Abstract
In this paper, by using p-adic Volkenborn integral, and generating functions, we give some properties of the Bernstein basis functions, the Apostol-Daehee numbers and polynomials, Apostol-Bernoulli polynomials, some special numbers including the Stirling numbers, the Euler numbers, the Daehee numbers, and the Changhee numbers. By using an integral equation and functional equations of the generating functions and their partial differential equations (PDEs), we give a recurrence relation for the Apostol-Daehee polynomials. We also give some identities, relations, and integral representations for these numbers and polynomials. By using these relations, we compute these numbers and polynomials. We make further remarks and observations for special polynomials and numbers, which are used to study elementary word problems in engineering and in medicine.
Highlights
1 Introduction The special numbers and polynomials have been used in various applications in such diverse areas as mathematics, probability and statistics, mathematical physics, and engineering
In [ ], the Stirling number of the second kind S (n, k) are defined in combinatorics: the Stirling numbers of the second kind are the number of ways to partition a set of n objects into k groups
The Daehee polynomials are defined by means of the following generating function: FD(t, x) tn Dn(x) n!, n=
Summary
The special numbers and polynomials have been used in various applications in such diverse areas as mathematics, probability and statistics, mathematical physics, and engineering. In order to give some results including identities, relations, and formulas for special numbers and polynomials, we use the p-adic Volkenborn integral and generating function methods. Let f ∈ UD(Zp), the set of uniformly differentiable functions on Zp. The p-adic qVolkenborn integration of f on Zp is defined by Kim [ ] as follows: Zp f (x) dμq(x). For instance on the set of complex numbers, we assume that λ ∈ C and on set of p-adic numbers or p-adic integrals, we assume that λ ∈ Zp. The Apostol-Bernoulli polynomials Bn(x; λ) are defined by means of the following generating function: FA(t, x; λ). In [ ], the Stirling number of the second kind S (n, k) are defined in combinatorics: the Stirling numbers of the second kind are the number of ways to partition a set of n objects into k groups These numbers are defined by means of the following generating function:.
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