Hypergeometric Bernoulli Polynomials Defined on Simplicial $d$-Polytopic Numbers
Combinatorics
2026-04-01 v1 Number Theory
Abstract
We introduce an -analogue of the hypergeometric Bernoulli polynomials and study their properties. To achieve this goal, we introduce a calculus defined on the simplicial -polytopic numbers. Two definitions of the -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 -Hoggatt binomial coefficients. -analogs of the exponential function and the hypergeometric functions are given.
Cite
@article{arxiv.2603.28940,
title = {Hypergeometric Bernoulli Polynomials Defined on Simplicial $d$-Polytopic Numbers},
author = {Ronald Orozco},
journal= {arXiv preprint arXiv:2603.28940},
year = {2026}
}