Recurrence relations for Apostol-Bernoulli , -Euler and -Genocchi polynomials of higher order

TL;DR

This paper derives recurrence relations for generalized Apostol-Bernoulli, Euler, and Genocchi polynomials of higher order using Padé approximation, extending previous results and establishing lacunary relations.

Contribution

It extends existing recurrence relations to generalized higher-order Apostol polynomials using Padé approximation techniques.

Findings

Derived recurrence relations for higher-order Apostol-Bernoulli, Euler, and Genocchi polynomials.

Established lacunary relations for specific cases of these polynomials.

Extended previous results to more general polynomial classes.

Abstract

In \cite{luo2006,luosri2005}, Luo and Srivastava introduced some generalizations of the Apostol -Bernoulli polynomials and the Apostol-Euler polynomials. The main object of this paper is to extend the result of \cite{prevost2010} to these generalized polynomials. More precisely, using the Pad\'{e} approximation of the exponential function, we obtain recurrence relations for Apostol-Bernoulli, Euler and also Genocchi polynomials of higher order. As an application we prove lacunary relation for some particular cases.