On a family of special numbers and polynomials associated with Apostol-type numbers and polynomials and combinatorial numbers

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 λ-Apostol-Daehee numbers, special numbers, and finite sums.

On Generating Functions for Boole Type Polynomials and Numbers of Higher Order and Their Applications

“…For further information and generalization see also (cf. [11,[15][16][17][18]23]). Recently, the first author defined the following combinatorial numbers and polynomials, respectively:…”

“…In literature, there are various different and useful manuscripts related to not only Boole type polynomials and numbers, but also the Peters type polynomials and numbers. Some of those have been recently given by Boas [1], Jordan [2], Kim et al [3][4][5][6][7][8][9][10], Kucukoglu et al [11], Kruchinin [12], Roman [13], Simsek [14][15][16][17][18][19][20], Simsek and So [21], and also Srivastava et al [22,23]. By using generating function method, we give many important and fundamental properties of Boole type polynomials and numbers of higher order.…”

Identities, inequalities for Boole-type polynomials: approach to generating functions and infinite series

“…Similarly when λ = 1, the above equation reduces to the following Euler numbers: (1) n (1) (cf. [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] and the references cited therein).…”

Identities and Computation Formulas for Combinatorial Numbers Including Negative Order Changhee Polynomials

“…By using these functions, many computation formulas and relations including these sums and various kinds of special numbers and polynomials have been given (cf. References [1][2][3][4][5][6][7][8][9]).…”

“…where v ∈ N 0 and (z) v = (−1) v (−z) v = z(z + 1) · · · (z + v − 1) (cf. References [1,[3][4][5][6][7][8][9][10]). In order to give the results of this paper, we need the following generating functions for special polynomials and numbers.…”