Solution to Problem 2 in the SIAM Activity Group on Orthogonal Polynomials and Special Functions Newsletter

Author: Richard Askey and George Gasper

Title: Solution to Problem 2 in the SIAM Activity Group on Orthogonal Polynomials and Special Functions Newsletter

Abstract
We solve Problem 2 in the SIAM Activity Group on Orthogonal Polynomials and Special Functions Newsletter by proving that \[x^2 t^x {}_2F_1(x+1,x+1;2;1-t)\] is an even absolutely monotonic, and hence convex, function of $x$ when 0<t<1. Additional absolutely monotonic functions are obtained by using generating functions for the continuous dual Hahn polynomials $S_n(x^2;a,b,c)$ and for the Wilson polynomials $W_n(x^2;a,b,c,d).$

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Author Address
    George Gasper
    Department of Mathematics
    Northwestern University
    Evanston, IL 60208-2730
    george@math.nwu.edu