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<title>Wilson polynomials</title>
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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Wilson polynomials</span></span>
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<p>In mathematics, <b>Wilson polynomials</b> are a family of <a href="Orthogonal_polynomials" title="Orthogonal polynomials">orthogonal polynomials</a> introduced by <a href="James_A._Wilson" title="James A. Wilson">James A. Wilson</a>&nbsp;(<a href="#CITEREFWilson1980">1980</a>)
that generalize <a href="Jacobi_polynomials" title="Jacobi polynomials">Jacobi polynomials</a>, <a href="Hahn_polynomials" title="Hahn polynomials">Hahn polynomials</a>, and <a href="Charlier_polynomials" title="Charlier polynomials">Charlier polynomials</a>.
</p><p>They are defined in terms of the <a href="Generalized_hypergeometric_function" title="Generalized hypergeometric function">generalized hypergeometric function</a> and the <a href="Pochhammer_symbol" class="mw-redirect" title="Pochhammer symbol">Pochhammer symbols</a> by
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<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 p_{n}(t^{2})=(a+b)_{n}(a+c)_{n}(a+d)_{n}{}_{4}F_{3}\left({\begin{matrix}-n&amp;a+b+c+d+n-1&amp;a-t&amp;a+t\\a+b&amp;a+c&amp;a+d\end{matrix}};1\right).}">
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<annotation encoding="application/x-tex">{\displaystyle p_{n}(t^{2})=(a+b)_{n}(a+c)_{n}(a+d)_{n}{}_{4}F_{3}\left({\begin{matrix}-n&amp;a+b+c+d+n-1&amp;a-t&amp;a+t\\a+b&amp;a+c&amp;a+d\end{matrix}};1\right).}</annotation>
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</math></span><img src="./c5e6a9b79415c6cca820b470dc53883d4266b3bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; margin-left: -0.089ex; width:88.627ex; height:6.176ex;" alt="{\displaystyle p_{n}(t^{2})=(a+b)_{n}(a+c)_{n}(a+d)_{n}{}_{4}F_{3}\left({\begin{matrix}-n&amp;a+b+c+d+n-1&amp;a-t&amp;a+t\\a+b&amp;a+c&amp;a+d\end{matrix}};1\right).}" loading="lazy"></span></dd></dl>
<div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div>
<ul><li><a href="Askey%E2%80%93Wilson_polynomial" class="mw-redirect" title="Askey–Wilson polynomial">Askey–Wilson polynomials</a> are a q-analogue of Wilson polynomials.</li></ul>
<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
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</style><cite id="CITEREFWilson1980" class="citation cs2">Wilson, James A. (1980), "Some hypergeometric orthogonal polynomials", <i>SIAM Journal on Mathematical Analysis</i>, <b>11</b> (4): <span class="nowrap">690–</span>701, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1137%2F0511064">10.1137/0511064</a>, <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a>&nbsp;<a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0036-1410">0036-1410</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a>&nbsp;<a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0579561">0579561</a></cite></li>
<li><cite id="CITEREFKoornwinder2001" class="citation cs2">Koornwinder, T.H. (2001) [1994], <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Wilson_polynomials">"Wilson polynomials"</a>, <i><a href="Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="European_Mathematical_Society" title="European Mathematical Society">EMS Press</a></cite></li></ul>
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