Parametric kinds of generalized Apostol-Bernoulli polynomials and their properties

TL;DR

This paper introduces new generalized Apostol-Bernoulli polynomials with cosine and sine parametric generating functions, exploring their properties, differential equations, and special cases, expanding the mathematical framework of these polynomials.

Contribution

It defines novel parametric Apostol-Bernoulli polynomials using trigonometric generating functions and investigates their properties and related polynomial families.

Findings

New cosine and sine parametric generating functions for Apostol-Bernoulli polynomials

Derivation of differential equations and summation formulas for these polynomials

Introduction of related polynomial families like Gould--Hopper and Hermite--Appell types

Abstract

The purpose of this paper is to define generalized Apostol--Bernoulli polynomials with including a new cosine and sine parametric type of generating function using the quasi-monomiality properties and trigonometric functions. In this study, the Apostol-Bernoulli polynomials with three variable are defined with two new generating functions cosine and sine parameters. Then, we investigate multiplicative and derivative operators, diffrential equations, some summation formulas and partial differential equations for these polynomials. Moreover, we introduce Gould--Hopper--Apostol--Bernoulli type polynomials, Hermite--Appell--Apostol--Bernoulli type polynomials and truncated exponential Apostol--Bernoulli type polynomials. Finally, the special cases of these new polynomials are investigated, and the corresponding results are expressed.