Sheffer sequences with zeros on a line

TL;DR

This paper extends known results about Sheffer sequences, showing that many have zeros aligned on a vertical line, and connects a specific sequence to the Riemann zeta function through Mellin transform representations.

Contribution

It generalizes the zero distribution of Sheffer sequences and links a particular sequence to the Riemann zeta function via Mellin transform techniques.

Findings

Most Sheffer sequences have zeros on a vertical line

A specific Sheffer sequence relates to the Riemann zeta function

Mellin transform representations connect sequences to special functions

Abstract

We extend a result of Bump et al. to show that a large family of Sheffer sequences has their zeros - up to perhaps a finite number of exceptions - on a vertical line. We connect a particular such sequence to the Riemann zeta function via a product representation of a scaled Mellin transform, analogously to the product decomposition of a Mellin transform involving the generalized Laguerre polynomials into factors of the Gamma function and Meixner polynomials.