Abstract In this paper, we introduce a kind of combinatorial numbers, D − Stirling numbers, and its special cases. An exponential generating function of the D − Stirling numbers is given. We also present recurrence relations, monotonicity, and limiting properties of the D−Stirling numbers. Applications to statistical probability function estimation and restricted occupancy theory are provided. ...

M. K. LAURENSON*†, R. A. NORMAN‡, L. GILBERT§, H. W. REID¶ and P. J. HUDSON†§ * CTVM, University of Edinburgh, Easter Bush, Roslin, Midlothian, EH25 9RG, UK; † The Game Conservancy Trust, Upland Research Group, Crubenmore Lodge, Newtonmore, PH20 1BE, UK; ‡ Department of Computing Science and Mathematics, University of Stirling, Stirling, FK9 4LA, UK; § Institute of Biology, University of Stirli...

Given positive integers n, k, and m, the (n, k)-th mrestrained Stirling number of the first kind is the number of permutations of an n-set with k disjoint cycles of length ≤ m. Inverting the matrix consisting of the (n, k)-th m-restrained Stirling number of the first kind as the (n+1, k +1)-th entry, the (n, k)-th m-restrained Stirling number of the second kind is defined. In this paper, the mu...

M. K. LAURENSON*†‡, R. A. NORMAN§, L. GILBERT†, H. W. REID¶ and P. J. HUDSON†‡ * CTVM, University of Edinburgh, Easter Bush, Roslin, Midlothian, EH25 9RG, UK, † Institute of Biology, University of Stirling, Stirling FK9 4LA, UK, ‡ The Game Conservancy Trust, Upland Research Group, Crubenmore Lodge, Newtonmore, PH20 1BE, UK, § Department of Computing Science and Mathematics, University of Stirli...

A modified approach via differential operator is given to derive a generalization of Stirling numbers of the first kind. This approach combines the two techniques given by Cakic [3] and Blasiak [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 fir...

This paper considers the generalized Stirling numbers of the first and second kinds. First, we show that the sequences of the above generalized Stirling numbers are both log-concave under some mild conditions. Then, we show that some polynomials related to the above generalized Stirling numbers are q-log-concave or q-log-convex under suitable conditions. We further discuss the log-convexity of ...

in recent years, using new methods in utilization of energy resources has become necessary due to environmental pollution and restriction of energy resources. the hybrid system presented in this article produced power with sofc and stirling engine. the purpose is to analyze a 50 kw solid oxide fuel cell that could produce enough thermal energy for a 10 kw stirling engine working in the hybrid s...

A series of inequalities involving Stirling numbers of the first and second kinds with adjacent indices are obtained. Some of them show log-concavity of Stirling numbers in three different directions. The inequalities are used to prove unimodality or strong unimodality of all the subfamilies of Stirling probability functions. Some additional applications are also presented.

We develop polynomials in z ∈ C for which some generalized harmonic numbers are special cases at z= 0. By using the Riordan array method, we explore interesting relationships between these polynomials, the generalized Stirling polynomials, the Bernoulli polynomials, the Cauchy polynomials and the Nörlund polynomials. © 2007 Elsevier B.V. All rights reserved.

The Jacobi-Stirling numbers of the first and second kinds were introduced in the spectral theory and are polynomial refinements of the Legendre-Stirling numbers. Andrews and Littlejohn have recently given a combinatorial interpretation for the second kind of the latter numbers. Noticing that these numbers are very similar to the classical central factorial numbers, we give combinatorial interpr...