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

TL;DR

This paper demonstrates that integrals of squares of specific real-valued special functions can be used to prove that certain entire functions, including Pólya's Xi* and K_{iz}(a), have only real zeros, extending previous methods.

Contribution

It introduces a novel approach using integrals of squares of special functions to establish the reality of zeros for new classes of entire functions.

Findings

Proves Pólya's Xi* function has only real zeros.

Shows K_{iz}(a) functions have only real zeros for a > 0.

Extends sum-of-squares methods to new entire functions.

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_\alpha(z) when \alpha \ge -1, confluent hypergeometric functions {}_0F_1(c; z) when c > 0 or 0 > c > -1, Laguerre polynomials L_n^\alpha(z) when \alpha \ge -2, Jacobi polynomials P_n^{(\alpha,\beta)}(z) when \alpha \ge -1 and \beta \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. Milojevi\'c and C. V. Stanojevi\'c, 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 P\'olya \Xi^*(z) function, the K_{iz}(a) functions when a > 0, and some other entire functions.