How to generalize (and not to generalize) the Chu--Vandermonde identity

TL;DR

This paper explores two interpretations of the Chu--Vandermonde identity, providing a comprehensive classification of all possible generalizations for each interpretation, thus deepening understanding of its algebraic and matrix forms.

Contribution

It offers a complete characterization of generalizations of the Chu--Vandermonde identity for both polynomial and matrix interpretations, revealing new structural insights.

Findings

Complete classification of polynomial generalizations

Complete classification of matrix-based generalizations

Unified framework for understanding identity extensions

Abstract

We consider two different interpretations of the Chu--Vandermonde identity: as an identity for polynomials, and as an identity for infinite matrices. Each interpretation leads to a class of possible generalizations, and in both cases we obtain a complete characterization of the solutions.