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.
Highlights
Let N and C denote respectively the set of natural numbers and the set of complex numbers and let N0 := f0; 1; 2; 3; : : : g = N [ f0g
By using functional equations containing the generating functions for the Cauchy numbers and the Riemann integrals of the generating functions for the -Apostol-Daehee numbers and polynomials, we derive some identities and formulas for these numbers and polynomials
The outline of this paper may brie‡y given as follows: In Section 2, we present various identities and computation formulas containing the -Apostol-Daehee numbers and polynomials, and Simsek numbers and polynomials, the Stirling numbers of the ...rst kind, the Daehee numbers and the Chu-Vandermonde identity
Summary
Let N and C denote respectively the set of natural numbers and the set of complex numbers and let N0 := f0; 1; 2; 3; : : : g = N [ f0g. Many studies on Apostol-type numbers and polynomials have been carried out by some researchers (see [4]-[34]). In this paper, we are mainly dealt with the -Apostol-Daehee numbers Dn ( ) and polynomials Dn (x; ) introduced and investigated by Simsek [27, 28] respectively as in the following generating functions: GD (t; log ) :=. Another family of Apostol-type numbers and polynomials is the family of the numbers Yn ( ) (so-called Simsek numbers) and the polynomials Yn (x; )
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