On a q-analog of some numbers and polynomials

Serkan Araci,Mehmet Acikgoz,Erkan Ağyüz

doi:10.1186/s13660-014-0542-y

Serkan Araci, Mehmet Acikgoz + Show 1 more

Open Access

https://doi.org/10.1186/s13660-014-0542-y

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Abstract

In this paper, we introduce new q-analogs of the Changhee numbers and polynomials of the first kind and of the second kind. We also derive some new interesting identities related to the Stirling numbers of the first kind and of the second kind, the Euler polynomials of higher order and the q-analogs of Euler polynomials by applying the p-adic integrals method and some summation transform techniques. It turns out that some well-known results are derived as special cases.

Highlights

1 Introduction In mathematics, special functions are known as ‘useful functions’. Because of their remarkable properties, special functions have been used for centuries
The theory of special functions has been continuously developed with contributions by a host of mathematicians, including Euler, Legendre, Laplace, Gauss, Kummer, Eisenstein, Riemann, Ramanujan, and so on
The development of new special functions and of applications of special functions to new areas of mathematics have initiated a resurgence of interest in the p-adic analysis, q-analysis, analytic number theory, combinatorics, and so on

Summary

Introduction

Special functions (or special polynomials) are known as ‘useful functions’ Because of their remarkable properties, special functions have been used for centuries. Srivastava and Todorov [ ] derived an interesting extension of a representation for the generalized Bernoulli numbers in order to obtain interesting special cases considered earlier by Gould [ ]. For more on these issues, e.g., see [ – ]. Let C(Zp) be the space of continuous functions on Zp. For f ∈ C(Zp), the fermionic padic q-invariant integral on Zp is defined by Kim [ , ], as follows: pn –.

Zp n!

Applying to gives

When we compare the coefficients tn n!

Zp tn n!