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

2008/01/19 by George Gasper, Gasper, George
Mathematics · #15A09 #33D15 #33E20 #Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #Differential Equations and Boundary Problems #FOS: Mathematics #Iterative Methods for Nonlinear Equations #Mathematical functions and polynomials

paper · doi:10.48550/arxiv.0801.2996

openalex publication_date 2008/01/19 · openalex created_date 2024/04/11 · openalex updated_date 2026/07/28

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≥ -1, confluent hypergeometric functions ₀F₁(c; z) when c > 0 or 0 > c > -1, Laguerre polynomials L_n^α(z) when α≥ -2, Jacobi polynomials P_n^(α,β)(z) when α≥ -1 and β≥ -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.

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