Convolutions and zeros of orthogonal polynomials

Abstract

In an attempt to answer a long standing open question of Al-Salam we generate various beautiful formulae for convolutions of orthogonal polynomials similar toU(n)(x) = Sigma(n)(k=0) P(k)(x)P(n-k)(x).where U(n)(x) are the Chebyshev polynomials of the second kind and P(k)(x) are the Legendre polynomials. The results are derived both via the generating functions approach and a new convolution formulae for hypergeometric functions. We apply some addition formulae similar to the well-known expansionH(n)(x + Y) = 2(-n/2) Sigma(n)(k=0) (n k) H(k)(root 2x) H(n-k)(root 2y)for the Hermite polynomials, due to Appell and Kampe de Feriet, to obtain new interesting inequalities about the zeros of the corresponding orthogonal polynomials. (C) 2011 IMACS. Published by Elsevier B.V. All rights reserved.