Applications on the Apostol-Daehee numbers and polynomials associated with special numbers, polynomials, and p-adic integrals

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.

Identities and recurrence relations of special numbers and polynomials of higher order by analysis of their generating functions

“…Therefore, researchers can use them. In work of Simsek and Yardimci [ 12 ], we see that polynomials are used as models, related to differential equations, for real word problems related to sciences, approximate or curve fit experimental data, calculate beam deflection under loading, represent some properties of gases, and perform computer-aided geometric design in engineering. Polynomials represent the characteristics of a linear dynamic system, and we also know that a ratio of two polynomials represents a transfer function of a linear dynamic system.…”

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

“…The main purpose of this section is to investigate the answer to this question. In [18] and [20], the function G (t, λ) are studied as generating functions which are related to the λ-Apostol-Daehee numbers D n (λ). That is,…”

The volkenborn integral of the 𝒑-adic gamma function

“…The Volkenborn integral can be used for defining theadic log gamma functions, the -adic Bernoulli numbers and polynomial, the -adic zeta and Lfunctions. Special numbers and polynomials have played role in almost all areas of mathematics, in mathematical physics, computer science, engineering problems and other areas of science (Araci and Açikgöz, 2015;Kim et al, 2013a;2013b;Simsek and Yardimci, 2016;Simsek, 2014;Srivastava et al, 2012).…”