Base-$b$ analogues of classic combinatorial objects

TL;DR

This paper explores base-$b$ analogues of classical combinatorial objects, including binomial coefficients, Stirling numbers, Fibonacci numbers, and the exponential function, extending their properties within a digital framework.

Contribution

It introduces a general summation formula and derives base-$b$ analogues of key combinatorial and mathematical functions, expanding their digital and algebraic understanding.

Findings

Derived base-$b$ analogues of Stirling numbers and Fibonacci numbers

Established properties of the base-$b$ binomial coefficient

Extended the exponential function to base-$b$ context

Abstract

We study the properties of the base-$b$ binomial coefficient defined by Jiu and the second author, introduced in the context of a digital binomial theorem. After introducing a general summation formula, we derive base-$b$ analogues of the Stirling numbers of the second kind, the Fibonacci numbers and the classical exponential function.