Article ID: MTJPAM-D-24-00099

Title: New approach to construct generating function for the Apostol-Bernoulli polynomials via Nörlund sum operator

Montes Taurus J. Pure Appl. Math. / ISSN: 2687-4814

Article ID: MTJPAM-D-24-00099; Volume 6 / Issue 2 / Year 2024, Pages 15-23

Document Type: Research Paper

Author(s): Elif Sukruoglu ^a , Yilmaz Simsek ^b

^aDepartment of Mathematics Faculty of Science, University of Akdeniz, TR-0705, Antalya, Turkey

^bDepartment of Mathematics Faculty of Science, University of Akdeniz, TR-0705, Antalya, Turkey

Received: 8 July 2024, Accepted: 24 July 2024, Published: 2 August 2024

Corresponding Author: Elif Sukruoglu (Email address: elifsukruoglu@akdeniz.edu.tr)

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Abstract

The polynomial family defined by Apostol in 1951 and known today as the Apostol-Bernoulli polynomials is known to be studied using many different methods including the Lerch zeta type functions, p-adic Volkenborn integral, spline curves, infinite series involving the Eulerian numbers, etc. The aim of this article is to construct a generating function for the Apostol-Bernoulli polynomials using a new approach technique related to the Nörlund sum operator, in contrast to these methods. The method we use here is to construct these polynomials with a different technique by applying the Nörlund sum operator to analytic function and periodic function with the Laplace transform and the difference operator approach. Furthermore, by applying the Euler differential operator (or Cauchy–Euler operator) to function G(λ; z)=λ^z, and using the Nörlund sum operator, we also give another novel formula involving the Apostol-Bernoulli polynomials. Finally, we give some applications of our theorems. These applications give some new formulas involving the array polynomials, the Eulerian polynomials and the Stirling numbers of the second kind. We also give an explicit computational formula for the Nörlund sum operator.

Keywords: Nörlund sum operator, Apostol-Bernoulli polynomials, generating function, difference operator, Laplace transform, array polynomials

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