KYUNGPOOK Math. J. 50(2010), 177-193 Special Function Inverse Series Pairs Salar Yaseen Alsardary * Department of Mathematics, Physics and Statistics, University of the Sciences in Philadelphia, 600 South 43 rd Street, Philadelphia, PA 19104, USA e-mail : s.alsard@usp.edu Henry Wadsworth Gould Department of Mathematics, West Virginia University, P. O. Box 6310, Morgan- town, WV 26506-6310, USA e-mail : gould@math.wvu.edu Abstract. Working with the various special functions of mathematical physics and ap- plied mathematics we often encounter inverse relations of the type F n (x)= n ∑ k=0 A n k G k (x) and Gn(x)= n ∑ k=0 B n k F k (x), where n =0, 1, 2, ··· . Here F n (x),G n (x) denote special polynomial functions, and A n k ,B n k denote coefficients found by use of the orthogonal properties of Fn(x) and Gn(x), or by skillful series manipulations. Typically Gn(x)= x n and Fn(x)= Pn(x), the n-th Legendre polynomial. We give a collection of inverse series pairs of the type f (n)= n ∑ k=0 A n k g(k) if and only if g(n)= n ∑ k=0 B n k f (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. 1. Introduction When working with the various special functions of mathematical physics and applied mathematics we often encounter inverse relations of the following type: (1) F n (x)= n ∑ k=0 A n k x k , (2) x n = n ∑ k=0 B n k F k (x), * Corresponding Author. Received June 13, 2007; accepted November 2, 2008. 2000 Mathematics Subject Classification:33C45, 33E30, 11B68, 05A19, 15A09, 41A27. Key words and phrases: Special functions, Series Inverses, Linear Algebra, Matrix Inverses, Bernoulli and Euler Polynomials, Combinatorial Identities. 177