Abstract
Numerous polynomials, their extensions, and variations have been thoroughly explored, owing to their potential applications in a wide variety of research fields. The purpose of this work is to provide a unified family of Legendre-based generalized Apostol-Bernoulli, Apostol-Euler, and Apostol-Genocchi polynomials, with appropriate constraints for the Maclaurin series. Then we look at the formulae and identities that are involved, including an integral formula, differential formulas, addition formulas, implicit summation formulas, and general symmetry identities. We also provide an explicit representation for these new polynomials. Due to the generality of the findings given here, various formulae and identities for relatively simple polynomials and numbers, such as generalized Bernoulli, Euler, and Genocchi numbers and polynomials, are indicated to be deducible. Furthermore, we employ the umbral calculus theory to offer some additional formulae for these new polynomials.
Highlights
An Explicit ExpressionWe afford an explicit expression for the generalized Legendre-based Apostol-type polynomials SPn(α,β)(x, y, z; k, a, b) in Definition 1
Introduction and PreliminariesCertain multivariate special polynomials enable the study of various classes of partial differential equations that are often encountered in physical problems
We introduced a unification of various polynomials which are called generalized Legendre-based Apostol-type polynomials
Summary
We afford an explicit expression for the generalized Legendre-based Apostol-type polynomials SPn(α,β)(x, y, z; k, a, b) in Definition 1. The restrictions are given as in (18). 5)) to f (t), we obtain an explicit expression for Pn,β(k, a, b):. 1 + (a/β)b 1 − (a/β)b (ii) Consider Özarslan’s generalized Apostol-type numbers Pn(α,β)(k, a, b) (29) (see [17]). Comparing the coefficients of tn on the right-most sides of (41) and (43), we get:. (iii) The polynomials SPn(α,β)(x, y, z; k, a, b) in Definition 1. Equating the coefficients of tn on the resulting single series and the right-hand sided series of (30), we obtain the following explicit expression:. It may be worthwhile to mention in passing that Özarslan [12] (Theorem 2.1) presented an explicit representation of the unified family (22) in terms of a terminating Gauss hypergeometric function 2F1(z) with the argument z depending on the inner summation index
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