On Binomial Identities in Arbitrary Bases

TL;DR

This paper generalizes the digital binomial identity to any base using b-ary binomial coefficients, exploring their properties and connections to Lucas' theorem and Pascal triangles.

Contribution

It introduces b-ary binomial coefficients and studies their properties, extending binomial identities to arbitrary bases.

Findings

Established orthogonality of b-ary binomial coefficients

Linked b-ary binomial coefficients to Lucas' theorem

Analyzed the structure of b-ary Pascal triangles

Abstract

We extend the digital binomial identity as given by Nguyen el al. to an identity in an arbitrary base $b$, by introducing the $b-$ary binomial coefficients. We then study the properties of these coefficients such as orthogonality, a link to Lucas' theorem and the corresponding $b-$ary Pascal triangles.