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

Ronald Orozco

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.

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

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.

Paper Structure

This paper contains 9 sections, 22 theorems, 123 equations.

Table of Contents

  1. Introduction
  2. Simplicial polytopic numbers
  3. Simplicial $d$-polytopic calculus
  4. Simplicial $d$-polytopic derivative
  5. Simplicial $d$-polytopic binomial theorem
  6. Simplicial $d$-polytopic-type exponential function
  7. Simplicial $d$-polytopic-type hypergeometric functions
  8. $\mathrm{S}_{d}$-Hypergeometric Bernoulli numbers and polynomials
  9. $\mathrm{S}_d$-Bernoulli numbers and polynomials

Key Result

Proposition 1

For $d\geq1$,

Theorems & Definitions (53)

  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Proposition 4
  • proof
  • Definition 1
  • Theorem 1
  • ...and 43 more