We define a new class of functions, connected to the classical Laguerre–Pólya class,
which we call the shifted Laguerre–Pólya 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
multiplier sequence
for every
,
which is equivalent to its shifted coefficients satisfying all of the higher Turán
inequalities. This mirrors a classical result of Pólya 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.
