Hyperdifferential representation of polynomial sequences

Abstract

We introduce a representation of polynomial sequences {p_n(z)}∞_n=0 in the form p_n(z) = e^G(z)zⁿ , where G(z) is some differential operator, a representation we refer to as hyperdifferential representation. Such representation exists for sequences with p₀(z) = 1. It is outlined how the generator of representation G(z) is obtained. The method is applied to the Laguerre polynomials and new identities involving sums of them are obtained.