New identities for binary Krawtchouk polynomials, binomial coefficients and Catalan numbers

TL;DR

This paper derives new combinatorial identities for binary Krawtchouk polynomials by analyzing their characters, leading to novel relations for binomial coefficients and Catalan numbers.

Contribution

It introduces new identities for binary Krawtchouk polynomials and related combinatorial quantities using representation theory methods.

Findings

New identities for binary Krawtchouk polynomials

Novel relations for binomial coefficients

New relations for Catalan numbers

Abstract

We obtain new combinatorial identities for integral values of binary Krawtchouk polynomials $K^{2m}_p(x)$, $0\le p\le 2m$, by computing the characters of the $p$-exterior representations on certain elements of order 2 of $\mathrm{SO}(2m)$. From this identities, we deduce several new relations for binomial coefficients and Catalan numbers.