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

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