Special Function Inverse Series Pairs

• Journal title : Kyungpook mathematical journal
• Volume 50, Issue 2,  2010, pp.177-193
• Publisher : Department of Mathematics, Kyungpook National University
• DOI : 10.5666/KMJ.2010.50.2.177
Title & Authors
Special Function Inverse Series Pairs
Alsardary, Salar Yaseen; Gould, Henry Wadsworth;

Abstract
Working with the various special functions of mathematical physics and applied mathematics we often encounter inverse relations of the type $\small{F_n(x)=\sum\limits_{k=0}^{n}A^n_kG_k(x)}$ and $\small{ G_n(x)=\sum\limits_{k=0}^{n}B_k^nF_k(x)}$, where 0, 1, 2,$\small{\cdots}$. Here $\small{F_n(x)}$, $\small{G_n(x)}$ denote special polynomial functions, and $\small{A_k^n}$, $\small{B_k^n}$ denote coefficients found by use of the orthogonal properties of $\small{F_n(x)}$ and $\small{G_n(x)}$, or by skillful series manipulations. Typically $\small{G_n(x)=x^n}$ and $\small{F_n(x)=P_n(x)}$, the n-th Legendre polynomial. We give a collection of inverse series pairs of the type $\small{f(n)=\sum\limits_{k=0}^{n}A_k^ng(k)}$ if and only if $\small{g(n)=\sum\limits_{k=0}^{n}B_k^nf(k)}$, each pair being based on some reasonably well-known special function. We also state and prove an interesting generalization of a theorem of Rainville in this form.
Keywords
Special functions;Series Inverses;Linear Algebra;Matrix Inverses;Bernoulli and Euler Polynomials;Combinatorial Identities;
Language
English
Cited by
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