The higher order asymptotic expansion of the Krawtchouk polynomials

TL;DR

This paper derives a uniform higher order asymptotic expansion of Krawtchouk polynomials in terms of Hermite polynomials, extending classical convergence results and motivated by ergodic sums in Pascal adic transformation.

Contribution

It provides explicit higher order asymptotic expansions of Krawtchouk polynomials in Hermite polynomials, including initial terms, advancing the understanding of their convergence behavior.

Findings

Explicit expressions for initial terms of the asymptotic expansion

Uniform asymptotic expansion in terms of Hermite polynomials

Extension of classical convergence results

Abstract

The paper extends the classical result on the convergence of the Krawtchouk polynomials to the Hermite polynomials. We provide the uniform asymptotic expansion in terms of the Hermite polynomials. We explicitly obtain expressions for a few initial terms of this expansion. The research is motivated by the study of ergodic sums of the Pascal adic transformation.