[7]</a> <a href="Heinz_G%C3%B6llnitz" title="Heinz Göllnitz">Heinz Göllnitz</a> (Göllnitz-Gordon-Identitäten).

Ramanujan führte insgesamt 40 Identitäten mit den Funktionen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G(q),H(q)}">

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<annotation encoding="application/x-tex">{\displaystyle G(q),H(q)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/82723fbae43e2d9496d4cb02a5b33c32a2b23c75.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.682ex; height:2.843ex;" alt="{\displaystyle G(q),H(q)}" loading="lazy"> auf (in seinen Notizbüchern).<a href="#cite_note-8">[8]</a>

<div class="mw-heading mw-heading2"><h2 id="Anwendung_auf_Partitionen">Anwendung auf Partitionen</h2></div>

Da die in der Identität vorkommenden Terme erzeugende Funktionen bestimmter <a href="Partitionsfunktion" title="Partitionsfunktion">Partitionen</a> sind, machen die Identitäten Aussagen über Partitionen (Zerfällungen) <a href="Nat%C3%BCrliche_Zahl" title="Natürliche Zahl">natürlicher Zahlen</a>. Die Zahlenfolgen, welche sich aus den Koeffizienten der Maclaurinschen Reihen von den Rogers-Ramanujan-Funktionen G und H ergeben, sind spezielle Partitionszahlenfolgen der Stufe 5:

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<annotation encoding="application/x-tex">{\displaystyle G(x)={\frac {1}{(x;x^{5})_{\infty }(x^{4};x^{5})_{\infty }}}=1+\sum _{n=1}^{\infty }P_{G}(n)x^{n}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aa01600fe1d48eaa2a80a101fd5e83c75b2f10ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:46.431ex; height:6.843ex;" alt="{\displaystyle G(x)={\frac {1}{(x;x^{5})_{\infty }(x^{4};x^{5})_{\infty }}}=1+\sum _{n=1}^{\infty }P_{G}(n)x^{n}}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle H(x)={\frac {1}{(x^{2};x^{5})_{\infty }(x^{3};x^{5})_{\infty }}}=1+\sum _{n=1}^{\infty }P_{H}(n)x^{n}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/949703034e710bf247798920a2d42fea168523e7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:47.889ex; height:6.843ex;" alt="{\displaystyle H(x)={\frac {1}{(x^{2};x^{5})_{\infty }(x^{3};x^{5})_{\infty }}}=1+\sum _{n=1}^{\infty }P_{H}(n)x^{n}}" loading="lazy"></dd></dl>

Die Zahlenfolge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{G}(n)}">

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<annotation encoding="application/x-tex">{\displaystyle P_{G}(n)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f16d5bae307eb4580ec34cc38f86f16edf4ed35f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.22ex; height:2.843ex;" alt="{\displaystyle P_{G}(n)}" loading="lazy"> (OEIS-Code: A003114<a href="#cite_note-9">[9]</a>) stellt für die betroffene natürliche Zahl n die Anzahl der Möglichkeiten dar, diese Zahl in Summanden der Muster 4a + 1 oder 4a + 4 mit a ∈ ℕ₀ zu zerlegen. Somit gibt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{G}(n)}">

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<annotation encoding="application/x-tex">{\displaystyle P_{G}(n)}</annotation>

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Und die Zahlenfolge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{H}(n)}">

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<annotation encoding="application/x-tex">{\displaystyle P_{H}(n)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a2929264dfcbd9a1ab8bfc3bd9eda6ffd70b35e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.388ex; height:2.843ex;" alt="{\displaystyle P_{H}(n)}" loading="lazy"> (OEIS-Code: A003106<a href="#cite_note-10">[10]</a>) stellt analog hierzu für die betroffene natürliche Zahl n die Anzahl der Möglichkeiten dar, diese Zahl in Summanden der Muster 4a + 2 oder 4a + 3 mit a ∈ ℕ₀ zu zerlegen. Somit gibt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{H}(n)}">

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<annotation encoding="application/x-tex">{\displaystyle P_{H}(n)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a2929264dfcbd9a1ab8bfc3bd9eda6ffd70b35e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.388ex; height:2.843ex;" alt="{\displaystyle P_{H}(n)}" loading="lazy"> die Anzahl der Zerfällungen einer ganzen Zahl n, bei denen sich benachbarte Teile der Partition um mindestens 2 unterscheiden und bei der der kleinste Teil größer oder gleich 2 ist, ist gleich der Anzahl der Zerfällungen, deren Teile gleich 2 oder 3 mod 5 sind. Dies soll in den folgenden zwei Tabellen exemplarisch veranschaulicht werden:

<caption>Partitionszahlenfolge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{G}(n)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P_{G}(n)}</annotation>

</semantics>

</caption>

<th>Natürliche Zahl n

</th>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P_{G}(n)}</annotation>

</semantics>

</th>

<th>Summandarstellungen mit dem genannten Kriterium

</th></tr>

<tr>

<td>1

</td>

<td>1

</td>

<td>1

</td></tr>

<tr>

<td>2

</td>

<td>1

</td>

<td>1+1

</td></tr>

<tr>

<td>3

</td>

<td>1

</td>

<td>1+1+1

</td></tr>

<tr>

<td>4

</td>

<td>2

</td>

<td>4, 1+1+1+1

</td></tr>

<tr>

<td>5

</td>

<td>2

</td>

<td>4+1, 1+1+1+1+1

</td></tr>

<tr>

<td>6

</td>

<td>3

</td>

<td>6, 4+1+1, 1+1+1+1+1+1

</td></tr>

<tr>

<td>7

</td>

<td>3

</td>

<td>6+1, 4+1+1+1, 1+1+1+1+1+1+1

</td></tr>

<tr>

<td>8

</td>

<td>4

</td>

<td>6+1+1, 4+4, 4+1+1+1+1, 1+1+1+1+1+1+1+1

</td></tr>

<tr>

<td>9

</td>

<td>5

</td>

<td>9, 6+1+1+1, 4+4+1, 4+1+1+1+1+1, 1+1+1+1+1+1+1+1+1

</td></tr>

<tr>

<td>10

</td>

<td>6

</td>

<td>9+1, 6+4, 6+1+1+1+1, 4+4+1+1, 4+1+1+1+1+1+1, 1+1+1+1+1+1+1+1+1+1

</td></tr>

<tr>

<td>11

</td>

<td>7

</td>

<td>11, 9+1+1, 6+4+1, 6+1+1+1+1+1, 4+4+1+1+1, 4+1+1+1+1+1+1+1, 1+1+1+1+1+1+1+1+1+1+1

</td></tr>

<tr>

<td>12

</td>

<td>9

</td>

<td>11+1, 9+1+1+1, 6+6, 6+4+1+1, 6+1+1+1+1+1+1, 4+4+4, 4+4+1+1+1+1, 4+1+1+1+1+1+1+1+1, 1+1+1+1+1+1+1+1+1+1+1+1

</td></tr>

<tr>

<td>13

</td>

<td>10

</td>

<td>11+1+1, 9+4, 9+1+1+1+1, 6+6+1, 6+4+1+1+1, 6+1+1+1+1+1+1+1, 4+4+4+1, 4+4+1+1+1+1+1, 4+1+1+1+1+1+1+1+1+1, 1+1+1+1+1+1+1+1+1+1+1+1+1

</td></tr>

<tr>

<td>14

</td>

<td>12

</td>

<td>14, 11+1+1+1, 9+4+1, 9+1+1+1+1+1, 6+6+1+1, 6+4+4, 6+4+1+1+1+1, 6+1+1+1+1+1+1+1+1, 4+4+4+1+1, 4+4+1+1+1+1+1+1, 4+1+1+1+1+1+1+1+1+1+1, 1+1+1+1+1+1+1+1+1+1+1+1+1+1

</td></tr>

<tr>

<td>15

</td>

<td>14

</td>

<td>14+1, 11+4, 11+1+1+1+1, 9+6, 9+4+1+1, 9+1+1+1+1+1+1, 6+6+1+1+1, 6+4+4+1, 6+4+1+1+1+1+1, 6+1+1+1+1+1+1+1+1+1, 4+4+4+1+1+1, 4+4+1+1+1+1+1+1+1, 4+1+1+1+1+1+1+1+1+1+1+1, 1+1+1+1+1+1+1+1+1+1+1+1+1+1+1

</td></tr>

<tr>

<td>16

</td>

<td>17

</td>

<td>16, 14+1+1, 11+4+1, 11+1+1+1+1+1, 9+6+1, 9+4+1+1+1, 9+1+1+1+1+1+1+1, 6+6+4, 6+6+1+1+1+1, 6+4+4+1+1, 6+4+1+1+1+1+1+1, 6+1+1+1+1+1+1+1+1+1+1, 4+4+4+4, 4+4+4+1+1+1+1, 4+4+1+1+1+1+1+1+1+1, 4+1+1+1+1+1+1+1+1+1+1+1+1, 1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1

</td></tr></tbody></table>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P_{H}(n)}</annotation>

</semantics>

</caption>

<th>Natürliche Zahl n

</th>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P_{H}(n)}</annotation>

</semantics>

</th>

<th>Summandarstellungen mit dem genannten Kriterium

</th></tr>

<tr>

<td>1

</td>

<td>0

</td>

<td>keine

</td></tr>

<tr>

<td>2

</td>

<td>1

</td>

<td>2

</td></tr>

<tr>

<td>3

</td>

<td>1

</td>

<td>3

</td></tr>

<tr>

<td>4

</td>

<td>1

</td>

<td>2+2

</td></tr>

<tr>

<td>5

</td>

<td>1

</td>

<td>3+2

</td></tr>

<tr>

<td>6

</td>

<td>2

</td>

<td>3+3, 2+2+2

</td></tr>

<tr>

<td>7

</td>

<td>2

</td>

<td>7, 3+2+2

</td></tr>

<tr>

<td>8

</td>

<td>3

</td>

<td>8, 3+3+2, 2+2+2+2

</td></tr>

<tr>

<td>9

</td>

<td>3

</td>

<td>7+2, 3+3+3, 3+2+2+2

</td></tr>

<tr>

<td>10

</td>

<td>4

</td>

<td>8+2, 7+3, 3+3+2+2, 2+2+2+2+2

</td></tr></tbody></table>

<div class="mw-heading mw-heading2"><h2 id="Kunde_über_die_Kettenbrüche">Kunde über die Kettenbrüche</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Modulär_abgewandelte_Funktionen_von_G_und_H">Modulär abgewandelte Funktionen von G und H</h3></div>

Setzt man <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q=e^{2\pi i\tau }}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q=e^{2\pi i\tau }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/436512744dca0d805b9f95346f2ae740174a54e2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.665ex; height:3.009ex;" alt="{\displaystyle q=e^{2\pi i\tau }}" loading="lazy"> (wobei der Imaginärteil von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau \in \mathbb {C} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tau \in \mathbb {C} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/509194ac747c91eeaf8a5d2a1198f4d7e8f5ec54.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.721ex; height:2.176ex;" alt="{\displaystyle \tau \in \mathbb {C} }" loading="lazy"> positiv ist), sind

<msub>

</mrow>

</msub>

<msup>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G_{M}(q)=q^{\frac {-1}{60}}G(q)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/09f57ff8cbd83d37b321495f6a5485ade1a51c0f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.322ex; height:4.176ex;" alt="{\displaystyle G_{M}(q)=q^{\frac {-1}{60}}G(q)}" loading="lazy"></dd></dl>

und

<msub>

</mrow>

</msub>

<msup>

<mfrac>

</mfrac>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H_{M}(q)=q^{\frac {11}{60}}H(q)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1fb4cd71841831a343c3ccae1a702b24c2401c23.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.293ex; height:4.009ex;" alt="{\displaystyle H_{M}(q)=q^{\frac {11}{60}}H(q)}" loading="lazy"></dd></dl>

<a href="Modulfunktion" class="mw-redirect" title="Modulfunktion">Modulfunktionen</a>!

Diese Funktionen haben für den <a href="Kehrwert" title="Kehrwert">Kehrwert</a> der Gelfondschen Konstante und für das Quadrat von diesem Kehrwert gelten folgende Werte:

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

<msqrt>

</msqrt>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

<mroot>

</mrow>

</mroot>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G_{M}{\bigl [}\exp(-\pi ){\bigr ]}=2^{-1/2}5^{-1/4}({\sqrt {5}}-1)^{1/4}({\sqrt[{4}]{5}}+1)^{1/2}R{\bigl [}\exp(-\pi ){\bigr ]}^{-1/2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1d07a3f2fcf3ccbdf81690a49b5e929547be9a32.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:67.777ex; height:3.843ex;" alt="{\displaystyle G_{M}{\bigl [}\exp(-\pi ){\bigr ]}=2^{-1/2}5^{-1/4}({\sqrt {5}}-1)^{1/4}({\sqrt[{4}]{5}}+1)^{1/2}R{\bigl [}\exp(-\pi ){\bigr ]}^{-1/2}}" loading="lazy"></dd>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

<msqrt>

</msqrt>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

<mroot>

</mrow>

</mroot>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H_{M}{\bigl [}\exp(-\pi ){\bigr ]}=2^{-1/2}5^{-1/4}({\sqrt {5}}-1)^{1/4}({\sqrt[{4}]{5}}+1)^{1/2}R{\bigl [}\exp(-\pi ){\bigr ]}^{1/2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/389ecd6ad80318ca96de224e73dbfb27b96e3831.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:66.603ex; height:3.843ex;" alt="{\displaystyle H_{M}{\bigl [}\exp(-\pi ){\bigr ]}=2^{-1/2}5^{-1/4}({\sqrt {5}}-1)^{1/4}({\sqrt[{4}]{5}}+1)^{1/2}R{\bigl [}\exp(-\pi ){\bigr ]}^{1/2}}" loading="lazy"></dd>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

<msqrt>

</msqrt>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G_{M}{\bigl [}\exp(-2\pi ){\bigr ]}=10^{-1/4}({\sqrt {5}}-1)^{1/4}R{\bigl [}\exp(-2\pi ){\bigr ]}^{-1/2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/59a399eace9278b6add5753093435ce68435c7af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:54.517ex; height:3.843ex;" alt="{\displaystyle G_{M}{\bigl [}\exp(-2\pi ){\bigr ]}=10^{-1/4}({\sqrt {5}}-1)^{1/4}R{\bigl [}\exp(-2\pi ){\bigr ]}^{-1/2}}" loading="lazy"></dd>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

<msqrt>

</msqrt>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H_{M}{\bigl [}\exp(-2\pi ){\bigr ]}=10^{-1/4}({\sqrt {5}}-1)^{1/4}R{\bigl [}\exp(-2\pi ){\bigr ]}^{1/2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c8d9c3dfb31745a7d513814808b68a1feb67a48f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:53.343ex; height:3.843ex;" alt="{\displaystyle H_{M}{\bigl [}\exp(-2\pi ){\bigr ]}=10^{-1/4}({\sqrt {5}}-1)^{1/4}R{\bigl [}\exp(-2\pi ){\bigr ]}^{1/2}}" loading="lazy"></dd></dl>

Der Rogers-Ramanujan-Kettenbruch nimmt für diese Abszissenwerte folgende Ordinatenwerte an:

<td>

<mtr>

<mtd>

</mrow>

</mtd>

<mtd>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<msqrt>

</msqrt>

</mrow>

<msqrt>

</msqrt>

</mrow>

<msqrt>

<msqrt>

</msqrt>

</mrow>

</msqrt>

</mrow>

<msqrt>

<msqrt>

</msqrt>

</mrow>

</msqrt>

</mrow>

<mroot>

</mrow>

</mroot>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mi>arctan</mi>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mi>arcsin</mi>

<msup>

</mrow>

</msup>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<msup>

</mrow>

</msup>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<msqrt>

</msqrt>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}R[\exp(-\pi )]{}&={\tfrac {1}{4}}({\sqrt {5}}+1)({\sqrt {5}}-{\sqrt {{\sqrt {5}}+2}})({\sqrt {{\sqrt {5}}+2}}+{\sqrt[{4}]{5}})=\\[4pt]&{}=\tan {\bigl [}{\tfrac {1}{4}}\arctan(2)+{\tfrac {1}{2}}\arcsin(\Phi ^{-2}){\bigr ]}=\\[4pt]&{}=\Phi ^{3/2}\operatorname {cl} ({\tfrac {1}{5}}\varpi )^{-3/2}\operatorname {cl} ({\tfrac {2}{5}}\varpi )^{3/2}\operatorname {cl} ({\tfrac {1}{10}}\varpi )^{2}\operatorname {cl} ({\tfrac {3}{10}}\varpi )\operatorname {slh} ({\tfrac {2}{5}}{\sqrt {2}}\,\varpi )\\[4pt]\end{aligned}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8545232e28820665ddb12d39284bf7e8e32026f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.671ex; width:71.111ex; height:14.509ex;" alt="{\displaystyle {\begin{aligned}R[\exp(-\pi )]{}&={\tfrac {1}{4}}({\sqrt {5}}+1)({\sqrt {5}}-{\sqrt {{\sqrt {5}}+2}})({\sqrt {{\sqrt {5}}+2}}+{\sqrt[{4}]{5}})=\\[4pt]&{}=\tan {\bigl [}{\tfrac {1}{4}}\arctan(2)+{\tfrac {1}{2}}\arcsin(\Phi ^{-2}){\bigr ]}=\\[4pt]&{}=\Phi ^{3/2}\operatorname {cl} ({\tfrac {1}{5}}\varpi )^{-3/2}\operatorname {cl} ({\tfrac {2}{5}}\varpi )^{3/2}\operatorname {cl} ({\tfrac {1}{10}}\varpi )^{2}\operatorname {cl} ({\tfrac {3}{10}}\varpi )\operatorname {slh} ({\tfrac {2}{5}}{\sqrt {2}}\,\varpi )\\[4pt]\end{aligned}}}" loading="lazy">

</td></tr>

<tr>

<td>

<mtr>

<mtd>

</mrow>

</mtd>

<mtd>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mi>arctan</mi>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}R[\exp(-2\pi )]{}&=4\sin({\tfrac {1}{20}}\pi )\sin({\tfrac {3}{20}}\pi )=\\[4pt]&{}=\tan {\bigl [}{\tfrac {1}{4}}\arctan(2){\bigr ]}\end{aligned}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aafcdedc58c15285c9d722ba77add17ebae688fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:37.916ex; height:8.509ex;" alt="{\displaystyle {\begin{aligned}R[\exp(-2\pi )]{}&=4\sin({\tfrac {1}{20}}\pi )\sin({\tfrac {3}{20}}\pi )=\\[4pt]&{}=\tan {\bigl [}{\tfrac {1}{4}}\arctan(2){\bigr ]}\end{aligned}}}" loading="lazy">

</td></tr></tbody></table></dd></dl>

Mit der <a href="Dedekindsche_Etafunktion" title="Dedekindsche Etafunktion">Dedekindschen Etafunktion</a> können die modulierten Funktionen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{M}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G_{M}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8ab6ce334c0de8fb3417356ce6ed775812acea73.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.786ex; height:2.509ex;" alt="{\displaystyle G_{M}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{M}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H_{M}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e04914a9a27382605c457434c25653c443b7145c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.89ex; height:2.509ex;" alt="{\displaystyle H_{M}}" loading="lazy"> direkt über den Kettenbruch R dargestellt werden:

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G_{M}(q)=\eta _{W}(q^{5})^{1/2}\eta _{W}(q)^{-1/2}R(q)^{-1/2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aa7ec98f82223bff0bef05c66ed14e859bb2f9fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.099ex; height:3.343ex;" alt="{\displaystyle G_{M}(q)=\eta _{W}(q^{5})^{1/2}\eta _{W}(q)^{-1/2}R(q)^{-1/2}}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle H_{M}(q)=\eta _{W}(q^{5})^{1/2}\eta _{W}(q)^{-1/2}R(q)^{1/2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3f31279f770a241ba33e2eb5dcd59a8794aeedcf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.925ex; height:3.343ex;" alt="{\displaystyle H_{M}(q)=\eta _{W}(q^{5})^{1/2}\eta _{W}(q)^{-1/2}R(q)^{1/2}}" loading="lazy"></dd></dl>

Für die Dedekindsche Etafunktion nach Weberscher Definition<a href="#cite_note-11">[11]</a> gelten diese Formeln:

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<annotation encoding="application/x-tex">{\displaystyle \eta _{W}(x)=2^{-1/6}\vartheta _{10}(x)^{1/6}\vartheta _{00}(x)^{1/6}\vartheta _{01}(x)^{2/3}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cdbd0c1402556e61cf49f39945edbe04e8f8164c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.748ex; height:3.343ex;" alt="{\displaystyle \eta _{W}(x)=2^{-1/6}\vartheta _{10}(x)^{1/6}\vartheta _{00}(x)^{1/6}\vartheta _{01}(x)^{2/3}}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle \eta _{W}(x)=2^{-1/3}\vartheta _{10}(x^{1/2})^{1/3}\vartheta _{00}(x^{1/2})^{1/3}\vartheta _{01}(x^{1/2})^{1/3}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/72582d80577a0e413c1d14aff1367c6632e346dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:49.843ex; height:3.343ex;" alt="{\displaystyle \eta _{W}(x)=2^{-1/3}\vartheta _{10}(x^{1/2})^{1/3}\vartheta _{00}(x^{1/2})^{1/3}\vartheta _{01}(x^{1/2})^{1/3}}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle \eta _{W}(x)=x^{1/24}\prod _{n=1}^{\infty }(1-x^{n})=x^{1/24}(x;x)_{\infty }}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/54d4ce2ee1b64ad5a290abbff86fafc4c5d010d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:41.357ex; height:6.843ex;" alt="{\displaystyle \eta _{W}(x)=x^{1/24}\prod _{n=1}^{\infty }(1-x^{n})=x^{1/24}(x;x)_{\infty }}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle \eta _{W}(x)=x^{1/24}{\biggl \{}1+\sum _{n=1}^{\infty }{\bigl [}-x^{{\text{Fn}}(2n-1)}-x^{{\text{Kr}}(2n-1)}+x^{{\text{Fn}}(2n)}+x^{{\text{Kr}}(2n)}{\bigr ]}{\biggr \}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/921df645dc18712bc30b2dc26b383ed568790118.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:69.349ex; height:6.843ex;" alt="{\displaystyle \eta _{W}(x)=x^{1/24}{\biggl \{}1+\sum _{n=1}^{\infty }{\bigl [}-x^{{\text{Fn}}(2n-1)}-x^{{\text{Kr}}(2n-1)}+x^{{\text{Fn}}(2n)}+x^{{\text{Kr}}(2n)}{\bigr ]}{\biggr \}}}" loading="lazy"></dd></dl>

Bei der zweiten dieser beiden Formeln wird der <a href="Pentagonalzahlensatz" title="Pentagonalzahlensatz">Pentagonalzahlensatz</a><a href="#cite_note-12">[12]</a> beschrieben.

Hierbei gelten für die <a href="F%C3%BCnfeckszahl" title="Fünfeckszahl">Fünfeckszahlen</a> und die Kartenhauszahlen diese grundlegenden Definitionen:

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<annotation encoding="application/x-tex">{\displaystyle {\text{Fn}}(z)={\tfrac {1}{2}}z(3z-1)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b112b9b4c91da28f63de424f12be2f3b0b17439e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:19.615ex; height:3.509ex;" alt="{\displaystyle {\text{Fn}}(z)={\tfrac {1}{2}}z(3z-1)}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle {\text{Kr}}(z)={\tfrac {1}{2}}z(3z+1)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e7f361ec2025ceecb67ec4ff6022982b6c0e1db2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:19.525ex; height:3.509ex;" alt="{\displaystyle {\text{Kr}}(z)={\tfrac {1}{2}}z(3z+1)}" loading="lazy"></dd></dl>

Mit den Pochhammer-Produkten alleine gelten dann für die nicht modulierten Funktionen G und H dann die folgende Identität:

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<annotation encoding="application/x-tex">{\displaystyle G(q)=(q;q^{5})_{\infty }^{-1}(q^{4};q^{5})_{\infty }^{-1}=(q^{5};q^{5})_{\infty }^{1/2}(q;q)_{\infty }^{-1/2}{\biggl [}{\frac {H(q)}{G(q)}}{\biggr ]}^{-1/2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0b5d28ca114344ae62de7fdd79f6e67ce1f35ec7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:59.703ex; height:6.843ex;" alt="{\displaystyle G(q)=(q;q^{5})_{\infty }^{-1}(q^{4};q^{5})_{\infty }^{-1}=(q^{5};q^{5})_{\infty }^{1/2}(q;q)_{\infty }^{-1/2}{\biggl [}{\frac {H(q)}{G(q)}}{\biggr ]}^{-1/2}}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle H(q)=(q^{2};q^{5})_{\infty }^{-1}(q^{3};q^{5})_{\infty }^{-1}=(q^{5};q^{5})_{\infty }^{1/2}(q;q)_{\infty }^{-1/2}{\biggl [}{\frac {H(q)}{G(q)}}{\biggr ]}^{1/2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4744780e486f5b783471a627dfe39aea205ef747.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:59.726ex; height:6.843ex;" alt="{\displaystyle H(q)=(q^{2};q^{5})_{\infty }^{-1}(q^{3};q^{5})_{\infty }^{-1}=(q^{5};q^{5})_{\infty }^{1/2}(q;q)_{\infty }^{-1/2}{\biggl [}{\frac {H(q)}{G(q)}}{\biggr ]}^{1/2}}" loading="lazy"></dd></dl>

Die Richtigkeit des Produkts der beiden nun genannten Formeln kann direkt anhand der Pochhammerschen Produktreihen erkannt werden. Man kann für die modulierten Funktionen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{M}}">

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<annotation encoding="application/x-tex">{\displaystyle G_{M}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle H_{M}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\eta _{W}(q^{5})}{\eta _{W}(q)}}={\frac {\eta _{W}(q^{2})^{4}}{\eta _{W}(q)^{4}}}\,{\frac {\vartheta _{01}(q^{5})}{\vartheta _{01}(q)}}{\biggl [}{\frac {5\,\vartheta _{01}(q^{5})^{2}}{4\,\vartheta _{01}(q)^{2}}}-{\frac {1}{4}}{\biggr ]}^{-1}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/64df8bb0e7d29427b9efa594854003a8f7cba686.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:48.606ex; height:6.676ex;" alt="{\displaystyle {\frac {\eta _{W}(q^{5})}{\eta _{W}(q)}}={\frac {\eta _{W}(q^{2})^{4}}{\eta _{W}(q)^{4}}}\,{\frac {\vartheta _{01}(q^{5})}{\vartheta _{01}(q)}}{\biggl [}{\frac {5\,\vartheta _{01}(q^{5})^{2}}{4\,\vartheta _{01}(q)^{2}}}-{\frac {1}{4}}{\biggr ]}^{-1}}" loading="lazy"></dd></dl>

Gegeben waren für die modulierten Funktionen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{M}}">

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<annotation encoding="application/x-tex">{\displaystyle G_{M}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle H_{M}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle G_{M}(q)=\eta _{W}(q^{5})^{1/2}\eta _{W}(q)^{-1/2}R(q)^{-1/2}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle H_{M}(q)=\eta _{W}(q^{5})^{1/2}\eta _{W}(q)^{-1/2}R(q)^{1/2}}</annotation>

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Die Kombination der drei zuletzt genannten Formeln ergibt folgendes Formelpaar:

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<annotation encoding="application/x-tex">{\displaystyle G_{M}(q)={\frac {\eta _{W}(q^{2})^{2}}{\eta _{W}(q)^{2}}}{\biggl [}{\frac {\vartheta _{01}(q^{5})}{\vartheta _{01}(q)}}{\biggr ]}^{1/2}{\biggl [}{\frac {5\,\vartheta _{01}(q^{5})^{2}}{4\,\vartheta _{01}(q)^{2}}}-{\frac {1}{4}}{\biggr ]}^{-1/2}R(q)^{-1/2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fa71bf7d59b0f6dd18300bf9f1d310c6e9ad6f01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:62.412ex; height:6.843ex;" alt="{\displaystyle G_{M}(q)={\frac {\eta _{W}(q^{2})^{2}}{\eta _{W}(q)^{2}}}{\biggl [}{\frac {\vartheta _{01}(q^{5})}{\vartheta _{01}(q)}}{\biggr ]}^{1/2}{\biggl [}{\frac {5\,\vartheta _{01}(q^{5})^{2}}{4\,\vartheta _{01}(q)^{2}}}-{\frac {1}{4}}{\biggr ]}^{-1/2}R(q)^{-1/2}}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle H_{M}(q)={\frac {\eta _{W}(q^{2})^{2}}{\eta _{W}(q)^{2}}}{\biggl [}{\frac {\vartheta _{01}(q^{5})}{\vartheta _{01}(q)}}{\biggr ]}^{1/2}{\biggl [}{\frac {5\,\vartheta _{01}(q^{5})^{2}}{4\,\vartheta _{01}(q)^{2}}}-{\frac {1}{4}}{\biggr ]}^{-1/2}R(q)^{1/2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/98b09ad01ff206298075b91270af4caed5ddefed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:61.238ex; height:6.843ex;" alt="{\displaystyle H_{M}(q)={\frac {\eta _{W}(q^{2})^{2}}{\eta _{W}(q)^{2}}}{\biggl [}{\frac {\vartheta _{01}(q^{5})}{\vartheta _{01}(q)}}{\biggr ]}^{1/2}{\biggl [}{\frac {5\,\vartheta _{01}(q^{5})^{2}}{4\,\vartheta _{01}(q)^{2}}}-{\frac {1}{4}}{\biggr ]}^{-1/2}R(q)^{1/2}}" loading="lazy">

</td></tr></tbody></table></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Rogers-Ramanujan-Kettenbrüche">Rogers-Ramanujan-Kettenbrüche</h3></div>

Folgender <a href="Kettenbruch" title="Kettenbruch">Kettenbruch</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R(q)}">

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<annotation encoding="application/x-tex">{\displaystyle R(q)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bbfdc5f3479a908aaa62c7b455f6ae1f314bef9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.643ex; height:2.843ex;" alt="{\displaystyle R(q)}" loading="lazy"> heißt <a href="Rogers-Ramanujan-Kettenbruch" title="Rogers-Ramanujan-Kettenbruch">Rogers-Ramanujan-Kettenbruch</a><a href="#cite_note-13">[13]</a><a href="#cite_note-14">[14]</a>, Kettenbruch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(q)}">

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<annotation encoding="application/x-tex">{\displaystyle S(q)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c5afeeb5ad0b7d09d9c2aa1795e6bcfd7f15e6ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.378ex; height:2.843ex;" alt="{\displaystyle S(q)}" loading="lazy"> heißt alternierender Rogers-Ramanujan-Kettenbruch!

<th>Standardisierter Kettenbruch

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<th>Alternierender Kettenbruch

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<annotation encoding="application/x-tex">{\displaystyle R(q)=q^{1/5}\left[1+{\frac {q}{1+{\frac {q^{2}}{1+{\frac {q^{3}}{1+\cdots }}}}}}\right]}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e4f7af859623f25abd646ef747f5e59c524098e7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:30.89ex; height:13.843ex;" alt="{\displaystyle R(q)=q^{1/5}\left[1+{\frac {q}{1+{\frac {q^{2}}{1+{\frac {q^{3}}{1+\cdots }}}}}}\right]}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle S(q)=q^{1/5}\left[1-{\frac {q}{1+{\frac {q^{2}}{1-{\frac {q^{3}}{1+\cdots }}}}}}\right]}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bb0648407cccbe6d31b610509a434b406b7c5d73.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:30.625ex; height:13.843ex;" alt="{\displaystyle S(q)=q^{1/5}\left[1-{\frac {q}{1+{\frac {q^{2}}{1-{\frac {q^{3}}{1+\cdots }}}}}}\right]}" loading="lazy">

</td></tr></tbody></table></dd></dl>

Durch den Faktor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q^{\frac {1}{5}}}">

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<annotation encoding="application/x-tex">{\displaystyle q^{\frac {1}{5}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/45a0812d2abd749e92673313d73a95a6974b047a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.815ex; height:3.843ex;" alt="{\displaystyle q^{\frac {1}{5}}}" loading="lazy"> entsteht so ein Quotient von Modulfunktionen:

Es gilt diese Definition<a href="#cite_note-15">[15]</a> für den genannten Kettenbruch:

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<annotation encoding="application/x-tex">{\displaystyle R(q)={\frac {H_{M}(q)}{G_{M}(q)}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/561762a401f0cfba107c01f808c4013dd20b2f51.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:15.347ex; height:6.509ex;" alt="{\displaystyle R(q)={\frac {H_{M}(q)}{G_{M}(q)}}}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle R(q)=q^{1/5}{\frac {(q;q^{5})_{\infty }(q^{4};q^{5})_{\infty }}{(q^{2};q^{5})_{\infty }(q^{3};q^{5})_{\infty }}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6871a6d308cf824323f5476b4ea9ed1e4109feb2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:30.327ex; height:6.676ex;" alt="{\displaystyle R(q)=q^{1/5}{\frac {(q;q^{5})_{\infty }(q^{4};q^{5})_{\infty }}{(q^{2};q^{5})_{\infty }(q^{3};q^{5})_{\infty }}}}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle R(q)=q^{1/5}\prod _{k=0}^{\infty }{\frac {(1-q^{5k+1})(1-q^{5k+4})}{(1-q^{5k+2})(1-q^{5k+3})}}=q^{1/5}{\frac {H(q)}{G(q)}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d35ed9f3d983770230adbc7b1e04f19dacae32a5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:50.559ex; height:7.176ex;" alt="{\displaystyle R(q)=q^{1/5}\prod _{k=0}^{\infty }{\frac {(1-q^{5k+1})(1-q^{5k+4})}{(1-q^{5k+2})(1-q^{5k+3})}}=q^{1/5}{\frac {H(q)}{G(q)}}}" loading="lazy"></dd></dl>

oder mit der <a href="Ramanujan-Thetafunktion" title="Ramanujan-Thetafunktion">Ramanujanschen Thetafunktion</a>

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<annotation encoding="application/x-tex">{\displaystyle f(a,b)=\sum _{k=-\infty }^{\infty }a^{\frac {k(k+1)}{2}}b^{\frac {k(k-1)}{2}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1d7f0d3eda38fe9c9f07177dc4ef908e01b6d9c9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:27.525ex; height:7.009ex;" alt="{\displaystyle f(a,b)=\sum _{k=-\infty }^{\infty }a^{\frac {k(k+1)}{2}}b^{\frac {k(k-1)}{2}}}" loading="lazy"></dd></dl>

ist

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<annotation encoding="application/x-tex">{\displaystyle R(q)=q^{1/5}{\frac {f(-q,-q^{4})}{f(-q^{2},-q^{3})}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d5385dc817da59f9775735129d6d642f11c7d972.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:24.36ex; height:6.676ex;" alt="{\displaystyle R(q)=q^{1/5}{\frac {f(-q,-q^{4})}{f(-q^{2},-q^{3})}}}" loading="lazy">.</dd></dl>

Der Zusammenhang des Kettenbruchs mit den Rogers-Ramanujan-Funktionen fand schon Rogers 1894 (und später unabhängig Ramanujan).

Der Kettenbruch lässt sich auch durch die <a href="Dedekindsche_%CE%B7-Funktion" class="mw-redirect" title="Dedekindsche η-Funktion">Dedekindsche η-Funktion</a><a href="#cite_note-16">[16]</a> ausdrücken:

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<mi>arccot</mi>

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<annotation encoding="application/x-tex">{\displaystyle R(q)=\tan {\biggl \{}{\frac {1}{2}}\operatorname {arccot} {\biggl [}{\frac {\eta _{W}(q^{1/5})}{2\eta _{W}(q^{5})}}+{\frac {1}{2}}{\biggr ]}{\biggr \}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7eae1fe4001f9563bdbc0ff1485d95395b49f0df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:40.78ex; height:6.843ex;" alt="{\displaystyle R(q)=\tan {\biggl \{}{\frac {1}{2}}\operatorname {arccot} {\biggl [}{\frac {\eta _{W}(q^{1/5})}{2\eta _{W}(q^{5})}}+{\frac {1}{2}}{\biggr ]}{\biggr \}}}" loading="lazy"></dd></dl>

Der alternierende Kettenbruch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(q)}">

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<annotation encoding="application/x-tex">{\displaystyle S(q)}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle S(q)=q^{1/5}{\frac {H(-q)}{G(-q)}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2853af0f7c7d8be3159be800a60307b085542a74.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:18.841ex; height:6.509ex;" alt="{\displaystyle S(q)=q^{1/5}{\frac {H(-q)}{G(-q)}}}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle S(q)=q^{1/5}{\frac {f(q,-q^{4})}{f(-q^{2},q^{3})}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/30145a02163ad890db07d7d84aca750d35cb25f5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:22.288ex; height:6.676ex;" alt="{\displaystyle S(q)=q^{1/5}{\frac {f(q,-q^{4})}{f(-q^{2},q^{3})}}}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle S(q)={\frac {R(q^{4})}{R(q)R(q^{2})}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68d1f809ccb9eff59c0d6dbcff3c950143708e07.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:18.662ex; height:6.676ex;" alt="{\displaystyle S(q)={\frac {R(q^{4})}{R(q)R(q^{2})}}}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle S(q)=q^{1/5}{\frac {G(q)G(q^{2})H(q^{4})}{H(q)H(q^{2})G(q^{4})}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e77da1af579d2d49b2d980366832f3fb37514a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:28.809ex; height:6.676ex;" alt="{\displaystyle S(q)=q^{1/5}{\frac {G(q)G(q^{2})H(q^{4})}{H(q)H(q^{2})G(q^{4})}}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Identitäten_mit_Jacobischen_Thetafunktionen">Identitäten mit Jacobischen Thetafunktionen</h3></div>

Folgende Definitionen sind für die <a href="Thetafunktion#Definition_vom_Theta-Nullwert" class="mw-redirect" title="Thetafunktion">Jacobischen Theta-Nullwertfunktionen</a> gültig:

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<annotation encoding="application/x-tex">{\displaystyle \vartheta _{00}(x)=1+2\sum _{n=1}^{\infty }x^{\Box (n)}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c30635d02ba0b1aba85a817700f815e9b70012e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:23.888ex; height:6.843ex;" alt="{\displaystyle \vartheta _{00}(x)=1+2\sum _{n=1}^{\infty }x^{\Box (n)}}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle \vartheta _{01}(x)=1-2\sum _{n=1}^{\infty }(-1)^{n+1}x^{\Box (n)}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d6003f7b73dc59cb94076ad2b7d5485c217b9aa6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:31.6ex; height:6.843ex;" alt="{\displaystyle \vartheta _{01}(x)=1-2\sum _{n=1}^{\infty }(-1)^{n+1}x^{\Box (n)}}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle \vartheta _{10}(x)=2x^{1/4}+2x^{1/4}\sum _{n=1}^{\infty }x^{2\bigtriangleup (n)}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/654a2182b6252650d50115f157b60edb9f8de707.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:32.948ex; height:6.843ex;" alt="{\displaystyle \vartheta _{10}(x)=2x^{1/4}+2x^{1/4}\sum _{n=1}^{\infty }x^{2\bigtriangleup (n)}}" loading="lazy"></dd></dl>

Und folgende Produktdefinitionen sind zu den genannten Summendefinitionen identisch:

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<annotation encoding="application/x-tex">{\displaystyle \vartheta _{00}(x)=\prod _{n=1}^{\infty }(1-x^{2n})(1+x^{2n-1})^{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3fc32e0f0ac51fbf98e85f371ad12ed2d1d01cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:34.094ex; height:6.843ex;" alt="{\displaystyle \vartheta _{00}(x)=\prod _{n=1}^{\infty }(1-x^{2n})(1+x^{2n-1})^{2}}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle \vartheta _{01}(x)=\prod _{n=1}^{\infty }(1-x^{2n})(1-x^{2n-1})^{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0fa78d91719132a9dfb36a90bd6dd5b1a1e61cd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:34.094ex; height:6.843ex;" alt="{\displaystyle \vartheta _{01}(x)=\prod _{n=1}^{\infty }(1-x^{2n})(1-x^{2n-1})^{2}}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle \vartheta _{10}(x)=2x^{1/4}\prod _{n=1}^{\infty }(1-x^{2n})(1+x^{2n})^{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aafdf20f8768c30b151a7fd8d464424c530914e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:37.57ex; height:6.843ex;" alt="{\displaystyle \vartheta _{10}(x)=2x^{1/4}\prod _{n=1}^{\infty }(1-x^{2n})(1+x^{2n})^{2}}" loading="lazy"></dd></dl>

Diese drei sogenannten Theta-Nullwert-Funktionen werden mit der Jacobischen Identität zueinander verknüpft:

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<annotation encoding="application/x-tex">{\displaystyle \vartheta _{10}(x)={\sqrt[{4}]{\vartheta _{00}(x)^{4}-\vartheta _{01}(x)^{4}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68c5ae92bc640738ba5a1e90fff0886f9124d337.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:29.538ex; height:4.843ex;" alt="{\displaystyle \vartheta _{10}(x)={\sqrt[{4}]{\vartheta _{00}(x)^{4}-\vartheta _{01}(x)^{4}}}}" loading="lazy"></dd></dl>

Die Mathematiker <a href="Edmund_Taylor_Whittaker" title="Edmund Taylor Whittaker">Edmund Taylor Whittaker</a> und <a href="George_Neville_Watson" title="George Neville Watson">George Neville Watson</a><a href="#cite_note-17">[17]</a><a href="#cite_note-18">[18]</a><a href="#cite_note-19">[19]</a> entdeckten diese Definitionsidentitäten.

Die Rogers-Ramanujan-Kettenbruchfunktionen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R(x)}">

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<annotation encoding="application/x-tex">{\displaystyle R(x)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cd5e851b43895fbe06436240dc7daa4d2033f082.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.903ex; height:2.843ex;" alt="{\displaystyle R(x)}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(x)}">

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<annotation encoding="application/x-tex">{\displaystyle S(x)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1886b5a535ed8f168a7c3a83afc8ca440edcdc6d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.638ex; height:2.843ex;" alt="{\displaystyle S(x)}" loading="lazy"> stehen zu den Theta-Nullwertfunktionen in diesen Beziehungen:

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<annotation encoding="application/x-tex">{\displaystyle R(x)=\tan {\biggl \langle }{\frac {1}{2}}\operatorname {arccot} {\biggl \{}{\frac {\vartheta _{01}(x^{1/5})[5\,\vartheta _{01}(x^{5})^{2}-\vartheta _{01}(x)^{2}]}{2\,\vartheta _{01}(x^{5})[\vartheta _{01}(x)^{2}-\vartheta _{01}(x^{1/5})^{2}]}}+{\frac {1}{2}}{\biggr \}}{\biggr \rangle }}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a9657598f8a6a9f10c1b9539438cb129162e4b01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:64.086ex; height:7.009ex;" alt="{\displaystyle R(x)=\tan {\biggl \langle }{\frac {1}{2}}\operatorname {arccot} {\biggl \{}{\frac {\vartheta _{01}(x^{1/5})[5\,\vartheta _{01}(x^{5})^{2}-\vartheta _{01}(x)^{2}]}{2\,\vartheta _{01}(x^{5})[\vartheta _{01}(x)^{2}-\vartheta _{01}(x^{1/5})^{2}]}}+{\frac {1}{2}}{\biggr \}}{\biggr \rangle }}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle S(x)=\tan {\biggl \langle }{\frac {1}{2}}\operatorname {arccot} {\biggl \{}{\frac {\vartheta _{00}(x^{1/5})[5\,\vartheta _{00}(x^{5})^{2}-\vartheta _{00}(x)^{2}]}{2\,\vartheta _{00}(x^{5})[\vartheta _{00}(x^{1/5})^{2}-\vartheta _{00}(x)^{2}]}}-{\frac {1}{2}}{\biggr \}}{\biggr \rangle }}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/299bb1183e03d5b242e6f4ea71794c316cc7710e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:63.821ex; height:7.009ex;" alt="{\displaystyle S(x)=\tan {\biggl \langle }{\frac {1}{2}}\operatorname {arccot} {\biggl \{}{\frac {\vartheta _{00}(x^{1/5})[5\,\vartheta _{00}(x^{5})^{2}-\vartheta _{00}(x)^{2}]}{2\,\vartheta _{00}(x^{5})[\vartheta _{00}(x^{1/5})^{2}-\vartheta _{00}(x)^{2}]}}-{\frac {1}{2}}{\biggr \}}{\biggr \rangle }}" loading="lazy"></dd></dl>

Das Element der fünften Wurzel kann auch vom Nomen der Thetafunktionen entfernt werden und auf die äußere Tangensfunktion übertragen werden. So kann eine Formel gebildet werden, welche nur mit einer von den drei Hauptthetafunktionen auskommt:

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<annotation encoding="application/x-tex">{\displaystyle R(x)=\tan {\biggl \{}{\frac {1}{2}}\arctan {\biggl [}{\frac {1}{2}}-{\frac {\vartheta _{01}(x)^{2}}{2\vartheta _{01}(x^{5})^{2}}}{\biggr ]}{\biggr \}}^{1/5}\tan {\biggl \{}{\frac {1}{2}}\operatorname {arccot} {\biggl [}{\frac {1}{2}}-{\frac {\vartheta _{01}(x)^{2}}{2\vartheta _{01}(x^{5})^{2}}}{\biggr ]}{\biggr \}}^{2/5}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b757250844d30ae1365b456dc76dededbef3a3b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:82.048ex; height:6.843ex;" alt="{\displaystyle R(x)=\tan {\biggl \{}{\frac {1}{2}}\arctan {\biggl [}{\frac {1}{2}}-{\frac {\vartheta _{01}(x)^{2}}{2\vartheta _{01}(x^{5})^{2}}}{\biggr ]}{\biggr \}}^{1/5}\tan {\biggl \{}{\frac {1}{2}}\operatorname {arccot} {\biggl [}{\frac {1}{2}}-{\frac {\vartheta _{01}(x)^{2}}{2\vartheta _{01}(x^{5})^{2}}}{\biggr ]}{\biggr \}}^{2/5}}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle S(x)=\tan {\biggl \{}{\frac {1}{2}}\arctan {\biggl [}{\frac {\vartheta _{00}(x)^{2}}{2\vartheta _{00}(x^{5})^{2}}}-{\frac {1}{2}}{\biggr ]}{\biggr \}}^{1/5}\cot {\biggl \{}{\frac {1}{2}}\operatorname {arccot} {\biggl [}{\frac {\vartheta _{00}(x)^{2}}{2\vartheta _{00}(x^{5})^{2}}}-{\frac {1}{2}}{\biggr ]}{\biggr \}}^{2/5}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/161f136318a79bbfe64427b40acfe6cbf56acb32.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:81.524ex; height:6.843ex;" alt="{\displaystyle S(x)=\tan {\biggl \{}{\frac {1}{2}}\arctan {\biggl [}{\frac {\vartheta _{00}(x)^{2}}{2\vartheta _{00}(x^{5})^{2}}}-{\frac {1}{2}}{\biggr ]}{\biggr \}}^{1/5}\cot {\biggl \{}{\frac {1}{2}}\operatorname {arccot} {\biggl [}{\frac {\vartheta _{00}(x)^{2}}{2\vartheta _{00}(x^{5})^{2}}}-{\frac {1}{2}}{\biggr ]}{\biggr \}}^{2/5}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Anwendung_bei_quintischen_Gleichungen">Anwendung bei quintischen Gleichungen</h3></div>

Der allgemeine Fall der quintischen Gleichungen in der <a href="Bringsches_Radikal" title="Bringsches Radikal">Bring-Jerrard-Form</a> hat eine nicht elementare Lösung basierend auf dem <a href="Satz_von_Abel-Ruffini" title="Satz von Abel-Ruffini">Satz von Abel-Ruffini</a> und soll nun unter Verwendung des <a href="Elliptisches_Nomen" title="Elliptisches Nomen">Elliptischen Nomens</a>, der <a href="Jacobische_Thetafunktion" title="Jacobische Thetafunktion">Jacobischen Thetafunktion</a>, den beiden <a href="Rogers-Ramanujan-Kettenbruch" title="Rogers-Ramanujan-Kettenbruch">Rogers-Ramanujan-Kettenbruchfunktionen</a> R und S und auch den Identitäten der <a href="Hyperbolisch_lemniskatischer_Sinus" title="Hyperbolisch lemniskatischer Sinus">Hyperbolischen Lemniskatischen Funktionen</a> behandelt werden:

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<annotation encoding="application/x-tex">{\displaystyle x^{5}+5\,x=4\,c}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dd266817fc1e59ca01230bb07db86c94ab4813e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.758ex; height:2.843ex;" alt="{\displaystyle x^{5}+5\,x=4\,c}" loading="lazy"></dd></dl>

Die reelle Lösung für alle reellen Werte <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c\in \mathbb {R} }">

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<annotation encoding="application/x-tex">{\displaystyle c\in \mathbb {R} }</annotation>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}x={}&{\frac {S{\bigl \langle }q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (c)]^{2}\}{\bigr \rangle }^{2}-R{\bigl \langle }q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (c)]^{2}\}^{2}{\bigr \rangle }}{S{\bigl \langle }q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (c)]^{2}\}{\bigr \rangle }^{2}}}\times \\[4pt]&{}\times {\frac {1-R{\bigl \langle }q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (c)]^{2}\}^{2}{\bigr \rangle }\,S{\bigl \langle }q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (c)]^{2}\}{\bigr \rangle }}{R{\bigl \langle }q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (c)]^{2}\}^{2}{\bigr \rangle }^{2}}}\times \\[4pt]&{}\times {\frac {\vartheta _{00}{\bigl \langle }q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (c)]^{2}\}^{5}{\bigr \rangle }\,\vartheta _{00}{\bigl \langle }q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (c)]^{2}\}^{1/5}{\bigr \rangle }^{2}-5\,\vartheta _{00}{\bigl \langle }q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (c)]^{2}\}^{5}{\bigr \rangle }^{3}}{4\,\vartheta _{10}{\bigl \langle }q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (c)]^{2}\}{\bigr \rangle }\,\vartheta _{01}{\bigl \langle }q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (c)]^{2}\}{\bigr \rangle }\,\vartheta _{00}{\bigl \langle }q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (c)]^{2}\}{\bigr \rangle }}}\end{aligned}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/814fceb335bf690f140efd7f9ba8be0d55c6c6af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -13.005ex; width:92.726ex; height:27.176ex;" alt="{\displaystyle {\begin{aligned}x={}&{\frac {S{\bigl \langle }q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (c)]^{2}\}{\bigr \rangle }^{2}-R{\bigl \langle }q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (c)]^{2}\}^{2}{\bigr \rangle }}{S{\bigl \langle }q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (c)]^{2}\}{\bigr \rangle }^{2}}}\times \\[4pt]&{}\times {\frac {1-R{\bigl \langle }q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (c)]^{2}\}^{2}{\bigr \rangle }\,S{\bigl \langle }q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (c)]^{2}\}{\bigr \rangle }}{R{\bigl \langle }q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (c)]^{2}\}^{2}{\bigr \rangle }^{2}}}\times \\[4pt]&{}\times {\frac {\vartheta _{00}{\bigl \langle }q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (c)]^{2}\}^{5}{\bigr \rangle }\,\vartheta _{00}{\bigl \langle }q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (c)]^{2}\}^{1/5}{\bigr \rangle }^{2}-5\,\vartheta _{00}{\bigl \langle }q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (c)]^{2}\}^{5}{\bigr \rangle }^{3}}{4\,\vartheta _{10}{\bigl \langle }q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (c)]^{2}\}{\bigr \rangle }\,\vartheta _{01}{\bigl \langle }q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (c)]^{2}\}{\bigr \rangle }\,\vartheta _{00}{\bigl \langle }q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (c)]^{2}\}{\bigr \rangle }}}\end{aligned}}}" loading="lazy"></dd></dl>

Alternativ hierzu kann dieselbe Lösung auch so dargestellt werden:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}x={}&{\frac {5\,\vartheta _{00}(Q^{5})^{3}-\vartheta _{00}(Q^{5})\,\vartheta _{00}(Q)^{2}}{4\,\vartheta _{10}(Q)\,\vartheta _{01}(Q)\,\vartheta _{00}(Q)}}\times {\frac {S(Q)^{2}+R(Q^{2})}{S(Q)}}\times {\bigl [}R(Q^{2})S(Q)+R(Q^{2})+S(Q)-1{\bigr ]}\\[4pt]&\mathrm {mit} \,\,Q=q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (c)]^{2}\}\end{aligned}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2ede1607c2283fffac8a2e6b75fe4aae61abe050.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.005ex; width:92.575ex; height:11.176ex;" alt="{\displaystyle {\begin{aligned}x={}&{\frac {5\,\vartheta _{00}(Q^{5})^{3}-\vartheta _{00}(Q^{5})\,\vartheta _{00}(Q)^{2}}{4\,\vartheta _{10}(Q)\,\vartheta _{01}(Q)\,\vartheta _{00}(Q)}}\times {\frac {S(Q)^{2}+R(Q^{2})}{S(Q)}}\times {\bigl [}R(Q^{2})S(Q)+R(Q^{2})+S(Q)-1{\bigr ]}\\[4pt]&\mathrm {mit} \,\,Q=q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (c)]^{2}\}\end{aligned}}}" loading="lazy"></dd></dl>

Der Mathematiker <a href="Charles_Hermite" title="Charles Hermite">Charles Hermite</a> ermittelte den Wert des elliptischen Moduls k im Verhältnis zum Koeffizienten des Absolutterms der Bring-Jerrard-Form. In seinem Aufsatz „Sur la résolution de l'Équation du cinquiéme degré Comptes rendus“ beschrieb er die Berechnungsmethode für den elliptischen Modul in Bezug auf den absoluten Term. Die italienische Version seines Aufsatzes „Sulla risoluzione delle equazioni del quinto grado“ enthält genau auf Seite 258 die obere Bring-Jerrard-Gleichungsformel, die direkt nach dem elliptischen Modul gelöst werden kann:

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<annotation encoding="application/x-tex">{\displaystyle k={\bigl (}2c^{2}+2+2{\sqrt {c^{4}+1}}{\bigr )}^{-1/2}{\bigl (}{\sqrt {{\sqrt {c^{4}+1}}+1}}+c{\bigr )}=\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (c)]^{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ce02c026c5f7e5f232dee21997ef77f7607cb8ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:69.043ex; height:4.843ex;" alt="{\displaystyle k={\bigl (}2c^{2}+2+2{\sqrt {c^{4}+1}}{\bigr )}^{-1/2}{\bigl (}{\sqrt {{\sqrt {c^{4}+1}}+1}}+c{\bigr )}=\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (c)]^{2}}" loading="lazy"></dd></dl>

Mit dem Kürzel ctlh wird die <a href="Hyperbolisch_lemniskatischer_Sinus" title="Hyperbolisch lemniskatischer Sinus">Hyperbolisch lemniskatische Funktion</a> Cotangens Lemniscatus Hyperbolicus ausgedrückt und das Kürzel aclh stellt den Areacosinus Lemniscatus Hyperbolicus dar.

Zwei Beispiele für diesen Lösungsalgorithmus seien nun erwähnt:

Erstes Rechenbeispiel:

<td>

Quintische Bring-Jerrard-Gleichung:

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<annotation encoding="application/x-tex">{\displaystyle x^{5}+5\,x=8}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7a9d5bb9780db1453458a0a373e0e2f650fd8a15.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.364ex; height:2.843ex;" alt="{\displaystyle x^{5}+5\,x=8}" loading="lazy"></dd></dl>

Lösungsformel:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}x={}&{\frac {S{\bigl \langle }q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (2)]^{2}\}{\bigr \rangle }^{2}-R{\bigl \langle }q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (2)]^{2}\}^{2}{\bigr \rangle }}{S{\bigl \langle }q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (2)]^{2}\}{\bigr \rangle }^{2}}}\times \\[4pt]&{}\times {\frac {1-R{\bigl \langle }q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (2)]^{2}\}^{2}{\bigr \rangle }\,S{\bigl \langle }q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (2)]^{2}\}{\bigr \rangle }}{R{\bigl \langle }q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (2)]^{2}\}^{2}{\bigr \rangle }^{2}}}\times \\[4pt]&{}\times {\frac {\vartheta _{00}{\bigl \langle }q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (2)]^{2}\}^{5}{\bigr \rangle }\,\vartheta _{00}{\bigl \langle }q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (2)]^{2}\}^{1/5}{\bigr \rangle }^{2}-5\,\vartheta _{00}{\bigl \langle }q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (2)]^{2}\}^{5}{\bigr \rangle }^{3}}{4\,\vartheta _{10}{\bigl \langle }q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (2)]^{2}\}{\bigr \rangle }\,\vartheta _{01}{\bigl \langle }q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (2)]^{2}\}{\bigr \rangle }\,\vartheta _{00}{\bigl \langle }q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (2)]^{2}\}{\bigr \rangle }}}\end{aligned}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ef53d58f4913f86148e32f10bc01fbadeef41abc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -13.005ex; width:93.193ex; height:27.176ex;" alt="{\displaystyle {\begin{aligned}x={}&{\frac {S{\bigl \langle }q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (2)]^{2}\}{\bigr \rangle }^{2}-R{\bigl \langle }q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (2)]^{2}\}^{2}{\bigr \rangle }}{S{\bigl \langle }q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (2)]^{2}\}{\bigr \rangle }^{2}}}\times \\[4pt]&{}\times {\frac {1-R{\bigl \langle }q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (2)]^{2}\}^{2}{\bigr \rangle }\,S{\bigl \langle }q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (2)]^{2}\}{\bigr \rangle }}{R{\bigl \langle }q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (2)]^{2}\}^{2}{\bigr \rangle }^{2}}}\times \\[4pt]&{}\times {\frac {\vartheta _{00}{\bigl \langle }q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (2)]^{2}\}^{5}{\bigr \rangle }\,\vartheta _{00}{\bigl \langle }q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (2)]^{2}\}^{1/5}{\bigr \rangle }^{2}-5\,\vartheta _{00}{\bigl \langle }q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (2)]^{2}\}^{5}{\bigr \rangle }^{3}}{4\,\vartheta _{10}{\bigl \langle }q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (2)]^{2}\}{\bigr \rangle }\,\vartheta _{01}{\bigl \langle }q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (2)]^{2}\}{\bigr \rangle }\,\vartheta _{00}{\bigl \langle }q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (2)]^{2}\}{\bigr \rangle }}}\end{aligned}}}" loading="lazy"></dd></dl>

Nachkommastellen des Nomens:

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<annotation encoding="application/x-tex">{\displaystyle q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (2)]^{2}\}=q{\bigl [}{\bigl (}{\sqrt {{\sqrt {17}}+1}}+2{\bigr )}{\bigl (}10+2{\sqrt {17}}{\bigr )}^{-1/2}{\bigr ]}=}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6acb1beedfede49b09628974da592baccd84b8b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:60.746ex; height:4.843ex;" alt="{\displaystyle q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (2)]^{2}\}=q{\bigl [}{\bigl (}{\sqrt {{\sqrt {17}}+1}}+2{\bigr )}{\bigl (}10+2{\sqrt {17}}{\bigr )}^{-1/2}{\bigr ]}=}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle =0{,}3063466544466074265361088194021326272090461143559097382981847144\ldots }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d95ea5e906fc40c4592c840378333b800f1c34ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:81.77ex; height:2.509ex;" alt="{\displaystyle =0{,}3063466544466074265361088194021326272090461143559097382981847144\ldots }" loading="lazy">

Nachkommastellen der Lösung:

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<annotation encoding="application/x-tex">{\displaystyle x=1{,}1670361837016430473110194319963961012975521104880199105205748723\ldots }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8d6fbe864104913f3d68df6661e6e597a22c1a86.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:83.745ex; height:2.509ex;" alt="{\displaystyle x=1{,}1670361837016430473110194319963961012975521104880199105205748723\ldots }" loading="lazy">

</td></tr></tbody></table></dd></dl>

Zweites Rechenbeispiel:

<td>

Quintische Bring-Jerrard-Gleichung:

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<annotation encoding="application/x-tex">{\displaystyle x^{5}+5\,x=12}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cc14215b9339725c4ed82202d840aabccf599092.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.527ex; height:2.843ex;" alt="{\displaystyle x^{5}+5\,x=12}" loading="lazy"></dd></dl>

Solution:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}x={}&{\frac {S{\bigl \langle }q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (3)]^{2}\}{\bigr \rangle }^{2}-R{\bigl \langle }q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (3)]^{2}\}^{2}{\bigr \rangle }}{S{\bigl \langle }q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (3)]^{2}\}{\bigr \rangle }^{2}}}\times \\[4pt]&{}\times {\frac {1-R{\bigl \langle }q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (3)]^{2}\}^{2}{\bigr \rangle }\,S{\bigl \langle }q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (3)]^{2}\}{\bigr \rangle }}{R{\bigl \langle }q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (3)]^{2}\}^{2}{\bigr \rangle }^{2}}}\times \\[4pt]&{}\times {\frac {\vartheta _{00}{\bigl \langle }q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (3)]^{2}\}^{5}{\bigr \rangle }\,\vartheta _{00}{\bigl \langle }q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (3)]^{2}\}^{1/5}{\bigr \rangle }^{2}-5\,\vartheta _{00}{\bigl \langle }q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (3)]^{2}\}^{5}{\bigr \rangle }^{3}}{4\,\vartheta _{10}{\bigl \langle }q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (3)]^{2}\}{\bigr \rangle }\,\vartheta _{01}{\bigl \langle }q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (3)]^{2}\}{\bigr \rangle }\,\vartheta _{00}{\bigl \langle }q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (3)]^{2}\}{\bigr \rangle }}}\end{aligned}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f4e7cd20adecd7d320eb790ac68c594bbede741e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -13.005ex; width:93.193ex; height:27.176ex;" alt="{\displaystyle {\begin{aligned}x={}&{\frac {S{\bigl \langle }q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (3)]^{2}\}{\bigr \rangle }^{2}-R{\bigl \langle }q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (3)]^{2}\}^{2}{\bigr \rangle }}{S{\bigl \langle }q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (3)]^{2}\}{\bigr \rangle }^{2}}}\times \\[4pt]&{}\times {\frac {1-R{\bigl \langle }q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (3)]^{2}\}^{2}{\bigr \rangle }\,S{\bigl \langle }q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (3)]^{2}\}{\bigr \rangle }}{R{\bigl \langle }q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (3)]^{2}\}^{2}{\bigr \rangle }^{2}}}\times \\[4pt]&{}\times {\frac {\vartheta _{00}{\bigl \langle }q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (3)]^{2}\}^{5}{\bigr \rangle }\,\vartheta _{00}{\bigl \langle }q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (3)]^{2}\}^{1/5}{\bigr \rangle }^{2}-5\,\vartheta _{00}{\bigl \langle }q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (3)]^{2}\}^{5}{\bigr \rangle }^{3}}{4\,\vartheta _{10}{\bigl \langle }q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (3)]^{2}\}{\bigr \rangle }\,\vartheta _{01}{\bigl \langle }q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (3)]^{2}\}{\bigr \rangle }\,\vartheta _{00}{\bigl \langle }q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (3)]^{2}\}{\bigr \rangle }}}\end{aligned}}}" loading="lazy"></dd></dl>

Nachkommastellen des Nomens:

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<annotation encoding="application/x-tex">{\displaystyle q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (3)]^{2}\}=q{\bigl [}{\bigl (}{\sqrt {{\sqrt {82}}+1}}+3{\bigr )}{\bigl (}20+2{\sqrt {82}}{\bigr )}^{-1/2}{\bigr ]}=}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/709f6e2c9146c782a96a167a7b1fb9fd92f8bf8f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:60.746ex; height:4.843ex;" alt="{\displaystyle q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (3)]^{2}\}=q{\bigl [}{\bigl (}{\sqrt {{\sqrt {82}}+1}}+3{\bigr )}{\bigl (}20+2{\sqrt {82}}{\bigr )}^{-1/2}{\bigr ]}=}" loading="lazy">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle =0{,}3706649511520240756244325221775686571518680899597473957509743879\ldots }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/48672a03eac1a0d820bb4c7fffbcf6af08fbdc2c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:81.77ex; height:2.509ex;" alt="{\displaystyle =0{,}3706649511520240756244325221775686571518680899597473957509743879\ldots }" loading="lazy">

Nachkommastellen der Lösung:

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<annotation encoding="application/x-tex">{\displaystyle x=1{,}3840917958231463592477551262671354748859350601806764501691889116\ldots }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3b0ea30eabce888fc63922839662cd0fbc422aca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:83.745ex; height:2.509ex;" alt="{\displaystyle x=1{,}3840917958231463592477551262671354748859350601806764501691889116\ldots }" loading="lazy">

</td></tr></tbody></table></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Anwendung_in_der_statistischen_Mechanik">Anwendung in der statistischen Mechanik</h2></div>

Die Identitäten haben Anwendung in der <a href="Statistische_Mechanik" title="Statistische Mechanik">statistischen Mechanik</a> bei der Lösung des Hard Hexagon Modells durch <a href="Rodney_Baxter" title="Rodney Baxter">Rodney Baxter</a> 1980.<a href="#cite_note-20">[20]</a> Das Hard Hexagon Modell ist ein Gas von Teilchen auf einem Dreiecksgitter, so dass keine zwei Teilchen auf dem Gitter benachbart sein dürfen. Sie finden auch in weiteren exakt lösbaren Modellen der statistischen Mechanik Anwendung.

<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>

<ul><li><a href="George_E._Andrews" class="mw-redirect" title="George E. Andrews">George E. Andrews</a>: The theory of partitions, Addison-Wesley 1976, Cambridge University Press 1998</li>

<li><a href="David_Bressoud" title="David Bressoud">David Bressoud</a>, Analytic and combinatorial generalizations of the Rogers-Ramanujan identities, American Mathematical Society 1980</li>

<li>David Bressoud: An easy proof of the Rogers-Ramanujan identities, J. of Number Theory, Band 16, 1983, S. 235–241.</li>

<li><a href="Godfrey_Harold_Hardy" title="Godfrey Harold Hardy">Godfrey Harold Hardy</a>, E. M. Wright: Introduction to the theory of numbers, Oxford, Clarendon Press 1975 (S. 290ff, Kapitel 19-13)</li>

<li><a href="George_E._Andrews" class="mw-redirect" title="George E. Andrews">George E. Andrews</a>, <a href="Rodney_J._Baxter" class="mw-redirect" title="Rodney J. Baxter">Rodney J. Baxter</a>: A motivated proof of the Rogers-Ramanujan identities, American Mathematical Monthly, Band 96, 1989, S. 401–409.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Weblinks">Weblinks</h2></div>

<ul><li><a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Rogers-RamanujanIdentities.html">Rogers-Ramanujan-Identities, Mathworld</a></li>

<li><a rel="nofollow" class="external text" href="https://www.math.ucdavis.edu/~anne/iap.pdf">Anne Schilling, The Rogers-Ramanujan identities at Y2K, 2000, pdf</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>

<li id="cite_note-1"><a href="#cite_ref-1">↑</a> Rogers, Second memoir on the expansion of certain infinite products, Proc. London Math. Soc., Band 25, 1894, S. 318–343.

</li>

<li id="cite_note-2"><a href="#cite_ref-2">↑</a> Er teilte sie <a href="Percy_Alexander_MacMahon" title="Percy Alexander MacMahon">Percy Alexander MacMahon</a> mit, der sie in seinem Buch Combinatory Analysis, Cambridge University Press, Band 2, 1916, veröffentlichte (ohne Beweis)

</li>

<li id="cite_note-3"><a href="#cite_ref-3">↑</a> Rogers, Ramanujan, Proof of certain identities in combinatory analysis, Cambr. Phil. Soc. Proc., Band 19, 1919, S. 211–216.

</li>

<li id="cite_note-4"><a href="#cite_ref-4">↑</a> Schur, Ein Beitrag zur additiven Zahlentheorie und zur Theorie der Kettenbrüche, Sitzungsberichte der Preuß. Akademie der Wissenschaften, Math.-Phys. Klasse, 1917, S. 302–321, auch in Schur, Gesammelte Abhandlungen, Band 2, Springer, 1973.

</li>

<li id="cite_note-5"><a href="#cite_ref-5">↑</a> Bailey, Generalized hypergeometric series, Cambridge University Press 1935.

</li>

<li id="cite_note-6"><a href="#cite_ref-6">↑</a> Slater, Further identities of the Rogers-Ramanujan type, Proceedings of the London Mathematical Society. Second Series, Band 54, 1952, S. 147–167.

</li>

<li id="cite_note-7"><a href="#cite_ref-7">↑</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Andrews-GordonIdentity.html">Andrews-Gordon Identity, Mathworld</a>

</li>

<li id="cite_note-8"><a href="#cite_ref-8">↑</a> <a rel="nofollow" class="external text" href="https://qseries.org/fgarvan/qmaple/theta-supplement/papers/BERNDT-CHOI-ET-AL-2007.pdf">Bruce Berndt u. a., Ramanujans forty identities for the Rogers-Ramanujan-functions, pdf</a>

</li>

<li id="cite_note-9"><a href="#cite_ref-9">↑</a> <a rel="nofollow" class="external text" href="http://oeis.org/A003114">A003114 - OEIS.</a> Abgerufen am 6. August 2022.<span style="display: none;" class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adc&rfr_id=info%3Asid%2Fde.wikipedia.org%3ARogers-Ramanujan-Identit%C3%A4ten&rft.title=A003114+-+OEIS&rft.description=A003114+-+OEIS&rft.identifier=http%3A%2F%2Foeis.org%2FA003114">

</li>

<li id="cite_note-10"><a href="#cite_ref-10">↑</a> <a rel="nofollow" class="external text" href="http://oeis.org/A003106">A003106 - OEIS.</a> Abgerufen am 6. August 2022.<span style="display: none;" class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adc&rfr_id=info%3Asid%2Fde.wikipedia.org%3ARogers-Ramanujan-Identit%C3%A4ten&rft.title=A003106+-+OEIS&rft.description=A003106+-+OEIS&rft.identifier=http%3A%2F%2Foeis.org%2FA003106">

</li>

<li id="cite_note-11"><a href="#cite_ref-11">↑</a> Eric W. Weisstein: <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/">Dedekind Eta Function.</a> Abgerufen am 2. April 2022 (englisch).<span style="display: none;" class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adc&rfr_id=info%3Asid%2Fde.wikipedia.org%3ARogers-Ramanujan-Identit%C3%A4ten&rft.title=Dedekind+Eta+Function&rft.description=Dedekind+Eta+Function&rft.identifier=https%3A%2F%2Fmathworld.wolfram.com%2F&rft.creator=Eric+W.+Weisstein&rft.language=en">

</li>

<li id="cite_note-12"><a href="#cite_ref-12">↑</a> <a rel="nofollow" class="external free" href="https://vdoc.pub/download/a-brief-introduction-to-theta-functions-6v41da306900">https://vdoc.pub/download/a-brief-introduction-to-theta-functions-6v41da306900</a>

</li>

<li id="cite_note-13"><a href="#cite_ref-13">↑</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Rogers-RamanujanContinuedFraction.html">Rogers-Ramanujan Continued Fraction, Mathworld</a>

</li>

<li id="cite_note-14"><a href="#cite_ref-14">↑</a> <a rel="nofollow" class="external text" href="https://faculty.math.illinois.edu/~berndt/articles/rrcf.pdf">Bruce Berndt u. a., The Rogers-Ramanujan continued fraction, pdf</a>

</li>

<li id="cite_note-15"><a href="#cite_ref-15">↑</a> Bruce C. Berndt, Heng Huat Chan, Sen-Shan Huang, Soon-Yi Kang, Jaebum Sohn, Seung Hwan Son: <cite style="font-style:italic">The Rogers–Ramanujan continued fraction</cite>. In: <cite style="font-style:italic">Journal of Computational and Applied Mathematics</cite>. Band 105, Nr. 1, 1. Mai 1999, <a href="Internationale_Standardnummer_f%C3%BCr_fortlaufende_Sammelwerke" title="Internationale Standardnummer für fortlaufende Sammelwerke">ISSN</a> <a rel="nofollow" class="external text" href="https://zdb-katalog.de/list.xhtml?t=iss%3D%220377-0427%22&key=cql">0377-0427</a>, S. 9–24, <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016/S0377-0427%2899%2900033-3">10.1016/S0377-0427(99)00033-3</a> (<a rel="nofollow" class="external text" href="https://www.sciencedirect.com/science/article/pii/S0377042799000333">sciencedirect.com</a> [abgerufen am 7. September 2023]).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rfr_id=info:sid/de.wikipedia.org:Rogers-Ramanujan-Identit%C3%A4ten&rft.atitle=The+Rogers-Ramanujan+continued+fraction&rft.au=Bruce+C.+Berndt%2C+Heng+Huat+Chan%2C+Sen-Shan+Huang%2C+...&rft.date=1999-05-01&rft.doi=10.1016%2FS0377-0427%2899%2900033-3&rft.genre=journal&rft.issn=0377-0427&rft.issue=1&rft.jtitle=Journal+of+Computational+and+Applied+Mathematics&rft.pages=9-24&rft.volume=105" style="display:none">

</li>

<li id="cite_note-16"><a href="#cite_ref-16">↑</a> <a href="Eric_Weisstein" title="Eric Weisstein">Eric W. Weisstein</a>: <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Rogers-RamanujanContinuedFraction.html">Rogers-Ramanujan Continued Fraction.</a> In: <a href="MathWorld" title="MathWorld">MathWorld</a> (englisch).

</li>

<li id="cite_note-17"><a href="#cite_ref-17">↑</a> <a href="Eric_Weisstein" title="Eric Weisstein">Eric W. Weisstein</a>: <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/JacobiThetaFunctions.html">Jacobi Theta Functions.</a> In: <a href="MathWorld" title="MathWorld">MathWorld</a> (englisch).

</li>

<li id="cite_note-18"><a href="#cite_ref-18">↑</a> <a rel="nofollow" class="external free" href="http://wayback.cecm.sfu.ca/~pborwein/TEMP_PROTECTED/pi-agm.pdf">http://wayback.cecm.sfu.ca/~pborwein/TEMP_PROTECTED/pi-agm.pdf</a>

</li>

<li id="cite_note-19"><a href="#cite_ref-19">↑</a> <a rel="nofollow" class="external text" href="https://dlmf.nist.gov/20.5">DLMF: 20.5 Infinite Products and Related Results.</a> Abgerufen am 13. August 2022.<span style="display: none;" class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adc&rfr_id=info%3Asid%2Fde.wikipedia.org%3ARogers-Ramanujan-Identit%C3%A4ten&rft.title=DLMF%3A+20.5+Infinite+Products+and+Related+Results&rft.description=DLMF%3A+20.5+Infinite+Products+and+Related+Results&rft.identifier=https%3A%2F%2Fdlmf.nist.gov%2F20.5">

</li>

<li id="cite_note-20"><a href="#cite_ref-20">↑</a> Baxter, Exactly solvable models in statistical mechanics, Academic Press 1982. Zuerst Baxter, Journal of Physics, A, Band 13, 1980, L61-L70. Siehe auch <a href="George_E._Andrews" class="mw-redirect" title="George E. Andrews">George E. Andrews</a>, The hard-hexagon model and Rogers-Ramanujan type identities, Proc. Nat. Acad. Sci., Band 78, 1981, S. 5290–5292, <a rel="nofollow" class="external text" href="https://www.pnas.org/content/pnas/78/9/5290.full.pdf">pdf</a>

</li>

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