Abstract In this paper, we discuss the properties of associated Stirling numbers. By means of the method of coefficients, we establish a series of identities involving associated Stirling numbers, Bernoulli numbers, harmonic numbers, and the Cauchy numbers of the first kind. In addition, we give the asymptotic expansion of certain sums involving 2-associated Stirling numbers of the second kind ...

Liquids can work in heat engine cycles that employ regeneration. Four such cycles are discussed: Stirling, Malone, Stirling-Malone, and Brayton. Both regeneration and the role of the second thermodynamic medium are treated, and the principles are verified by quantitative measurements with propylene in a Stirling-Malone cycle.

The concept of Lucky k-polynomials and in particular Lucky χ-polynomials was recently introduced. This paper introduces Stirling number of the fourth kind and Lucky partitions of a finite set in order to determine either the Lucky k- or Lucky χ-polynomial of a graph. The integer partitions influence Stirling partitions of the second kind.

1 Department of Computing Science and Mathematics, University of Stirling, Stirling, United Kingdom, 2 Department of Philosophy, University of Stirling, Stirling, United Kingdom, 3 Institute for Advanced Studies in the Humanities, University of Edinburgh, Edinburgh, United Kingdom, 4 School of Philosophy, Psychology and Language Sciences, University of Edinburgh, Edinburgh, United Kingdom, 5 De...

Marked mesh patterns are a very general type of permutation pattern. We examine a particular marked mesh pattern originally defined by Kitaev and Remmel, and show that its generating function is described by the r-Stirling numbers. We examine some ramifications of various properties of the r-Stirling numbers for this generating function, and find (seemingly new) formulas for the r-Stirling numb...

The Legendre-Stirling numbers are the coeffi cients in the integral Lagrangian symmetric powers of the classical Legendre second-order differential expression. In many ways, these numbers mimic the classical Stirling numbers of the second kind which play a similar role in the integral powers of the classical second-order Laguerre differential expression. In a recent paper, Andrews and Littlejoh...

A Stirling number of the second kind is a combinatorial function which yields interesting number theoretic properties with regard to primality. The Stirling number of the second kind, S(n; k) = 1 k! k P i=0 ( 1) k i (k i), counts the number of partitions of an n-element set into k non-empty subsets. A Stirling prime (of the second kind) is a prime p such that p = S(n; k) for some integers n and...

We present new short proofs for both Stirling’s formula and Stirling’s formula for the Gamma function. Our approach in the …rst case relies upon analysis of Wallis’formula, while the second result follows from the logconvexity property of the Gamma function. The well known formula of Stirling asserts that (1) n! p 2 ne n as n!1; in the sense that the ratio of the two sides tends to 1. This prov...