New Families of Special Polynomial Identities Based upon Combinatorial Sums Related to p-Adic Integrals

Yilmaz Simsek

doi:10.3390/sym13081484

Abstract

The aim of this paper is to study and investigate generating-type functions, which have been recently constructed by the author, with the aid of the Euler’s identity, combinatorial sums, and p-adic integrals. Using these generating functions with their functional equation, we derive various interesting combinatorial sums and identities including new families of numbers and polynomials, the Bernoulli numbers, the Euler numbers, the Stirling numbers, the Daehee numbers, the Changhee numbers, and other numbers and polynomials. Moreover, we present some revealing remarks and comments on the results of this paper.

Highlights

In [1], we recently constructed generating-type functions for some new families of special polynomials and numbers via the umbral calculus convention method. We showed that these new families of special polynomials and numbers are associated with finite calculus, combinatorial numbers and polynomials, polynomial of the chordal graph, and special functions and their applications
The Bernoulli polynomials of order n and degree k are defined by means of the following generating function: w ew − 1
The Euler polynomials of order n and degree k are defined by means of the following generating function: n

Summary

Introduction

In [1], we recently constructed generating-type functions for some new families of special polynomials and numbers via the umbral calculus convention method. The Bernoulli polynomials of order n and degree k are defined by means of the following generating function:. The Euler polynomials of order n and degree k are defined by means of the following generating function:. The Stirling numbers of the first kind, S1 (v, d), are defined by means of the following generating function: d!S1 (v, d) v w. The Daehee numbers, Dn , are defined by means of the following generating function: Dn n log(1 + w) = ∑. Y on Z , which n v v p denotes the set of p-adic integers These formulas include the Bernoulli numbers and polynomials, the Euler numbers and polynomials, the Stirling numbers, the Daehee numbers, and the Changhee numbers. Applying the p-adic fermionic integral to Equation (6) and Equation (20) on Z p , and using (45), (46), (48), and (29), after some elementary calculations, we obtain the following p-adic fermionic integral of the polynomials of θn z; −

Combinatorial Sums and Identities Derived from p-Adic Integrals

Conclusions

Methods