Abstract
The main purpose and motivation of this work is to investigate and provide some new identities, inequalities and relations for combinatorial numbers and polynomials, and for Peters type polynomials with the help of their generating functions. The results of this paper involve some special numbers and polynomials such as Stirling numbers, the Apostol–Euler numbers and polynomials, Peters polynomials, Boole polynomials, Changhee numbers and the other well-known combinatorial numbers and polynomials. Finally, in the light of Boole’s inequality (Bonferroni’s inequalities) and bounds of the Stirling numbers of the second kind, some inequalities for a combinatorial finite sum are derived. We mention an open problem including bounds for our numbers. Some remarks and observations are presented.
Highlights
Some well-known notations and definitions are given first: N = {1, 2, 3, . . . }, N0 = {0, 1, 2, 3, . . . }
The motivation of this paper as regards generating functions for combinatorial numbers is related to the work of Simsek [16]
We summarize our paper as follows: In Sect. 2, we give generating functions for Peters type combinatorial numbers and polynomials
Summary
When x = 0, the above equation reduces to the following Apostol–Euler numbers: En(λ) = En(0, λ). The Stirling numbers of the second kind, S2(n, k), are defined as follows: (et – 1)k ∞ We give some special values of this polynomials as follows: When x = 0, we have the Peters numbers [15, 20]: sn(λ, μ) = sn(0; λ, μ). The motivation of this paper as regards generating functions for combinatorial numbers is related to the work of Simsek [16]. By using the above integral equation, the first author gave the generalized Apostol– Changhee numbers and polynomials by means of the following generating functions, respectively:
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