Generating functions for generalization Simsek numbers and their applications

Mouloud Goubi

doi:10.2298/aadm200522005g

Abstract

Our purpose in this work is the complete the study of Simsek numbers. We give answer to some open problems concerning polynomial representations and associated generating function. At the end of the study we investigate a new generalization of these numbers and obtain useful identities which connect Simsek numbers and Stirling numbers of second kind.

Highlights

The Stirling numbers of the second kind S2(n, k) are given by means of the generating function 1 et − 1 k = k!tn S2(n, k) n!, n≥0It is obvious to prove that 1k S2(n, k) = k!k (−1)k−j jn. j j=1 Just writing1 et − 1 k = 1 k k (−1)k−j ejt k! k! j j=02010 Mathematics Subject Classification. 11B83, 11B68, 26C05, 26C10
Introducing the notion of generating function of functions, we have proved that the expression of the generating function fn (λ, x) = B(n, k, λ)xk k=0 is given by successive derivatives of the function
One explicit formula of the numbers B (n, k, λ) as a polynomial on k is established in the following theorem

Summary

INTRODUCTION

The Stirling numbers of the second kind S2(n, k) are given by means of the generating function. Simsek numbers; generating functions; exponential partial Bell polynomials. S2(n, k) are special case of Bell polynomials of second kind Bn,k so called exponential partial Bell polynomials (see [1]) these are defined by k tm tn xm m! Some well-known explicit formulae of Bell polynomials (see [1]) are. Let λ a complex number, we consider the numbers B (n, k, λ) defined by the following generating function (see [7]). In our recent work [3] we have extended the problem to numbers B (n, k, λ) and provide a positive answer. In this paper we use techniques from advanced algebra where the composition law (see [5]) and the Cauchy product of functions play an important role, to give another polynomial representation of B(n, k, λ) and compute explicitly the function fn(λ, x) and obtain the corresponding polynomial Pn,λ(x) such that fn(λ, x)

EXPLICIT FORMULA OF SIMSEK NUMBERS

EXPLICIT FORMULA OF THE GENERATING FUNCTION

SIMSEK NUMBERS OF HIGHER ORDER