Two Useful Facts About Generating Functions
Alex Kasman, Robert Milson
TL;DR
The paper links generating functions to both differential and difference operators: if a generating function $\psi(z)$ is an eigenfunction of a differential operator $L$, then the sequence of coefficients $(a_n)$ is an eigensequence of the associated difference operator $\\Omega_L$. It also provides a practical orthogonality test: if $\langle \psi(w),\psi(z)\rangle_{\mathcal{R}}$ depends only on the product $wz$ via some $\theta$, then $\theta$ generates the diagonal inner products $\omega_n=\langle a_n,a_n\rangle_{\mathcal{R}}$, certifying orthogonality. The authors illustrate these ideas with a simple numerical sequence, Exceptional Hermite Polynomials (XHPs) and their rich commutative algebra of recurrence relations, and a non-commutative matrix example, highlighting how recurrences emerge from the eigenoperator framework. The approach provides a unified method to derive recurrences and verify orthogonality from generating functions, with potential generalizations to weighted or Laurent generating functions and bispectral settings.
Abstract
Sequences are often conveniently encoded in the form of a generating function depending on a formal variable. This note presents two observations that allow one to draw conclusions about the generated sequence from the generating function. The first constructively produces "recursion relations" for the sequence from differential operators in the formal variable having the generating function as an eigenfunction. The second allows one to determine whether the sequence is orthogonal with respect to some inner product by considering the result of taking the inner product of the generating function with itself. Examples presented to demonstrate the use and value of these methods include a sequence of numbers, a family of Exceptional Hermite Polynomials, and an example illustrating the result in a non-commutative setting.