A Note on Extended Binomial Coefficients

TL;DR

This paper provides a detailed asymptotic expansion of extended binomial coefficients, using a local central limit theorem and explicit formulas involving Hermite polynomials and Bernoulli numbers.

Contribution

It introduces a comprehensive asymptotic expansion for extended binomial coefficients with explicit coefficients and uniform error bounds.

Findings

Derived a complete asymptotic expansion with explicit coefficients.

Used a local central limit theorem to obtain the expansion.

Provided explicit formulas involving Hermite polynomials and Bernoulli numbers.

Abstract

We study the distribution of the extended binomial coefficients by deriving a complete asymptotic expansion with uniform error terms. We obtain the expansion from a local central limit theorem and we state all coefficients explicitly as sums of Hermite polynomials and Bernoulli numbers.