Analysis of the p-adic q-Volkenborn Integrals: an approach to Apostol-type special numbers and polynomials

TL;DR

This paper explores p-adic q-Volkenborn integrals to define and analyze generalized Apostol-type special numbers and polynomials, deriving identities, relations, and integral representations using p-adic integrals and generating functions.

Contribution

It introduces new generating functions for generalized Apostol-type numbers via p-adic integrals and investigates their properties and relations, expanding the understanding of these special numbers.

Findings

Derived identities involving Apostol-Bernoulli, Apostol-Euler, and other numbers.

Established integral representations for generalized Apostol-Daehee and Changhee numbers.

Connected these numbers with Stirling, Bernoulli, and Euler numbers through generating functions.

Abstract

By applying the p-adic q-Volkenborn Integrals including the bosonic and the fermionic p-adic integrals on p-adic integers, we define generating functions, attached to the Dirichlet character, for the generalized Apostol-Bernoulli numbers and polynomials, the generalized Apostol-Euler numbers and polynomials, generalized Apostol-Daehee numbers and polynomials, and also generalized Apostol-Changhee numbers and polynomials. We investigate some properties of these numbers and polynomials with their generating functions. By using these generating functions and their functional equation, we give some identities and relations including the generalized Apostol-Daehee and Apostol-Changhee numbers and polynomials, the Stirling numbers, the Bernoulli numbers of the second kind, Frobenious-Euler polynomials, the generalized Bernoulli numbers and the generalized Euler numbers and the Frobenious-Euler polynomials. By using the bosonic and the fermionic p-adic integrals, we derive integral represantations for the generalized Apostol-type Daehee numbers and the generalized Apostol-type Changhee numbers.