相关论文: Some identities and properties on degenerate Stirl…
We present various identities in the form of convolutions involving Stirling numbers of both kinds, Lah numbers, and binomial coefficients. Certain convolution polynomials are discussed also. The proofs are based on several series…
The Stirling number of the second kind $n\brace k$ counts the number of ways to partition a set of $n$ labeled balls into $k$ non-empty unlabeled cells. As an extension of this, we consider $b_1+b_2+\ldots+b_n$ balls with $b_1$ balls…
Exponentiating the hypergeometric series gives a recursion relation for integer sequences which are generalizations of conventional Bell numbers. The corresponding associated Stirling numbers of the second kind are also generated and…
We show the classical $q$-Stirling numbers of the second kind can be expressed compactly 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…
In this paper, we introduce the degenerate gamma random variables which are connected with the degenerate gamma functions and the degenerate exponential functions, and deduce the expectation and variance of those random variables.
This is a historical introduction to the theory of Stirling numbers of the second kind S(n,k) from the point of view of analysis. We tell the story of their birth in the book of James Stirling (1730) and show how they mature in the works of…
A triangular field of rational numbers is characterized, with relations to Stirling numbers 2nd, Hyperbolic functions, and centered Binomial distribution. A Generating function is given.
Generalizations of Bell polynomials, Bell numbers, and Stirling numbers of the second kind have been introduced and their generating functions were evaluated.
We consider generalized Stirling numbers of the second kind $% S_{a,b,r}^{\alpha_{s},\beta_{s},r_{s},p_{s}}\left( p,k\right) $, $% k=0,1,\ldots .rp+\sum_{s=2}^{L}r_{s}p_{s}$, where $a,b,\alpha_{s},\beta_{s} $ are complex numbers, and…
The Stirling numbers of type $B$ of the second kind count signed set partitions. In this paper we provide new combinatorial and analytical identities regarding these numbers as well as Broder's $r$-version of these numbers. Among these…
The 2-adic valuation of the Stirling numbres is examined. We conjecture pattrens about the distributions of these valuations in residue classes modulo powers of 2.
In this paper, we introduce the extended r-central factorial numbers of the second and first kinds and the extended r-central Bell polynomials, as extended versions and central analogues of some previously introduced numbers and…
In the paper, the author finds an explicit formula for computing Bernoulli numbers of the second kind in terms of Stirling numbers of the first kind.
This paper investigates the Stirling numbers of the first and second kind associated with a delta series f (t). These numbers provide a robust framework that satisfies the orthogonality and inverse relations, often lacking in recent…
In this paper, we introduce multi-Lah numbers and multi-Stirling numbers of the first kind and recall multi-Bernoulli numbers, all of whose generating functions are given with the help of multiple logarithm. The aim of this paper is to…
In this short note we provide some algebraic identity with a proof exploiting its probabilistic interpretation. We show several consequences of the identity, in particular we obtain a new representation of a Stirling number of second kind,…
In the paper, the author presents diagonal recurrence relations for the Stirling numbers of the first kind. As by-products, the author also recovers three explicit formulas for special values of the Bell polynomials of the second kind.
Recently, the degenerate harmonic and the degenerate hyperharmonic numbers are introduced respectively as degenerate versions of the harmonic and the hyperharmonic numbers. The aim of this paper is to introduce the degenerate…
In this paper, we introduce poly-Bernoulli numbers with level $2$, related to the Stirling numbers of the second kind with level $2$, and study several properties of poly-Bernoulli numbers with level $2$ from their expressions, relations,…
In the paper, the author derives several "diagonal" recurrence relations, constructs some inequalities, finds monotonicity, and poses a conjecture related to Stirling numbers of the second kind.