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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Polynomial sequence</span></span>
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<p>In <a href="Mathematics" title="Mathematics">mathematics</a>, a <b>polynomial sequence</b> is a <a href="Sequence" title="Sequence">sequence</a> of <a href="Polynomial" title="Polynomial">polynomials</a> indexed by the nonnegative <a href="Integer" title="Integer">integers</a> 0, 1, 2, 3, ..., in which each <a href="Indexed_family" title="Indexed family">index</a> is equal to the <a href="Degree_of_a_polynomial" title="Degree of a polynomial">degree</a> of the corresponding polynomial. Polynomial sequences are a topic of interest in <a href="Enumerative_combinatorics" title="Enumerative combinatorics">enumerative combinatorics</a> and <a href="Algebraic_combinatorics" title="Algebraic combinatorics">algebraic combinatorics</a>, as well as <a href="Applied_mathematics" title="Applied mathematics">applied mathematics</a>.
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<div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2></div>
<p>Some polynomial sequences arise in <a href="Physics" title="Physics">physics</a> and <a href="Approximation_theory" title="Approximation theory">approximation theory</a> as the solutions of certain <a href="Ordinary_differential_equation" title="Ordinary differential equation">ordinary differential equations</a>:
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<ul><li><a href="Laguerre_polynomials" title="Laguerre polynomials">Laguerre polynomials</a></li>
<li><a href="Chebyshev_polynomials" title="Chebyshev polynomials">Chebyshev polynomials</a></li>
<li><a href="Legendre_polynomials" title="Legendre polynomials">Legendre polynomials</a></li>
<li><a href="Jacobi_polynomials" title="Jacobi polynomials">Jacobi polynomials</a></li></ul>
<p>Others come from <a href="Statistics" title="Statistics">statistics</a>:
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<ul><li><a href="Hermite_polynomials" title="Hermite polynomials">Hermite polynomials</a></li></ul>
<p>Many are studied in <a href="Algebra" title="Algebra">algebra</a> and combinatorics:
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<ul><li><a href="Monomial" title="Monomial">Monomials</a></li>
<li><a href="Rising_factorial" class="mw-redirect" title="Rising factorial">Rising factorials</a></li>
<li><a href="Falling_factorial" class="mw-redirect" title="Falling factorial">Falling factorials</a></li>
<li><a href="All-one_polynomial" class="mw-redirect" title="All-one polynomial">All-one polynomials</a></li>
<li><a href="Abel_polynomials" title="Abel polynomials">Abel polynomials</a></li>
<li><a href="Bell_polynomials" title="Bell polynomials">Bell polynomials</a></li>
<li><a href="Bernoulli_polynomials" title="Bernoulli polynomials">Bernoulli polynomials</a></li>
<li><a href="Cyclotomic_polynomial" title="Cyclotomic polynomial">Cyclotomic polynomials</a></li>
<li><a href="Dickson_polynomial" title="Dickson polynomial">Dickson polynomials</a></li>
<li><a href="Fibonacci_polynomials" title="Fibonacci polynomials">Fibonacci polynomials</a></li>
<li><a href="Lagrange_polynomials" class="mw-redirect" title="Lagrange polynomials">Lagrange polynomials</a></li>
<li><a href="Lucas_polynomials" class="mw-redirect" title="Lucas polynomials">Lucas polynomials</a></li>
<li><a href="Spread_polynomials" class="mw-redirect" title="Spread polynomials">Spread polynomials</a></li>
<li><a href="Touchard_polynomials" title="Touchard polynomials">Touchard polynomials</a></li>
<li><a href="Rook_polynomials" class="mw-redirect" title="Rook polynomials">Rook polynomials</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="Classes_of_polynomial_sequences">Classes of polynomial sequences</h2></div>
<ul><li>Polynomial sequences of <a href="Binomial_type" title="Binomial type">binomial type</a></li>
<li><a href="Orthogonal_polynomials" title="Orthogonal polynomials">Orthogonal polynomials</a></li>
<li><a href="Secondary_polynomials" title="Secondary polynomials">Secondary polynomials</a></li>
<li><a href="Sheffer_sequence" title="Sheffer sequence">Sheffer sequence</a></li>
<li><a href="Sturm_sequence" class="mw-redirect" title="Sturm sequence">Sturm sequence</a></li>
<li><a href="Generalized_Appell_polynomials" title="Generalized Appell polynomials">Generalized Appell polynomials</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div>
<ul><li><a href="Umbral_calculus" title="Umbral calculus">Umbral calculus</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
<ul><li>Aigner, Martin. "A course in enumeration", GTM Springer, 2007, <style data-mw-deduplicate="TemplateStyles:r1238218222">
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</style><a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>3-540-39032-4</bdi> p21.</li>
<li>Roman, Steven "The Umbral Calculus", Dover Publications, 2005, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-486-44139-9</bdi>.</li>
<li>Williamson, S. Gill "Combinatorics for Computer Science", Dover Publications, (2002) p177.</li></ul></div><!--htdig_noindex--><div><div class="zim-footer">
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