On a new class of Laguerre-P\'{o}lya type functions with applications in number theory

TL;DR

This paper introduces the shifted Laguerre-Pólya class, linking it to number theory and classical inequalities, and explores its properties and implications for functions like the Riemann Xi function.

Contribution

It defines a new class of functions related to Laguerre-Pólya, characterizes it via multiplier sequences and inequalities, and connects it to number theory conjectures.

Findings

Riemann Xi function belongs to the shifted Laguerre-Pólya class.

Membership in this class is equivalent to certain multiplier sequence conditions.

Functions in this class satisfy extended Laguerre inequalities.

Abstract

We define a new class of functions, connected to the classical Laguerre-P\'{o}lya class, which we call the shifted Laguerre-P\'{o}lya class. Recent work of Griffin, Ono, Rolen, and Zagier shows that the Riemann Xi function is in this class. We prove that a function being in this class is equivalent to the Taylor coefficients, once shifted, being a degree $d$ multiplier sequence for every $d$, which is equivalent to shifted coefficients satisfying all of the higher T\'{u}ran inequalities. This mirrors a classical result of P\'{o}lya and Schur. We further show some order derivative of a function in this class satisfies each extended Laguerre inequality. Finally, we discuss some old and new conjectures about iterated inequalities for functions in this class.