Hypergeometric Bernoulli Polynomials Defined on Simplicial $d$-Polytopic Numbers

TL;DR

This paper introduces an ${ m S}_d$-analogue of hypergeometric Bernoulli polynomials using a new calculus on simplicial $d$-polytopic numbers, establishing identities and analogs of classical functions.

Contribution

It develops a novel ${ m S}_d$-calculus framework and defines ${ m S}_d$-analogs of exponential and hypergeometric functions, expanding the theory of Bernoulli polynomials.

Findings

Derived an identity linking Kummer confluent hypergeometric functions and Touchard polynomials.

Introduced ${ m S}_d$-analogs of exponential and hypergeometric functions.

Established properties of the ${ m S}_d$-Bernoulli polynomials.

Abstract

We introduce an ${\rm S}_d$-analogue of the hypergeometric Bernoulli polynomials and study their properties. To achieve this goal, we introduce a calculus defined on the simplicial $d$-polytopic numbers. Two definitions of the ${\rm S}_d$-derivatives are given. These two definitions allow us to derive an identity relating Kummer confluent hypergeometric function and Touchard polynomials. This calculus is closely related to the $d$-Hoggatt binomial coefficients. ${\rm S}_d$-analogs of the exponential function and the hypergeometric functions are given.