Asymptotic Estimates for Second Kind Generalized Stirling Numbers

2014

Here presented is a unified approach to Stirling numbers and their generalizations as well as generalized Stirling functions by using generalized factorial functions, k-Gamma functions, and generalized divided difference. Previous well-known extensions of Stirling numbers due to Riordan, Carlitz, Howard, Charalambides-Koutras, Gould-Hopper, Hsu-Shiue, Tsylova Todorov, Ahuja-Enneking, and Stirling functions introduced by Butzer and Hauss, Butzer, Kilbas, and Trujilloet and others are included as particular cases of our generalization. Some basic properties related to our general pattern such as their recursive relations and generating functions are discussed. Some asymptotic expansions for the generalized Stirling functions and generalized Stirling numbers are established. In addition, four algorithms for calculating the Stirling numbers based on our generalization are also given. Tian-Xiao He Dept. Math, Illinois Wesleyan University Bloomington, IL, USA Asymptotic Expansions and Com...

arXiv (Cornell University), 2011

Here presented is a unified approach to Stirling numbers and their generalizations as well as generalized Stirling functions by using generalized factorial functions, k-Gamma functions, and generalized divided difference. Previous well-known extensions of Stirling numbers due to Riordan, Carlitz, Howard, Charalambides-Koutras, Gould-Hopper, Hsu-Shiue, Tsylova Todorov, Ahuja-Enneking, and Stirling functions introduced by Butzer and Hauss, Butzer, Kilbas, and Trujilloet and others are included as particular cases of our generalization. Some basic properties related to our general pattern such as their recursive relations and generating functions are discussed. Three algorithms for calculating the Stirling numbers based on our generalization are also given, which include a comprehensive algorithm using the characterization of Riordan arrays.

Journal Human Research in Rehabilitation, 2013

In this paper we give a representation of Stirling numbers of the second kind, we obtain explicit formulas for some cases of Stirling numbers of the second kind and illustrate a method for founding other such formulas. 2010 Mathematics Subject Classification. 11B73, 05A10.

The journal of combinatorial mathematics and combinatorial computing, 2013

Here presented is a unified expression of Stirling numbers and their generalizations by using generalized factorial functions and generalized divided difference. Three algorithms for calculating the Stirling numbers and their generalizations based on our unified form are also given, which include a comprehensive algorithm using the characterization of Riordan arrays.

A combinatorial methods are used to investigate some properties of certain generalized Stirling numbers, including explicit formula and recurrence relations. Furthermore, an expression of these numbers with symmetric function is deduced.

Studies in Applied Mathematics, 2014

For the Chebyshev-Stirling numbers, a special case of the Jacobi-Stirling numbers, asymptotic formulae are derived in terms of a local central limit theorem. The underlying probabilistic approach also applies to the classical Stirling numbers of the second kind. Thereby a supplement of the asymptotic analysis for these numbers is established.

Mathematical and Computer Modelling, 2011

In this paper new explicit expressions for both kinds of Comtet numbers and some interesting special cases are derived. Moreover, we define and study the generalized multiparameter non-central Stirling numbers and generalized Comtet numbers via differential operators. Furthermore, recurrence relations and new explicit formulas for those numbers are obtained. Finally some interesting special cases, new combinatorial identities and a connection between these numbers and some interesting polynomials are deduced.

Applied Mathematics and Computation, 2010

The aim of this paper is to introduce an approximations family of the factorial function that contains Stirling's formula, Burnside's formula and Gosper's formula. The parameters which provide the best approximations are indicated. Finally, numerical computations are made to show the superiority of our formulas over other known formulas.

Applied Mathematics, 2014

A modified approach via differential operator is given to derive a new family of generalized Stirling numbers of the first kind. This approach gives us an extension of the techniques given by El-Desouky [1] and Gould [2]. Some new combinatorial identities and many relations between different types of Stirling numbers are found. Furthermore, some interesting special cases of the generalized Stirling numbers of the first kind are deduced. Also, a connection between these numbers and the generalized harmonic numbers is derived. Finally, some applications in coherent states and matrix representation of some results obtained are given.

Journal of Applied Mathematics and Decision Sciences, 1997

An inductive method has been presented for nding Stirling numbers of the second kind. Applications to some discrete probability distributions for nding higher order moments have been discussed.