Mellin transforms with only critical zeros: Chebyshev and Gegenbauer functions

TL;DR

This paper investigates Mellin transforms of Chebyshev and Gegenbauer functions, revealing that their polynomial factors have zeros exclusively on the critical line or real axis, with implications for special functions and number theory.

Contribution

It identifies new classes of polynomials with zeros only on the critical line, expressed via hypergeometric functions, and extends these results to Gegenbauer functions.

Findings

Polynomials with zeros only on the critical line are characterized.

Extension of results to a family of polynomials with functional equations.

Generalization to Mellin transforms of Gegenbauer functions.

Abstract

We consider the Mellin transforms of certain Chebyshev functions based upon the Chebyshev polynomials. We show that the transforms have polynomial factors whose zeros lie all on the critical line or on the real line. The polynomials with zeros only on the critical line are identified in terms of certain $_3F_2(1)$ hypergeometric functions. Furthermore, we extend this result to a 1-parameter family of polynomials with zeros only on the critical line. These polynomials possess the functional equation $p_n(s;\beta)=(-1)^{\lfloor n/2 \rfloor} p_n(1-s;\beta)$. We then present the generalization to the Mellin transform of certain Gegenbauer functions. The results should be of interest to special function theory, combinatorics, and analytic number theory.