The notion of the negative q-binomial was recently introduced by Fu, Reiner, Stanton and Thiem. Mirroring the negative q-binomial, we show the classical q-Stirling numbers of the second kind can be expressed as a pair of statistics on a subset of restricted growth words. The resulting expressions are polynomials in q and 1 + q. We extend this enumerative result via a decomposition of the Stirli...

Thermoacoustic (TA) heat engines are engines which convert heat into acoustic power. This relatively new class of heat engines gains commercial interest, because of their low number of moving parts and high potential efficiency [1]. Especially traveling wave thermoacoustic engines are capable of reaching this high efficiency, because traveling wave engines perform a thermodynamic cycle similar ...

A combinatorial interpretation of the earlier studied generalized Stirling numbers, emerging in a normal ordering problem and its inversion, is given. It involves unordered forests of certain types of labeled trees. Partition number arrays related to such forests are also presented.

Let S(n, k) denote the Stirling number of the second kind, and let Kn be such that S(n,Kn − 1) < S(n,Kn) ≥ S(n,Kn + 1). Using a probabilistic argument, we show that, for all n ≥ 2, ⌊e⌋ − 2 ≤ Kn ≤ ⌊e ⌋+ 1, where ⌊x⌋ denotes the integer part of x, and w(n) denotes Lambert’s W function.

We introduce a family of polynomials that generalizes the Bell polynomials, in connection with the combinatorics of the central moments of the Poisson distribution. We show that these polynomials are dual of the Charlier polynomials by the Stirling transform, and we study the resulting combinatorial identities for the number of partitions of a set into subsets of size at least 2.

For well-known three orderings (normal ordering, anti-normal ordering, and Weyl ordering) in Quantum Physics, we give a simple proof for the intertwining formulas among them. Moreover, for the creation operator a† and the annihilation operator a of a harmonic oscillator, we show Weyl ordering expression of (a†a)n as different types of polynomials with respect to the number operator N ≡ a†a. Non...

IN THE COURSE of war service in the Solomon Islands the writer was stationed on Stirling Island in the Treasuries from December 10, 1943, ro January 13, 1944. A collection of land snails was made and subsequently this was supplemented by a collection made on Mono Island at about the same time by Me. N. Gardner. Smith (1885) recorded a number of species collected by _Guppy from the group, but si...

Using a series transformation, Stirling-De Moivre asymptotic series approximation to the Gamma function is converted into a new one with better convergence properties. The new formula is compared with those of Stirling, Laplace and Ramanujan for real arguments greater than 0.5 and turns out to be, for equal number of ’correction’ terms, numerically superior to all of them. As a side benefit, a ...

In this paper we characterize the divisibility by 2 of the Stirling number of the second kind, S(n; k); where n is a su ciently high power of 2. Let 2(r) denote the highest power of 2 which divides r. We show that there exists a function L(k) such that for all n L(k); 2 k!S(2; k) = k 1 hold, independently from n: (Here the independence follows from the periodicity of the Stirling numbers modulo...

We investigate the diagonal generating function of the Jacobi-Stirling numbers of the second kind JS(n+ k, n; z) by generalizing the analogous results for the Stirling and Legendre-Stirling numbers. More precisely, letting JS(n + k, n; z) = pk,0(n) + pk,1(n)z + · · ·+ pk,k(n)z, we show that (1− t)3k−i+1 ∑ n≥0 pk,i(n)t n is a polynomial in t with nonnegative integral coefficients and provide com...