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

Abstract

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