Applications of q-Umbral Calculus to Modified Apostol Type q-Bernoulli Polynomials

Mehmet Acikgoz,Resul Ates,Ugur Duran,Serkan Araci

doi:10.3844/jmssp.2018.7.15

Abstract

This article aims to identify the generating function of modified Apostol type q-Bernoulli polynomials. With the aid of this generating function, some properties of modified Apostol type q-Bernoulli polynomials are given. It is shown that aforementioned polynomials are q-Appell. Hence, we make use of these polynomials to have applications on q-Umbral calculus. From those applications, we derive some theorems in order to get Apostol type modified q-Bernoulli polynomials as a linear combination of some known polynomials which we stated in the paper.

Highlights

Throughout this paper, we make use of the following standard notations: N:= {1,2,3,⋯} and N 0 = N∪ {0} as usual,Z denotes the set of integers, R denotes the set of real numbers and C denotes the set of complex numbers.We begin with the fundamental properties of q-calculus
As usual, Z denotes the set of integers, R denotes the set of real numbers and C
We review briefly the concept of q-umbral calculus

Summary

Introduction

By using an exponential function eq(x), Kupershmidt (2005) defined the following q-Bernoulli polynomials: Very recently, Kurt (2016) defined Apostol type qBernoulli polynomials of order α by making use of the following generating function: and: cL | p( x) = c L | p( x) for any constant c in C. For the properties of q-umbral calculus, we refer the reader to see the references (Araci et al, 2007; Choi et al, 2008; Kac and Cheung, 2002; Kim and Kim, 2014a; Kim et al, 2013; Mahmudov and Keleshteri, 2013; Roman, 1985). Let P be the algebra of polynomials in the single variable x over the field complex numbers and let P∗ be the vector space of all linear functionals on P .

Objectives

Results

Conclusion