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

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.