Abstract
In this article, we examine a family of some special numbers and polynomials not only with their generating functions, but also with computation algorithms for these numbers and polynomials. By using these algorithms, we provide several values of these numbers and polynomials. Furthermore, some new identities, formulas and combinatorial sums are obtained by using relations derived from the functional equations of these generating functions. These identities and formulas include the Apostol-type numbers and polynomials, and also the Stirling numbers. Finally, we give further remarks and observations on the generating function including ?-Apostol-Daehee numbers, special numbers, and finite sums.
Highlights
By using a computation formula including the Apostol-Bernoulli numbers and the Stirling numbers of the first kind, we define a new family of special numbers and polynomials associated with the Apostol-type numbers and polynomials
With the help of this aforementioned computation formula, computation algorithms are presented for calculating values of the numbers and the polynomials belonging to this new family
Since the generating functions and their functional equations are used effectively in order to obtain results of this paper, we recall some definitions associated with the well-known special numbers and polynomials with their generating functions as follows: The Apostol-Bernoulli polynomials Bn(x; λ) are defined as follows tetx
Summary
By using a computation formula including the Apostol-Bernoulli numbers and the Stirling numbers of the first kind, we define a new family of special numbers and polynomials associated with the Apostol-type numbers and polynomials. By using p-adic integral equation, the second author [16] defined the following generating functions for the family of special numbers Yn,χ(λ, q) and polynomials Yn,χ(z; λ, q) including the generalized Apostol-type numbers and polynomials attached to Dirichlet character χ, respectively (6). The motivation of this paper, in the light of the above knowledge regarding to the numbers Yn(λ), is to define a new family of special numbers and polynomials associated with the Apostol-type numbers and polynomials with their computation algorithms, and derive miscellaneous novel identities, relations and combinatorial sums including the Stirling numbers and Apostol-type numbers and polynomials. We provide an explicit formula for the λ-ApostolDaehee numbers with their several values
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