Binomial arrays and generalized Vandermonde identities

TL;DR

This paper introduces binomial arrays and extends Vandermonde identities, revealing new combinatorial formulas and connecting them to the representation theory of SL(2, F).

Contribution

It develops a generalized Vandermonde identity and a framework for binomial arrays, linking combinatorics with representation theory.

Findings

Infinite families of summation formulas for Catalan numbers

Generalization of the classical Vandermonde Identity

Connection to finite-dimensional representations of SL(2, F)

Abstract

In previous work on Clebsch-Gordan coefficients, certain remarkable hexagonal arrays of integers are constructed that display behaviors found in Pascal's Triangle. We explain these behaviors further using the binomial transform and discrete convolution. Here we begin by introducing the notion of a binomial array and develop several "hockey stick" rules. Then we give an algorithm that generalizes the classical Vandermonde Identity; this produces infinite families of summation formulas, which we use to expand and prove certain combinatorial identities for the Catalan numbers. Finally, we recast the theory in terms of the finite-dimensional representation theory of $SL(2, F)$.