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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Fox H-function</span></span>
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</style><div role="note" class="hatnote navigation-not-searchable">"H function" redirects here; not to be confused with <a href="Harmonic_number" title="Harmonic number">Harmonic number</a>.</div>
<p>In mathematics, the <b>Fox H-function</b> <i>H</i>(<i>x</i>) is a generalization of the <a href="Meijer_G-function" title="Meijer G-function">Meijer G-function</a> and the <a href="Fox%E2%80%93Wright_function" title="Fox–Wright function">Fox–Wright function</a> introduced by <a href="Charles_Fox_(mathematician)" title="Charles Fox (mathematician)">Charles Fox</a> (<a href="#CITEREFFox1961">1961</a>).
It is defined by a <a href="Mellin%E2%80%93Barnes_integral" class="mw-redirect" title="Mellin–Barnes integral">Mellin–Barnes integral</a>
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{p,q}^{\,m,n}\!\left[z\left|{\begin{matrix}(a_{1},A_{1})&(a_{2},A_{2})&\ldots &(a_{p},A_{p})\\(b_{1},B_{1})&(b_{2},B_{2})&\ldots &(b_{q},B_{q})\end{matrix}}\right.\right]={\frac {1}{2\pi i}}\int _{L}{\frac {\prod _{j=1}^{m}\Gamma (b_{j}+B_{j}s)\,\prod _{j=1}^{n}\Gamma (1-a_{j}-A_{j}s)}{\prod _{j=m+1}^{q}\Gamma (1-b_{j}-B_{j}s)\,\prod _{j=n+1}^{p}\Gamma (a_{j}+A_{j}s)}}z^{-s}\,ds,}">
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<annotation encoding="application/x-tex">{\displaystyle H_{p,q}^{\,m,n}\!\left[z\left|{\begin{matrix}(a_{1},A_{1})&(a_{2},A_{2})&\ldots &(a_{p},A_{p})\\(b_{1},B_{1})&(b_{2},B_{2})&\ldots &(b_{q},B_{q})\end{matrix}}\right.\right]={\frac {1}{2\pi i}}\int _{L}{\frac {\prod _{j=1}^{m}\Gamma (b_{j}+B_{j}s)\,\prod _{j=1}^{n}\Gamma (1-a_{j}-A_{j}s)}{\prod _{j=m+1}^{q}\Gamma (1-b_{j}-B_{j}s)\,\prod _{j=n+1}^{p}\Gamma (a_{j}+A_{j}s)}}z^{-s}\,ds,}</annotation>
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</math></span><img src="./c1e0af72e03a0f3221f7f5241446802e999ff5b4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:107.315ex; height:7.509ex;" alt="{\displaystyle H_{p,q}^{\,m,n}\!\left[z\left|{\begin{matrix}(a_{1},A_{1})&(a_{2},A_{2})&\ldots &(a_{p},A_{p})\\(b_{1},B_{1})&(b_{2},B_{2})&\ldots &(b_{q},B_{q})\end{matrix}}\right.\right]={\frac {1}{2\pi i}}\int _{L}{\frac {\prod _{j=1}^{m}\Gamma (b_{j}+B_{j}s)\,\prod _{j=1}^{n}\Gamma (1-a_{j}-A_{j}s)}{\prod _{j=m+1}^{q}\Gamma (1-b_{j}-B_{j}s)\,\prod _{j=n+1}^{p}\Gamma (a_{j}+A_{j}s)}}z^{-s}\,ds,}" loading="lazy"></span></dd></dl>
<p>where <i>L</i> is a certain contour separating the poles of the two factors in the numerator.
</p>
<meta property="mw:PageProp/toc">
<div class="mw-heading mw-heading2"><h2 id="Relation_to_other_functions">Relation to other functions</h2></div>
<div class="mw-heading mw-heading3"><h3 id="Lambert_W-function">Lambert W-function</h3></div>
<p>A relation of the Fox H-Function to the -1 branch of the <a href="Lambert_W_function" title="Lambert W function">Lambert W-function</a> is given by
</p><p><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {\operatorname {W} _{-1}\left(-\alpha \cdot z\right)}}={\begin{cases}\lim _{\beta \to \alpha ^{-}}\left[{\frac {\alpha ^{2}\cdot \left(\left(\alpha -\beta \right)\cdot z\right)^{\frac {\alpha }{\beta }}}{\beta }}\cdot \operatorname {H} _{1,\,2}^{1,\,1}\left({\begin{matrix}\left({\frac {\alpha +\beta }{\beta }},\,{\frac {\alpha }{\beta }}\right)\\\left(0,\,1\right),\,\left(-{\frac {\alpha }{\beta }},\,{\frac {\alpha -\beta }{\beta }}\right)\\\end{matrix}}\mid -\left(\left(\alpha -\beta \right)\cdot z\right)^{{\frac {\alpha }{\beta }}-1}\right)\right],\,{\text{for}}\left|z\right|<{\frac {1}{e\left|\alpha \right|}}\\\lim _{\beta \to \alpha ^{-}}\left[{\frac {\alpha ^{2}\cdot \left(\left(\alpha -\beta \right)\cdot z\right)^{-{\frac {\alpha }{\beta }}}}{\beta }}\cdot \operatorname {H} _{2,\,1}^{1,\,1}\left({\begin{matrix}\left(1,\,1\right),\,\left({\frac {\beta -\alpha }{\beta }},\,{\frac {\alpha -\beta }{\beta }}\right)\\\left(-{\frac {\alpha }{\beta }},\,{\frac {\alpha }{\beta }}\right)\\\end{matrix}}\mid -\left(\left(\alpha -\beta \right)\cdot z\right)^{1-{\frac {\alpha }{\beta }}}\right)\right],\,{\text{otherwise}}\\\end{cases}}}">
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<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>α<!-- α --></mi>
<mi>β<!-- β --></mi>
</mfrac>
</mrow>
</mrow>
</msup>
</mrow>
<mi>β<!-- β --></mi>
</mfrac>
</mrow>
<mo>⋅<!-- ⋅ --></mo>
<msubsup>
<mi mathvariant="normal">H</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mn>1</mn>
</mrow>
</msubsup>
<mo><!-- --></mo>
<mrow>
<mo>(</mo>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mrow>
<mo>(</mo>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>β<!-- β --></mi>
<mo>−<!-- − --></mo>
<mi>α<!-- α --></mi>
</mrow>
<mi>β<!-- β --></mi>
</mfrac>
</mrow>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>α<!-- α --></mi>
<mo>−<!-- − --></mo>
<mi>β<!-- β --></mi>
</mrow>
<mi>β<!-- β --></mi>
</mfrac>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>(</mo>
<mrow>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>α<!-- α --></mi>
<mi>β<!-- β --></mi>
</mfrac>
</mrow>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>α<!-- α --></mi>
<mi>β<!-- β --></mi>
</mfrac>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>∣<!-- ∣ --></mo>
<mo>−<!-- − --></mo>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>α<!-- α --></mi>
<mo>−<!-- − --></mo>
<mi>β<!-- β --></mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>⋅<!-- ⋅ --></mo>
<mi>z</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>α<!-- α --></mi>
<mi>β<!-- β --></mi>
</mfrac>
</mrow>
</mrow>
</msup>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>]</mo>
</mrow>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mrow class="MJX-TeXAtom-ORD">
<mtext>otherwise</mtext>
</mrow>
</mtd>
</mtr>
</mtable>
<mo fence="true" stretchy="true" symmetric="true"></mo>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\overline {\operatorname {W} _{-1}\left(-\alpha \cdot z\right)}}={\begin{cases}\lim _{\beta \to \alpha ^{-}}\left[{\frac {\alpha ^{2}\cdot \left(\left(\alpha -\beta \right)\cdot z\right)^{\frac {\alpha }{\beta }}}{\beta }}\cdot \operatorname {H} _{1,\,2}^{1,\,1}\left({\begin{matrix}\left({\frac {\alpha +\beta }{\beta }},\,{\frac {\alpha }{\beta }}\right)\\\left(0,\,1\right),\,\left(-{\frac {\alpha }{\beta }},\,{\frac {\alpha -\beta }{\beta }}\right)\\\end{matrix}}\mid -\left(\left(\alpha -\beta \right)\cdot z\right)^{{\frac {\alpha }{\beta }}-1}\right)\right],\,{\text{for}}\left|z\right|<{\frac {1}{e\left|\alpha \right|}}\\\lim _{\beta \to \alpha ^{-}}\left[{\frac {\alpha ^{2}\cdot \left(\left(\alpha -\beta \right)\cdot z\right)^{-{\frac {\alpha }{\beta }}}}{\beta }}\cdot \operatorname {H} _{2,\,1}^{1,\,1}\left({\begin{matrix}\left(1,\,1\right),\,\left({\frac {\beta -\alpha }{\beta }},\,{\frac {\alpha -\beta }{\beta }}\right)\\\left(-{\frac {\alpha }{\beta }},\,{\frac {\alpha }{\beta }}\right)\\\end{matrix}}\mid -\left(\left(\alpha -\beta \right)\cdot z\right)^{1-{\frac {\alpha }{\beta }}}\right)\right],\,{\text{otherwise}}\\\end{cases}}}</annotation>
</semantics>
</math></span><img src="./5a843b58899573865ff26491541a1c75e99869d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.171ex; width:106.673ex; height:19.509ex;" alt="{\displaystyle {\overline {\operatorname {W} _{-1}\left(-\alpha \cdot z\right)}}={\begin{cases}\lim _{\beta \to \alpha ^{-}}\left[{\frac {\alpha ^{2}\cdot \left(\left(\alpha -\beta \right)\cdot z\right)^{\frac {\alpha }{\beta }}}{\beta }}\cdot \operatorname {H} _{1,\,2}^{1,\,1}\left({\begin{matrix}\left({\frac {\alpha +\beta }{\beta }},\,{\frac {\alpha }{\beta }}\right)\\\left(0,\,1\right),\,\left(-{\frac {\alpha }{\beta }},\,{\frac {\alpha -\beta }{\beta }}\right)\\\end{matrix}}\mid -\left(\left(\alpha -\beta \right)\cdot z\right)^{{\frac {\alpha }{\beta }}-1}\right)\right],\,{\text{for}}\left|z\right|<{\frac {1}{e\left|\alpha \right|}}\\\lim _{\beta \to \alpha ^{-}}\left[{\frac {\alpha ^{2}\cdot \left(\left(\alpha -\beta \right)\cdot z\right)^{-{\frac {\alpha }{\beta }}}}{\beta }}\cdot \operatorname {H} _{2,\,1}^{1,\,1}\left({\begin{matrix}\left(1,\,1\right),\,\left({\frac {\beta -\alpha }{\beta }},\,{\frac {\alpha -\beta }{\beta }}\right)\\\left(-{\frac {\alpha }{\beta }},\,{\frac {\alpha }{\beta }}\right)\\\end{matrix}}\mid -\left(\left(\alpha -\beta \right)\cdot z\right)^{1-{\frac {\alpha }{\beta }}}\right)\right],\,{\text{otherwise}}\\\end{cases}}}" loading="lazy"></span>where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {z}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>z</mi>
<mo accent="false">¯<!-- ¯ --></mo>
</mover>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\overline {z}}}</annotation>
</semantics>
</math></span><img src="./64281d029a1d4bef9545644f01821c713f876f76.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.208ex; height:2.343ex;" alt="{\displaystyle {\overline {z}}}" loading="lazy"></span> is the complex conjugate of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>z</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle z}</annotation>
</semantics>
</math></span><img src="./bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy"></span>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>
</p>
<div class="mw-heading mw-heading3"><h3 id="Meijer_G-function">Meijer G-function</h3></div>
<p>Compare to the Meijer G-function
</p>
<dl><dd></dd></dl>
<p><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{p,q}^{\,m,n}\!\left(\left.{\begin{matrix}a_{1},\dots ,a_{p}\\b_{1},\dots ,b_{q}\end{matrix}}\;\right|\,z\right)={\frac {1}{2\pi i}}\int _{L}{\frac {\prod _{j=1}^{m}\Gamma (b_{j}-s)\,\prod _{j=1}^{n}\Gamma (1-a_{j}+s)}{\prod _{j=m+1}^{q}\Gamma (1-b_{j}+s)\,\prod _{j=n+1}^{p}\Gamma (a_{j}-s)}}\,z^{s}\,ds.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msubsup>
<mi>G</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>p</mi>
<mo>,</mo>
<mi>q</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mspace width="thinmathspace"></mspace>
<mi>m</mi>
<mo>,</mo>
<mi>n</mi>
</mrow>
</msubsup>
<mspace width="negativethinmathspace"></mspace>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mo fence="true" stretchy="true" symmetric="true"></mo>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>p</mi>
</mrow>
</msub>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mi>b</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<msub>
<mi>b</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>q</mi>
</mrow>
</msub>
</mtd>
</mtr>
</mtable>
</mrow>
<mspace width="thickmathspace"></mspace>
</mrow>
<mo>|</mo>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>z</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mrow>
<mn>2</mn>
<mi>π<!-- π --></mi>
<mi>i</mi>
</mrow>
</mfrac>
</mrow>
<msub>
<mo>∫<!-- ∫ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>L</mi>
</mrow>
</msub>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<munderover>
<mo>∏<!-- ∏ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>m</mi>
</mrow>
</munderover>
<mi mathvariant="normal">Γ<!-- Γ --></mi>
<mo stretchy="false">(</mo>
<msub>
<mi>b</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
<mo>−<!-- − --></mo>
<mi>s</mi>
<mo stretchy="false">)</mo>
<mspace width="thinmathspace"></mspace>
<munderover>
<mo>∏<!-- ∏ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</munderover>
<mi mathvariant="normal">Γ<!-- Γ --></mi>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>−<!-- − --></mo>
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
<mo>+</mo>
<mi>s</mi>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<munderover>
<mo>∏<!-- ∏ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
<mo>=</mo>
<mi>m</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>q</mi>
</mrow>
</munderover>
<mi mathvariant="normal">Γ<!-- Γ --></mi>
<mo stretchy="false">(</mo>
<mn>1</mn>
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<msub>
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<mrow class="MJX-TeXAtom-ORD">
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</mrow>
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<mi>s</mi>
<mo stretchy="false">)</mo>
<mspace width="thinmathspace"></mspace>
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<mo>∏<!-- ∏ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
<mo>=</mo>
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>p</mi>
</mrow>
</munderover>
<mi mathvariant="normal">Γ<!-- Γ --></mi>
<mo stretchy="false">(</mo>
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
<mo>−<!-- − --></mo>
<mi>s</mi>
<mo stretchy="false">)</mo>
</mrow>
</mfrac>
</mrow>
<mspace width="thinmathspace"></mspace>
<msup>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>s</mi>
</mrow>
</msup>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mi>s</mi>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle G_{p,q}^{\,m,n}\!\left(\left.{\begin{matrix}a_{1},\dots ,a_{p}\\b_{1},\dots ,b_{q}\end{matrix}}\;\right|\,z\right)={\frac {1}{2\pi i}}\int _{L}{\frac {\prod _{j=1}^{m}\Gamma (b_{j}-s)\,\prod _{j=1}^{n}\Gamma (1-a_{j}+s)}{\prod _{j=m+1}^{q}\Gamma (1-b_{j}+s)\,\prod _{j=n+1}^{p}\Gamma (a_{j}-s)}}\,z^{s}\,ds.}</annotation>
</semantics>
</math></span><img src="./11fa5e5a906356bde05528111f2114238852bd30.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:79.095ex; height:7.509ex;" alt="{\displaystyle G_{p,q}^{\,m,n}\!\left(\left.{\begin{matrix}a_{1},\dots ,a_{p}\\b_{1},\dots ,b_{q}\end{matrix}}\;\right|\,z\right)={\frac {1}{2\pi i}}\int _{L}{\frac {\prod _{j=1}^{m}\Gamma (b_{j}-s)\,\prod _{j=1}^{n}\Gamma (1-a_{j}+s)}{\prod _{j=m+1}^{q}\Gamma (1-b_{j}+s)\,\prod _{j=n+1}^{p}\Gamma (a_{j}-s)}}\,z^{s}\,ds.}" loading="lazy"></span>
</p><p>The special case for which the Fox H reduces to the Meijer G is <i>A</i><sub><i>j</i></sub> = <i>B</i><sub><i>k</i></sub> = <i>C</i>, <i>C</i> > 0 for <i>j</i> = 1...<i>p</i> and <i>k</i> = 1...<i>q</i> :<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{p,q}^{\,m,n}\!\left[z\left|{\begin{matrix}(a_{1},C)&(a_{2},C)&\ldots &(a_{p},C)\\(b_{1},C)&(b_{2},C)&\ldots &(b_{q},C)\end{matrix}}\right.\right]={\frac {1}{C}}G_{p,q}^{\,m,n}\!\left(\left.{\begin{matrix}a_{1},\dots ,a_{p}\\b_{1},\dots ,b_{q}\end{matrix}}\;\right|\,z^{1/C}\right).}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msubsup>
<mi>H</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>p</mi>
<mo>,</mo>
<mi>q</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mspace width="thinmathspace"></mspace>
<mi>m</mi>
<mo>,</mo>
<mi>n</mi>
</mrow>
</msubsup>
<mspace width="negativethinmathspace"></mspace>
<mrow>
<mo>[</mo>
<mrow>
<mi>z</mi>
<mrow>
<mo>|</mo>
<mrow class="MJX-TeXAtom-ORD">
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mo stretchy="false">(</mo>
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<mi>C</mi>
<mo stretchy="false">)</mo>
</mtd>
<mtd>
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<mn>2</mn>
</mrow>
</msub>
<mo>,</mo>
<mi>C</mi>
<mo stretchy="false">)</mo>
</mtd>
<mtd>
<mo>…<!-- … --></mo>
</mtd>
<mtd>
<mo stretchy="false">(</mo>
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>p</mi>
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<mo>,</mo>
<mi>C</mi>
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</mtd>
</mtr>
<mtr>
<mtd>
<mo stretchy="false">(</mo>
<msub>
<mi>b</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<mi>C</mi>
<mo stretchy="false">)</mo>
</mtd>
<mtd>
<mo stretchy="false">(</mo>
<msub>
<mi>b</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>,</mo>
<mi>C</mi>
<mo stretchy="false">)</mo>
</mtd>
<mtd>
<mo>…<!-- … --></mo>
</mtd>
<mtd>
<mo stretchy="false">(</mo>
<msub>
<mi>b</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>q</mi>
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<mo>,</mo>
<mi>C</mi>
<mo stretchy="false">)</mo>
</mtd>
</mtr>
</mtable>
</mrow>
<mo fence="true" stretchy="true" symmetric="true"></mo>
</mrow>
</mrow>
<mo>]</mo>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mi>C</mi>
</mfrac>
</mrow>
<msubsup>
<mi>G</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>p</mi>
<mo>,</mo>
<mi>q</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mspace width="thinmathspace"></mspace>
<mi>m</mi>
<mo>,</mo>
<mi>n</mi>
</mrow>
</msubsup>
<mspace width="negativethinmathspace"></mspace>
<mrow>
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<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>p</mi>
</mrow>
</msub>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mi>b</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<msub>
<mi>b</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>q</mi>
</mrow>
</msub>
</mtd>
</mtr>
</mtable>
</mrow>
<mspace width="thickmathspace"></mspace>
</mrow>
<mo>|</mo>
</mrow>
<mspace width="thinmathspace"></mspace>
<msup>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mi>C</mi>
</mrow>
</msup>
</mrow>
<mo>)</mo>
</mrow>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle H_{p,q}^{\,m,n}\!\left[z\left|{\begin{matrix}(a_{1},C)&(a_{2},C)&\ldots &(a_{p},C)\\(b_{1},C)&(b_{2},C)&\ldots &(b_{q},C)\end{matrix}}\right.\right]={\frac {1}{C}}G_{p,q}^{\,m,n}\!\left(\left.{\begin{matrix}a_{1},\dots ,a_{p}\\b_{1},\dots ,b_{q}\end{matrix}}\;\right|\,z^{1/C}\right).}</annotation>
</semantics>
</math></span><img src="./8a258456f0dee32c310e13948bff8e3c8f4bb2de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:73.587ex; height:6.509ex;" alt="{\displaystyle H_{p,q}^{\,m,n}\!\left[z\left|{\begin{matrix}(a_{1},C)&(a_{2},C)&\ldots &(a_{p},C)\\(b_{1},C)&(b_{2},C)&\ldots &(b_{q},C)\end{matrix}}\right.\right]={\frac {1}{C}}G_{p,q}^{\,m,n}\!\left(\left.{\begin{matrix}a_{1},\dots ,a_{p}\\b_{1},\dots ,b_{q}\end{matrix}}\;\right|\,z^{1/C}\right).}" loading="lazy"></span></dd></dl>
<p>A generalization of the Fox H-function was given by <a href="Ram_Kishore_Saxena" title="Ram Kishore Saxena">Ram Kishore Saxena</a>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> A further generalization of this function, useful in physics and statistics, was provided by A.M. Mathai and <a href="Ram_Kishore_Saxena" title="Ram Kishore Saxena">Ram Kishore Saxena</a>.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup>
</p>
<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
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<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">
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</style><cite id="CITEREFRathie_and_Ozelim" class="citation web cs1">Rathie and Ozelim, Pushpa Narayan and Luan Carlos de Sena Monteiro. <a rel="nofollow" class="external text" href="https://www.researchgate.net/publication/365706509">"On the Relation between Lambert W-Function and Generalized Hypergeometric Functions"</a>. <i>Researchgate</i><span class="reference-accessdate">. Retrieved <span class="nowrap">1 March</span> 2023</span>.</cite></span>
</li>
<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">(<a href="#CITEREFSrivastavaManocha1984">Srivastava & Manocha 1984</a>, p. 50)</span>
</li>
<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite id="CITEREFMathaiSaxenaSaxena1973" class="citation book cs1">Mathai, A. M.; Saxena, R. K.; Saxena, Ram Kishore (1973). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=MvZUAAAAYAAJ"><i>Generalized Hypergeometric Functions with Applications in Statistics and Physical Sciences</i></a>. Springer. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-06482-6</bdi>.</cite></span>
</li>
<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><a href="#CITEREFInnayat-Hussain1987a">Innayat-Hussain (1987a)</a></span>
</li>
<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><cite id="CITEREFMathaiSaxena1978" class="citation book cs1">Mathai, A. M.; Saxena, Rajendra Kumar (1978). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=DkLvAAAAMAAJ"><i>The H-function with Applications in Statistics and Other Disciplines</i></a>. Wiley. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-470-26380-8</bdi>.</cite></span>
</li>
<li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><a href="#CITEREFRathie1997">Rathie (1997)</a></span>
</li>
</ol></div></div>
<ul><li><cite id="CITEREFFox1961" class="citation cs2">Fox, Charles (1961), "The G and H functions as symmetrical Fourier kernels", <i><a href="Transactions_of_the_American_Mathematical_Society" title="Transactions of the American Mathematical Society">Transactions of the American Mathematical Society</a></i>, <b>98</b> (3): <span class="nowrap">395–</span>429, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1993339">10.2307/1993339</a>, <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0002-9947">0002-9947</a>, <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1993339">1993339</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0131578">0131578</a></cite></li>
<li><cite id="CITEREFInnayat-Hussain1987a" class="citation cs2">Innayat-Hussain, AA (1987a), "New properties of hypergeometric series derivable from Feynman integrals. I: Transformation and reduction formulae", <i>J. Phys. A: Math. Gen.</i>, <b>20</b> (13): <span class="nowrap">4109–</span>4117, <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1987JPhA...20.4109I">1987JPhA...20.4109I</a>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F0305-4470%2F20%2F13%2F019">10.1088/0305-4470/20/13/019</a></cite></li>
<li><cite id="CITEREFInnayat-Hussain1987b" class="citation cs2">Innayat-Hussain, AA (1987b), "New properties of hypergeometric series derivable from Feynman integrals. II: A generalization of the H-function", <i>J. Phys. A: Math. Gen.</i>, <b>20</b> (13): <span class="nowrap">4119–</span>4128, <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1987JPhA...20.4119I">1987JPhA...20.4119I</a>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F0305-4470%2F20%2F13%2F020">10.1088/0305-4470/20/13/020</a></cite></li>
<li><cite id="CITEREFKilbas2004" class="citation cs2">Kilbas, Anatoly A. (2004), <i>H-Transforms: Theory and Applications</i>, CRC Press, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0415299169</bdi></cite></li></ul>
<ul><li><cite id="CITEREFMathaiSaxena1978" class="citation cs2">Mathai, A. M.; Saxena, Ram Kishore (1978), <i>The H-function with applications in statistics and other disciplines</i>, Halsted Press [John Wiley & Sons], New York-London-Sidney, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-470-26380-8</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0513025">0513025</a></cite></li>
<li><cite id="CITEREFMathaiSaxenaHaubold2010" class="citation cs2">Mathai, A. M.; Saxena, Ram Kishore; Haubold, Hans J. (2010), <i>The H-function</i>, Berlin, New York: <a href="Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-4419-0915-2</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2562766">2562766</a></cite></li>
<li><cite id="CITEREFRathie1997" class="citation cs2">Rathie, Arjun K. (1997), "A new generalization of generalized hypergeometric function", <i>Le Matematiche</i>, <b>LII</b>: <span class="nowrap">297–</span>310</cite>.</li>
<li><cite id="CITEREFSrivastavaGuptaGoyal1982" class="citation cs2">Srivastava, H. M.; Gupta, K. C.; Goyal, S. P. (1982), <i>The H-functions of one and two variables</i>, New Delhi: South Asian Publishers Pvt. Ltd., <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0691138">0691138</a></cite></li>
<li><cite id="CITEREFSrivastavaManocha1984" class="citation book cs1">Srivastava, H. M.; Manocha, H. L. (1984). <i>A treatise on generating functions</i>. E. Horwood. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-470-20010-3</bdi>.</cite></li></ul>
<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>
<ul><li><a rel="nofollow" class="external text" href="https://gitlab.com/RZ-FZJ/hypergeom">hypergeom</a> on <a href="GitLab" title="GitLab">GitLab</a></li>
<li><a rel="nofollow" class="external text" href="https://mathoverflow.net/questions/407760/is-there-a-specific-named-function-that-is-the-inverse-of-xxa-for-x-real/407777#407777">Use in solving <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x+x^{a}=y}">
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