On a new class of Laguerre–Pólya type functions with applications in number theory

Abstract

. We define a new class of functions, connected to the classical Laguerre-P´olya class, which we call the shifted Laguerre-P´olya 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 its Taylor coefficients, once shifted, being a degree d multiplier sequence for every d , which is equivalent to its shifted coefficients satisfying all of the higher Tur´an inequalities. This mirrors a classical result of P´olya and Schur. For each function in this class we show some order derivative satisfies each extended Laguerre inequality. Finally, we discuss some old and new conjectures about iterated inequalities for functions in this class.