Implementation of computation formulas for certain classes of Apostol-type polynomials and some properties associated with these polynomials

TL;DR

This paper develops identities, formulas, and a computational implementation for Apostol-type polynomials, including $ ext{lambda}$-Apostol-Daehee and related numbers, with applications to series representations and graphical analysis.

Contribution

It introduces new computation formulas and identities for Apostol-type polynomials, including a Mathematica implementation and graphical exploration of their behavior.

Findings

Derived infinite series for $ ext{lambda}$-Apostol-Daehee polynomials.

Implemented a Mathematica code for computing these polynomials.

Presented plots illustrating polynomial behavior for various parameters.

Abstract

The main purpose of this paper is to present various identities and computation formulas for certain classes of Apostol-type numbers and polynomials. The results of this paper contain not only the $\lambda$-Apostol-Daehee numbers and polynomials, but also Simsek numbers and polynomials, the Stirling numbers of the first kind, the Daehee numbers, and the Chu-Vandermonde identity. Furthermore, we derive an infinite series representation for the $\lambda$-Apostol-Daehee polynomials. By using functional equations containing the generating functions for the Cauchy numbers and the Riemann integrals of the generating functions for the $\lambda$-Apostol-Daehee numbers and polynomials, we also derive some identities and formulas for these numbers and polynomials. Moreover, we give implementation of a computation formula for the $\lambda$-Apostol-Daehee polynomials in Mathematica by Wolfram language. By this implementation, we also present some plots of these polynomials in order to investigate their behaviour some randomly selected special cases of its parameters. Finally, we conclude the paper with some comments and observations on our results.