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

We introduce an ${\mathrm{S}}_d$-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 ${\mathrm{S}}_d$-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. ${\mathrm{S}}_d$-analogs of the exponential function and the hypergeometric functions are given.