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Fibonacci polynomials

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In <a href="Mathematics" title="Mathematics">mathematics</a>, the Fibonacci polynomials are a <a href="Polynomial_sequence" title="Polynomial sequence">polynomial sequence</a> which can be considered as a generalization of the <a href="Fibonacci_number" class="mw-redirect" title="Fibonacci number">Fibonacci numbers</a>. The polynomials generated in a similar way from the <a href="Lucas_numbers" class="mw-redirect" title="Lucas numbers">Lucas numbers</a> are called Lucas polynomials.

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

These Fibonacci <a href="Polynomial" title="Polynomial">polynomials</a> are defined by a <a href="Recurrence_relation" title="Recurrence relation">recurrence relation</a>:<a href="#cite_note-BQ141-1">[1]</a>

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<annotation encoding="application/x-tex">{\displaystyle F_{n}(x)={\begin{cases}0,&{\mbox{if }}n=0\\1,&{\mbox{if }}n=1\\xF_{n-1}(x)+F_{n-2}(x),&{\mbox{if }}n\geq 2\end{cases}}}</annotation>

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</math><img src="./fc234536bb9cf7390812a82b5f103726e82a3e62.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:42.408ex; height:8.509ex;" alt="{\displaystyle F_{n}(x)={\begin{cases}0,&{\mbox{if }}n=0\\1,&{\mbox{if }}n=1\\xF_{n-1}(x)+F_{n-2}(x),&{\mbox{if }}n\geq 2\end{cases}}}" loading="lazy"></dd></dl>

The Lucas polynomials use the same recurrence with different starting values:<a href="#cite_note-2">[2]</a>

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<annotation encoding="application/x-tex">{\displaystyle L_{n}(x)={\begin{cases}2,&{\mbox{if }}n=0\\x,&{\mbox{if }}n=1\\xL_{n-1}(x)+L_{n-2}(x),&{\mbox{if }}n\geq 2.\end{cases}}}</annotation>

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</math><img src="./596baa024cd9de246c1beb21e8177c7b9cfbc228.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:43.319ex; height:8.509ex;" alt="{\displaystyle L_{n}(x)={\begin{cases}2,&{\mbox{if }}n=0\\x,&{\mbox{if }}n=1\\xL_{n-1}(x)+L_{n-2}(x),&{\mbox{if }}n\geq 2.\end{cases}}}" loading="lazy"></dd></dl>

They can be defined for negative indices by<a href="#cite_note-Springer-3">[3]</a>

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<annotation encoding="application/x-tex">{\displaystyle F_{-n}(x)=(-1)^{n-1}F_{n}(x),}</annotation>

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</math><img src="./2c3ce10791e2d32b6928177e93c58a34f5274efe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.827ex; height:3.176ex;" alt="{\displaystyle F_{-n}(x)=(-1)^{n-1}F_{n}(x),}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle L_{-n}(x)=(-1)^{n}L_{n}(x).}</annotation>

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</math><img src="./c0d0f325e82500a7fc5399f9eeb15745c485b90c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.903ex; height:2.843ex;" alt="{\displaystyle L_{-n}(x)=(-1)^{n}L_{n}(x).}" loading="lazy"></dd></dl>

The Fibonacci polynomials form a sequence of <a href="Orthogonal_polynomials#Recurrence_relation" title="Orthogonal polynomials">orthogonal polynomials</a> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{n}=C_{n}=1}">

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<annotation encoding="application/x-tex">{\displaystyle A_{n}=C_{n}=1}</annotation>

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</math><img src="./f0dd776e1e839e78b66f7e5922b732edf8a1b322.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.201ex; height:2.509ex;" alt="{\displaystyle A_{n}=C_{n}=1}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{n}=0}">

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<annotation encoding="application/x-tex">{\displaystyle B_{n}=0}</annotation>

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</math><img src="./c7358c944cec4dc747db7b0a6add6aa5cb97d0ca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.243ex; height:2.509ex;" alt="{\displaystyle B_{n}=0}" loading="lazy">.

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

The first few Fibonacci polynomials are:

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<annotation encoding="application/x-tex">{\displaystyle F_{0}(x)=0\,}</annotation>

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</math><img src="./9d2d4a278c6545d836f5c366ea821bb8b6fdb7cb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.336ex; height:2.843ex;" alt="{\displaystyle F_{0}(x)=0\,}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle F_{1}(x)=1\,}</annotation>

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</math><img src="./658facd8ef73879e20b3143b4615c4b9f749b26d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.336ex; height:2.843ex;" alt="{\displaystyle F_{1}(x)=1\,}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle F_{2}(x)=x\,}</annotation>

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</math><img src="./a29f80479602de8440e796c9d6224d1c7cde2e09.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.503ex; height:2.843ex;" alt="{\displaystyle F_{2}(x)=x\,}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle F_{3}(x)=x^{2}+1\,}</annotation>

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</math><img src="./9ef6aa18ca8d576b4329459ea51eb3dd67bca70d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.56ex; height:3.176ex;" alt="{\displaystyle F_{3}(x)=x^{2}+1\,}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle F_{4}(x)=x^{3}+2x\,}</annotation>

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</math><img src="./764aadf16474f43df83cc768073d216036d545d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.89ex; height:3.176ex;" alt="{\displaystyle F_{4}(x)=x^{3}+2x\,}" loading="lazy"></dd>

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

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</math><img src="./93f6dcfb45e870d88d29ba6ef9fb7c28106438bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.947ex; height:3.176ex;" alt="{\displaystyle F_{5}(x)=x^{4}+3x^{2}+1\,}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle F_{6}(x)=x^{5}+4x^{3}+3x\,}</annotation>

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</math><img src="./31b06bda879fee7755b9a905a1aa27b2e7131744.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.277ex; height:3.176ex;" alt="{\displaystyle F_{6}(x)=x^{5}+4x^{3}+3x\,}" loading="lazy"></dd></dl>

The first few Lucas polynomials are:

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<annotation encoding="application/x-tex">{\displaystyle L_{0}(x)=2\,}</annotation>

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</math><img src="./ac2f7c8d50f1a0223d113a01d0535fabbcc5a298.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.424ex; height:2.843ex;" alt="{\displaystyle L_{0}(x)=2\,}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle L_{1}(x)=x\,}</annotation>

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</math><img src="./e37ee360f5e529e274da698748b1e26db9914ebe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.591ex; height:2.843ex;" alt="{\displaystyle L_{1}(x)=x\,}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle L_{2}(x)=x^{2}+2\,}</annotation>

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</math><img src="./19edf53ec1dc9fda8ffc2dcf763a4b3dd716a02e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.648ex; height:3.176ex;" alt="{\displaystyle L_{2}(x)=x^{2}+2\,}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle L_{3}(x)=x^{3}+3x\,}</annotation>

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</math><img src="./1ae75fb213b67a5801c8c0defab7b7d2fa3b6346.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.978ex; height:3.176ex;" alt="{\displaystyle L_{3}(x)=x^{3}+3x\,}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle L_{4}(x)=x^{4}+4x^{2}+2\,}</annotation>

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</math><img src="./55d7619c106c9c758bcd8ac1c773e48029d1ef0a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.035ex; height:3.176ex;" alt="{\displaystyle L_{4}(x)=x^{4}+4x^{2}+2\,}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle L_{5}(x)=x^{5}+5x^{3}+5x\,}</annotation>

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</math><img src="./e66c54a1e8c5ccc66018bdac6f297d68e14d6340.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.365ex; height:3.176ex;" alt="{\displaystyle L_{5}(x)=x^{5}+5x^{3}+5x\,}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle L_{6}(x)=x^{6}+6x^{4}+9x^{2}+2.\,}</annotation>

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</math><img src="./9cc344ac936063b15fcb433f9677736cbc10c109.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.069ex; height:3.176ex;" alt="{\displaystyle L_{6}(x)=x^{6}+6x^{4}+9x^{2}+2.\,}" loading="lazy"></dd></dl>

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

<ul><li>The degree of Fn is n − 1 and the degree of Ln is n.</li></ul>

<ul><li>The Fibonacci and Lucas numbers are recovered by evaluating the polynomials at x = 1; <a href="Pell_numbers" class="mw-redirect" title="Pell numbers">Pell numbers</a> are recovered by evaluating Fn at x = 2.</li></ul>

<ul><li>The <a href="Generating_function#Ordinary_generating_function" title="Generating function">ordinary generating functions</a> for the sequences are:<a href="#cite_note-4">[4]</a>

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<annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }F_{n}(x)t^{n}={\frac {t}{1-xt-t^{2}}}}</annotation>

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</math><img src="./d038a5c07d4c090385692faff192ddc6dc83165b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:26.493ex; height:6.843ex;" alt="{\displaystyle \sum _{n=0}^{\infty }F_{n}(x)t^{n}={\frac {t}{1-xt-t^{2}}}}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }L_{n}(x)t^{n}={\frac {2-xt}{1-xt-t^{2}}}.}</annotation>

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</math><img src="./ec99dae9cf8f4ca4a4e910ef2b4d8425dc52ef83.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:27.228ex; height:6.843ex;" alt="{\displaystyle \sum _{n=0}^{\infty }L_{n}(x)t^{n}={\frac {2-xt}{1-xt-t^{2}}}.}" loading="lazy"></dd></dl></li></ul>

<ul><li>The polynomials can be expressed in terms of <a href="Lucas_sequence" title="Lucas sequence">Lucas sequences</a> as

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<annotation encoding="application/x-tex">{\displaystyle F_{n}(x)=U_{n}(x,-1),\,}</annotation>

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</math><img src="./c01dacf8dce0a6da965138331adf6dbda9a0c1b9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.934ex; height:2.843ex;" alt="{\displaystyle F_{n}(x)=U_{n}(x,-1),\,}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle L_{n}(x)=V_{n}(x,-1).\,}</annotation>

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</math><img src="./e868530da7a0e366912ad2710d6dfc9f8ba34f53.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.79ex; height:2.843ex;" alt="{\displaystyle L_{n}(x)=V_{n}(x,-1).\,}" loading="lazy"></dd></dl></li></ul>

<ul><li>They can also be expressed in terms of <a href="Chebyshev_polynomials" title="Chebyshev polynomials">Chebyshev polynomials</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {T}}_{n}(x)}">

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

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

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<annotation encoding="application/x-tex">{\displaystyle F_{n}(x)=i^{n-1}\cdot {\mathcal {U}}_{n-1}({\tfrac {-ix}{2}}),\,}</annotation>

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</math><img src="./7d5fe03ce499c99b64d486ba0be8a605745aeebb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:25.991ex; height:3.676ex;" alt="{\displaystyle F_{n}(x)=i^{n-1}\cdot {\mathcal {U}}_{n-1}({\tfrac {-ix}{2}}),\,}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle L_{n}(x)=2\cdot i^{n}\cdot {\mathcal {T}}_{n}({\tfrac {-ix}{2}}),\,}</annotation>

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</math><img src="./dc745c3ca27fb528be3630867ecaf7c4afd128e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:24.531ex; height:3.676ex;" alt="{\displaystyle L_{n}(x)=2\cdot i^{n}\cdot {\mathcal {T}}_{n}({\tfrac {-ix}{2}}),\,}" loading="lazy"></dd></dl></li></ul>

<dl><dd>where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

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

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</math><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"> is the <a href="Imaginary_unit" title="Imaginary unit">imaginary unit</a>.</dd></dl>

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

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</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Lucas_sequence" title="Lucas sequence">Lucas sequence</a></div>

As particular cases of Lucas sequences, Fibonacci polynomials satisfy a number of identities, such as<a href="#cite_note-Springer-3">[3]</a>

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<annotation encoding="application/x-tex">{\displaystyle F_{m+n}(x)=F_{m+1}(x)F_{n}(x)+F_{m}(x)F_{n-1}(x)\,}</annotation>

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</math><img src="./bcff625acc03325b962e7bf8ea6836b09dbe2308.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.422ex; height:2.843ex;" alt="{\displaystyle F_{m+n}(x)=F_{m+1}(x)F_{n}(x)+F_{m}(x)F_{n-1}(x)\,}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle L_{m+n}(x)=L_{m}(x)L_{n}(x)-(-1)^{n}L_{m-n}(x)\,}</annotation>

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</math><img src="./f69f7d6b0ae45b0ad7a6ab3bbe07ef14fcbc16c0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.985ex; height:2.843ex;" alt="{\displaystyle L_{m+n}(x)=L_{m}(x)L_{n}(x)-(-1)^{n}L_{m-n}(x)\,}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle F_{n+1}(x)F_{n-1}(x)-F_{n}(x)^{2}=(-1)^{n}\,}</annotation>

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</math><img src="./6c2bbdbb3dc0c06f08d53de614e43e48a718c463.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.136ex; height:3.176ex;" alt="{\displaystyle F_{n+1}(x)F_{n-1}(x)-F_{n}(x)^{2}=(-1)^{n}\,}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle F_{2n}(x)=F_{n}(x)L_{n}(x).\,}</annotation>

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</math><img src="./b6d9bc2b217f5288c01ec1c3ac265d9b006c1b16.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.599ex; height:2.843ex;" alt="{\displaystyle F_{2n}(x)=F_{n}(x)L_{n}(x).\,}" loading="lazy"></dd></dl>

Closed form expressions, similar to Binet's formula are:<a href="#cite_note-Springer-3">[3]</a>

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<annotation encoding="application/x-tex">{\displaystyle F_{n}(x)={\frac {\alpha (x)^{n}-\beta (x)^{n}}{\alpha (x)-\beta (x)}},\,L_{n}(x)=\alpha (x)^{n}+\beta (x)^{n},}</annotation>

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</math><img src="./739884cd39dbfc665ad0af2fddf88b08cb9efc92.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:49.643ex; height:6.509ex;" alt="{\displaystyle F_{n}(x)={\frac {\alpha (x)^{n}-\beta (x)^{n}}{\alpha (x)-\beta (x)}},\,L_{n}(x)=\alpha (x)^{n}+\beta (x)^{n},}" loading="lazy"></dd></dl>

where

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<annotation encoding="application/x-tex">{\displaystyle \alpha (x)={\frac {x+{\sqrt {x^{2}+4}}}{2}},\,\beta (x)={\frac {x-{\sqrt {x^{2}+4}}}{2}}}</annotation>

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</math><img src="./5ea20efedf563e0591572c93d8f3ff1722dfd8ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:44.149ex; height:6.176ex;" alt="{\displaystyle \alpha (x)={\frac {x+{\sqrt {x^{2}+4}}}{2}},\,\beta (x)={\frac {x-{\sqrt {x^{2}+4}}}{2}}}" loading="lazy"></dd></dl>

are the solutions (in t) of

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<annotation encoding="application/x-tex">{\displaystyle t^{2}-xt-1=0.\,}</annotation>

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</math><img src="./771aa261c58c3b864621bd1714d445919be32295.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:16.201ex; height:2.843ex;" alt="{\displaystyle t^{2}-xt-1=0.\,}" loading="lazy"></dd></dl>

For Lucas Polynomials n > 0, we have

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<annotation encoding="application/x-tex">{\displaystyle L_{n}(x)=\sum _{k=0}^{\lfloor n/2\rfloor }{\frac {n}{n-k}}{\binom {n-k}{k}}x^{n-2k}.}</annotation>

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</math><img src="./1fdb51ed494a4955b5a5fa7e1f3dfa32f30cc5f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:34.818ex; height:7.843ex;" alt="{\displaystyle L_{n}(x)=\sum _{k=0}^{\lfloor n/2\rfloor }{\frac {n}{n-k}}{\binom {n-k}{k}}x^{n-2k}.}" loading="lazy"></dd></dl>

A relationship between the Fibonacci polynomials and the standard basis polynomials is given by<a href="#cite_note-5">[5]</a>

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<annotation encoding="application/x-tex">{\displaystyle x^{n}=F_{n+1}(x)+\sum _{k=1}^{\lfloor n/2\rfloor }(-1)^{k}\left[{\binom {n}{k}}-{\binom {n}{k-1}}\right]F_{n+1-2k}(x).}</annotation>

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</math><img src="./9a5b8d7a17f74e975d5f66d3bc9674ae1a21e2e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:57.475ex; height:7.676ex;" alt="{\displaystyle x^{n}=F_{n+1}(x)+\sum _{k=1}^{\lfloor n/2\rfloor }(-1)^{k}\left[{\binom {n}{k}}-{\binom {n}{k-1}}\right]F_{n+1-2k}(x).}" loading="lazy"></dd></dl>

For example,

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

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</math><img src="./5d11e0699af3ee6064883347bc89d7df939488a8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.939ex; height:3.176ex;" alt="{\displaystyle x^{4}=F_{5}(x)-3F_{3}(x)+2F_{1}(x)\,}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle x^{5}=F_{6}(x)-4F_{4}(x)+5F_{2}(x)\,}</annotation>

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</math><img src="./b5fb1abcf10e32ef12a11a81b18731230673bbb0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.939ex; height:3.176ex;" alt="{\displaystyle x^{5}=F_{6}(x)-4F_{4}(x)+5F_{2}(x)\,}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle x^{6}=F_{7}(x)-5F_{5}(x)+9F_{3}(x)-5F_{1}(x)\,}</annotation>

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</math><img src="./8492aeeabe9bc14f39e6870c8803f1fd537c2a83.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.629ex; height:3.176ex;" alt="{\displaystyle x^{6}=F_{7}(x)-5F_{5}(x)+9F_{3}(x)-5F_{1}(x)\,}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle x^{7}=F_{8}(x)-6F_{6}(x)+14F_{4}(x)-14F_{2}(x)\,}</annotation>

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</math><img src="./421438e837e4e436def2969a04e9ae6c6d67727a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.954ex; height:3.176ex;" alt="{\displaystyle x^{7}=F_{8}(x)-6F_{6}(x)+14F_{4}(x)-14F_{2}(x)\,}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Combinatorial_interpretation">Combinatorial interpretation</h2></div>

If F(n,k) is the coefficient of xk in Fn(x), namely

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<annotation encoding="application/x-tex">{\displaystyle F_{n}(x)=\sum _{k=0}^{n}F(n,k)x^{k},\,}</annotation>

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</math><img src="./234b205896dea55fee4b0a3ae8dbabf3c738d158.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:23.335ex; height:7.009ex;" alt="{\displaystyle F_{n}(x)=\sum _{k=0}^{n}F(n,k)x^{k},\,}" loading="lazy"></dd></dl>

then F(n,k) is the number of ways an n−1 by 1 rectangle can be tiled with 2 by 1 <a href="Domino" class="mw-redirect" title="Domino">dominoes</a> and 1 by 1 squares so that exactly k squares are used.<a href="#cite_note-BQ141-1">[1]</a> Equivalently, F(n,k) is the number of ways of writing n−1 as an <a href="Composition_(number_theory)" class="mw-redirect" title="Composition (number theory)">ordered sum</a> involving only 1 and 2, so that 1 is used exactly k times. For example F(6,3)=4 and 5 can be written in 4 ways, 1+1+1+2, 1+1+2+1, 1+2+1+1, 2+1+1+1, as a sum involving only 1 and 2 with 1 used 3 times. By counting the number of times 1 and 2 are both used in such a sum, it is evident that

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<annotation encoding="application/x-tex">{\displaystyle F(n,k)={\begin{cases}\displaystyle {\binom {{\frac {1}{2}}(n+k-1)}{k}}&{\text{if }}n\not \equiv k{\pmod {2}},\\[12pt]0&{\text{else}}.\end{cases}}}</annotation>

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</math><img src="./aa4a7dce904fe9477a6eff0ded434ef50c56121b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.505ex; width:49.614ex; height:12.176ex;" alt="{\displaystyle F(n,k)={\begin{cases}\displaystyle {\binom {{\frac {1}{2}}(n+k-1)}{k}}&{\text{if }}n\not \equiv k{\pmod {2}},\\[12pt]0&{\text{else}}.\end{cases}}}" loading="lazy">

This gives a way of reading the coefficients from <a href="Pascal's_triangle" title="Pascal's triangle">Pascal's triangle</a> as shown on the right.

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

/* start https://en.wikipedia.org/ */

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<li id="cite_note-BQ141-1">^ <a href="#cite_ref-BQ141_1-0">a</a> <a href="#cite_ref-BQ141_1-1">b</a> Benjamin & Quinn p. 141

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> Benjamin & Quinn p. 142

</li>

<li id="cite_note-Springer-3">^ <a href="#cite_ref-Springer_3-0">a</a> <a href="#cite_ref-Springer_3-1">b</a> <a href="#cite_ref-Springer_3-2">c</a> Springer

</li>

/* start https://en.wikipedia.org/ */

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</style><cite id="CITEREFWeisstein" class="citation web cs1"><a href="Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/FibonacciPolynomial.html">"Fibonacci Polynomial"</a>. <a href="MathWorld" title="MathWorld">MathWorld</a>.</cite>

</li>

<li id="cite_note-5"><a href="#cite_ref-5">^</a> A proof starts from page 5 in <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170202051159/http://cmimc.org/Documents/Archive/AlgebraSolutions_2016.pdf">Algebra Solutions Packet (no author)</a>.

</li>

</ol></div></div>

<ul><li><cite id="CITEREFBenjaminQuinn2003" class="citation book cs1"><a href="Arthur_T._Benjamin" title="Arthur T. Benjamin">Benjamin, Arthur T.</a>; <a href="Jennifer_Quinn" title="Jennifer Quinn">Quinn, Jennifer J.</a> (2003). "Fibonacci and Lucas Polynomial". Proofs that Really Count: The Art of Combinatorial Proof. Dolciani Mathematical Expositions. Vol. 27. <a href="Mathematical_Association_of_America" title="Mathematical Association of America">Mathematical Association of America</a>. p. <a rel="nofollow" class="external text" href="https://archive.org/details/proofsthatreally0000benj/page/141">141</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-88385-333-7</bdi>.</cite></li>

<li><cite id="CITEREFPhilippou2001" class="citation cs2">Philippou, Andreas N. (2001) [1994], <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Fibonacci_polynomials">"Fibonacci polynomials"</a>, <a href="Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a>, <a href="European_Mathematical_Society" title="European Mathematical Society">EMS Press</a></cite></li>

<li><cite id="CITEREFPhilippou2001" class="citation cs2">Philippou, Andreas N. (2001) [1994], <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Lucas_polynomials">"Lucas polynomials"</a>, <a href="Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a>, <a href="European_Mathematical_Society" title="European Mathematical Society">EMS Press</a></cite></li>

<li><cite id="CITEREFWeisstein" class="citation web cs1"><a href="Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/LucasPolynomial.html">"Lucas Polynomial"</a>. <a href="MathWorld" title="MathWorld">MathWorld</a>.</cite></li>

<li>Jin, Z. On the Lucas polynomials and some of their new identities. Advances in Differential Equations 2018, 126 (2018). <a rel="nofollow" class="external free" href="https://doi.org/10.1186/s13662-018-1527-9">https://doi.org/10.1186/s13662-018-1527-9</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

<ul><li><cite id="CITEREFHoggattBicknell1973" class="citation journal cs1"><a href="Verner_Emil_Hoggatt%2C_Jr." class="mw-redirect" title="Verner Emil Hoggatt, Jr.">Hoggatt, V. E.</a>; Bicknell, Marjorie (1973). "Roots of Fibonacci polynomials". <a href="Fibonacci_Quarterly" title="Fibonacci Quarterly">Fibonacci Quarterly</a>. 11: 271–274. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0015-0517">0015-0517</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=0332645">0332645</a>.</cite></li>

<li><cite id="CITEREFHoggattLong1974" class="citation journal cs1">Hoggatt, V. E.; Long, Calvin T. (1974). "Divisibility properties of generalized Fibonacci Polynomials". <a href="Fibonacci_Quarterly" title="Fibonacci Quarterly">Fibonacci Quarterly</a>. 12: 113. <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=0352034">0352034</a>.</cite></li>

<li><cite id="CITEREFRicci1995" class="citation journal cs1">Ricci, Paolo Emilio (1995). "Generalized Lucas polynomials and Fibonacci polynomials". Rivista di Matematica della Università di Parma. V. Ser. 4: 137–146. <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=1395332">1395332</a>.</cite></li>

<li><cite id="CITEREFYuanZhang2002" class="citation journal cs1">Yuan, Yi; Zhang, Wenpeng (2002). "Some identities involving the Fibonacci Polynomials". Fibonacci Quarterly. 40 (4): 314. <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=1920571">1920571</a>.</cite></li>

<li><cite id="CITEREFCigler2003" class="citation journal cs1">Cigler, Johann (2003). "q-Fibonacci polynomials". Fibonacci Quarterly (41): 31–40. <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=1962279">1962279</a>.</cite></li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><abbr title="On-Line Encyclopedia of Integer Sequences">OEIS</abbr> <a rel="nofollow" class="external text" href="https://oeis.org/A162515">sequence A162515 (Triangle of coefficients of polynomials defined by Binet form)</a></li>

<li><abbr title="On-Line Encyclopedia of Integer Sequences">OEIS</abbr> <a rel="nofollow" class="external text" href="https://oeis.org/A011973">sequence A011973 (Triangle of coefficients of Fibonacci polynomials)</a></li></ul></div><div><div class="zim-footer">

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