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Formulas for p-adic q-integrals including falling-rising factorials, combinatorial sums and special numbers

Yilmaz Simsek

Abstract

The main purpose of this paper is to provide a novel approach to deriving formulas for the p-adic q-integral including the Volkenborn integral and the p-adic fermionic integral. By applying integral equations and these integral formulas to the falling factorials, the rising factorials and binomial coefficients, we derive some new and old identities and relations related to various combinatorial sums, well-known special numbers such as the Bernoulli and Euler numbers, the harmonic numbers, the Stirling numbers, the Lah numbers, the Harmonic numbers, the Fubini numbers, the Daehee numbers and the Changhee numbers. Applying these identities and formulas, we give some new combinatorial sums. Finally, by using integral equations, we derive generating functions for new families of special numbers and polynomials. We also give further comments and remarks on these functions, numbers and integral formulas.

Formulas for p-adic q-integrals including falling-rising factorials, combinatorial sums and special numbers

Abstract

The main purpose of this paper is to provide a novel approach to deriving formulas for the p-adic q-integral including the Volkenborn integral and the p-adic fermionic integral. By applying integral equations and these integral formulas to the falling factorials, the rising factorials and binomial coefficients, we derive some new and old identities and relations related to various combinatorial sums, well-known special numbers such as the Bernoulli and Euler numbers, the harmonic numbers, the Stirling numbers, the Lah numbers, the Harmonic numbers, the Fubini numbers, the Daehee numbers and the Changhee numbers. Applying these identities and formulas, we give some new combinatorial sums. Finally, by using integral equations, we derive generating functions for new families of special numbers and polynomials. We also give further comments and remarks on these functions, numbers and integral formulas.

Paper Structure

This paper contains 12 sections, 86 theorems, 339 equations.

Table of Contents

  1. Introduction
  2. Integral equations for the Volkenborn integral, the $p$-adic $q$ -integrals and the fermionic integral
  3. Properties of the $p$-adic Volkenborn integral
  4. Properties of the $p$-adic fermionic integral
  5. Application of the Volkenborn integral to the falling and rising factorials
  6. Formulas for the sequence $Y_{1}(n:B)$
  7. Formulas for the sequence $Y_{2}(n:B)$
  8. Integral formulas for the Volkenborn integral
  9. Application of the $p$-adic fermionic integral to the falling and rising factorials
  10. Formulas for the sequence $y_{1}(n:E)$
  11. Formulas for the sequence $y_{2}(n:E)$
  12. Identities for combinatorial sums including special numbers

Key Result

Theorem \oldthetheorem

Theorems & Definitions (147)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Theorem \oldthetheorem
  • Remark 5
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • ...and 137 more