Using integrals of squares of certain real-valued special functions to prove that the Pólya Ξ^*(z) function, the functions K_{iz}(a), a > 0, and some other entire functions have only real zeros

Abstract

Analogous to the use of sums of squares of certain real-valued special functions to prove the reality of the zeros of the Bessel functions J_α(z) when α\ge -1, confluent hypergeometric functions {}_0F_1(c; z) when c > 0 or 0 > c > -1, Laguerre polynomials L_n^α(z) when α\ge -2, Jacobi polynomials P_n^{(α,β)}(z) when α\ge -1 and β\ge -1, and some other entire special functions considered in G. Gasper [Using sums of squares to prove that certain entire functions have only real zeros, in Fourier Analysis: Analytic and Geometric Aspects, W. O. Bray, P. S. Milojević and C. V. Stanojević, eds., Marcel Dekker, Inc., 1994, 171--186.], integrals of squares of certain real-valued special functions are used to prove the reality of the zeros of the Pólya Ξ^*(z) function, the K_{iz}(a) functions when a > 0, and some other entire functions.