Special Numbers and Polynomials Including Their Generating Functions in Umbral Analysis Methods

Yilmaz Simsek

doi:10.3390/axioms7020022

Abstract

In this paper, by applying umbral calculus methods to generating functions for the combinatorial numbers and the Apostol type polynomials and numbers of order k, we derive some identities and relations including the combinatorial numbers, the Apostol-Bernoulli polynomials and numbers of order k and the Apostol-Euler polynomials and numbers of order k. Moreover, by using p-adic integral technique, we also derive some combinatorial sums including the Bernoulli numbers, the Euler numbers, the Apostol-Euler numbers and the numbers y 1 n , k ; λ . Finally, we make some remarks and observations regarding these identities and relations.

Highlights

In order to give the results presented in this paper, we use two techniques which are p-adic integral technique and the umbral calculus technique
In [1,2,3,4,5], we constructed generating functions for families of combinatorial numbers which are used in counting techniques and problems and computing negative order of the first and the second kind Euler numbers and other combinatorial sums
The Bernoulli numbers and the Euler numbers are related to the following p-adic integrals representations, respectively, Z

Summary

Introduction

In order to give the results presented in this paper, we use two techniques which are p-adic integral technique and the umbral calculus technique. The Apostol-Euler polynomials En ( x, λ) of order k are defined by FE (t, x; λ, k) =. En (0, λ) denote the Apostol-Euler numbers of order k. Order k and En := En denote the Euler numbers (cf see, for details, [6,8,9,10,11,12,13,14,15], and the references cited therein). In (cf Equation (8) [1]), we defined the combinatorial numbers y1 (n, k; λ) by means of the following generating function: Fy1 (t, k; λ) =. The Bernoulli numbers and the Euler numbers are related to the following p-adic integrals representations, respectively, Bn =.

Umbral Algebra and Calculus

Combinatorial Sums via p-Adic Integral