Mellin transforms with only critical zeros: Legendre functions

TL;DR

This paper studies Mellin transforms of Legendre functions, revealing polynomial factors with zeros exclusively on the critical line, identified via hypergeometric functions, with implications for special functions and number theory.

Contribution

It introduces new polynomials with zeros only on the critical line, expressed through hypergeometric functions, and explores their properties and related Mellin transforms.

Findings

Polynomials with zeros on the critical line are identified.

Hypergeometric representations of these polynomials are provided.

Connections to Mellin transforms of fractional functions are established.

Abstract

We consider the Mellin transforms of certain Legendre functions based upon the ordinary and associated Legendre polynomials. We show that the transforms have polynomial factors whose zeros lie all on the critical line Re $s=1/2$. The polynomials with zeros only on the critical line are identified in terms of certain $_3F_2(1)$ hypergeometric functions. These polynomials possess the functional equation $p_n(s)=(-1)^{\lfloor n/2 \rfloor} p_n(1-s)$. Other hypergeometric representations are presented, as well as certain Mellin transforms of fractional part and fractional part-integer part functions. The results should be of interest to special function theory, combinatorial geometry, and analytic number theory.