Two Useful Facts About Generating Functions

TL;DR

This paper introduces two methods for analyzing sequences via their generating functions, enabling the derivation of recursion relations and orthogonality properties, with applications to various mathematical sequences and polynomials.

Contribution

It presents novel techniques to extract recursion relations and orthogonality criteria directly from generating functions, expanding analytical tools in sequence analysis.

Findings

Derived recursion relations from differential operators acting on generating functions.

Provided criteria to determine orthogonality of sequences using inner products of generating functions.

Demonstrated methods on sequences, Exceptional Hermite Polynomials, and non-commutative examples.

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.