The purpose of this article is to introduce q-deformed Stirling numbers of the first and second kinds. Relations between these numbers, Riemann zeta function and q-Bernoulli numbers of higher order are given. Some relations related to the classical Stirling numbers and Bernoulli numbers of higher order are found. By using derivative operator to the generating function of the q-deformed Stirling numbers of the second kinds, a new function is defined which interpolates the q-deformed Stirling numbers of the second kinds at negative integers. The recurrence relations of the Stirling numbers of the first and second kind are given. In addition, relation between q-deformed Stirling numbers and q-Bell numbers is obtained.
We review and discuss some results on the representation of Bernoulli, poly-Bernoulli numbers, and Bernoulli and Cauchy polynomials in terms of Stirling numbers of the first or second kind, or in terms of r-Stirling numbers.
The aim of this paper is by using generating functions to further study some identities and properties on the degenerate Stirling numbers of the second kind, the degenerate $r$-Stirling numbers of the second kind, the degenerate Stirling numbers of the first kind and the unsigned degenerate Stirling numbers of the first kind.
Recently, authors studied the unsigned degenerate r-Stirling number of the first kind and the degenerate r-Stirling number of the second kind, respectively of which are the degenerate versions of the unsigned r-Stirling numbers of the first kind and those of the r-Stirling numbers of the second kind. The aim of this paper is to derive some identities involving such special numbers from the inverse relations for them.
The multi-Stirling numbers of the second kind, the unsigned multi-Stirling numbers of the first kind, the multi-Lah numbers and the multi-Bernoulli numbers are all defined with the help of the multiple logarithm, and generalize respectively the Stirling numbers of the second kind, the unsigned Stirling numbers of the first kind, the unsigned Lah numbers and the higher-order Bernoulli numbers . The aim of this paper is to introduce the multi-Stirling numbers of the second kind and to find several identities involving those four numbers defined by means of the multiple logarithm and some other s…
We give some formulas of poly-Cauchy numbers by the $r$-Stirling transform. In the case of the classical or poly-Bernoulli numbers, the formulas are with Stirling numbers of the first kind. In our case of the classical or poly-Cauchy numbers, the formulas are with Stirling numbers of the second kind. We also discuss annihilation formulas for poly-Cauchy number with negative indices.
A number of identities are proved by using Stirling transforms. These identities involve Stirling numbers of the first and second kinds, hyperharmonic and derangement numbers, Bernoulli and Euler numbers and polynomials, powers, power sums, and factorials.
The degenerate Stirling numbers of the second kind and of the first kind, which are respectively degenerate versions of the Stirling numbers of the second kind and of the first kind, appear frequently when we study various degenerate versions of some special numbers and polynomials. The aim of this paper is to consider the r-truncated degenerate Stirling numbers of the second kind, which reduce to the degenerate Stirling numbers of the second for r = 1, and to investigate their explicit expressions, some properties and related identities, in connection with several other degenerate special num…
In this article we use an interplay between Newton series and binomial formulas in order to generate a number of series identities involving Cauchy numbers, harmonic numbers, Laguerre polynomials, and Stirling numbers of the first kind.
In this paper, we derive some identities and recurrence relations for the degenerate Stirling numbers of the first kind and of the second kind which are degenerate versions of the ordinary Stirling numbers of the first kind and of the second kind. They are deduced from the normal oderings of degenerate integral powers of the number operator and their inversions, certain relations of boson operators and from the recurrence relations of the Stirling numbers themselves. Here we note that, while the normal ordering of an integral power of the number operator is expressed with the help of the Stirl…
For any positive integer r, the r-associated Stirling number of the second kind enumerates the number of partitions of the set{1,2,3,...,n} into k non-empty disjoint subsets such that each subset contains at least r elements. We introduce the degenerate r-associated Stirling numbers of the second kind and of the first kind. They are degenerate versions of the r-associated Stirling numbers of the second kind and of the first kind, and reduce to the degenerate Stirling numbers of the second kind and of the first kind for r=1. The aim of this paper is to derive recurrence relations for both of th…
Recently, the Cauchy-Carlitz number was defined as the counterpart of the Bernoulli-Carlitz number. Both numbers can be expressed explicitly in terms of so-called Stirling-Carlitz numbers. In this paper, we study the second analogue of Stirling-Carlitz numbers and give some general formulae, including Bernoulli and Cauchy numbers in formal power series with complex coefficients, and Bernoulli-Carlitz and Cauchy-Carlitz numbers in function fields. We also give some applications of Hasse-Teichmüller derivative to hypergeometric Bernoulli and Cauchy numbers in terms of associated Stirling numbers…
Recently, the $λ$-analogues of $r$-Stirling numbers of the first kind were studied by Kim-Kim. The aim of this paper is to introduce the $λ$-analogues of $r$-Stirling numbers of the second kind and to investigate some properties, recurrence relations and certain identities on those numbers. We also introduce the $λ$-analogues of Whitney-type $r$-Stirling numbers of the second and derive similar results to the case of the $λ$-analogues of r-Stirling numbers of the second kind. In addition, we consider the $λ$-analogues of Dowling polynomials and deduce a Dobinski-like formula.
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 study several relations among those three numbers. In more detail, we represent the multi-Bernoulli numbers in terms of the multi-Stirling numbers of the first kind and vice versa, and the multi-Lah numbers in terms of multi-Stirling numbers. In addition, we deduce a recurrence relation for multi-Lah numbers
We study inverse factorial series and their relation to Stirling numbers of the first kind. We prove a special representation of the polylogarithm function in terms of series with such numbers. Using various identities for Stirling numbers of the first kind we construct a number of expansions of functions in terms of inverse factorial series where the coefficients are special numbers. These results are used to prove/reprove the asymptotic expansion of some classical functions. We also prove a binomial formula involving inverse factorials.
In this paper, applying the Faà di Bruno formula and some properties of Bell polynomials, several closed formulas and determinantal expressions involving Stirling numbers of the second kind for higher-order Bernoulli and Euler polynomials are presented.
This work deals with a new generalization of $r$-Stirling numbers using $l$-tuple of permutations and partitions called $(l,r)$-Stirling numbers of both kinds. We study various properties of these numbers using combinatorial interpretations and symmetric functions. Also, we give a limit representation of the multiple zeta function using $(l,r)$-Stirling of the first kind.
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.
The $p$-adic Analysis of Stirling Numbers via Higher Order Bernoulli Numbers
本文利用我们以前对高阶Bernoulli数$B_n^{(l)}$的研究，研究了第二类Stirling数$S(n,k)$的$p$ -adic性质.例如，我们给出了公式$ν_2(S(2^h,k))=d_2(k)-1$的一个新的，大大简化的证明，如果$1\le k \le 2^h$，并推广到任意素数$p$。我们还考虑了第一类Stirling数$s(n,k)$，得到了与第二类Stirling数类似的新结果.给出了这两类Stirling数的模$p$同余式.
The Stirling numbers of the second kind are related to normal orderings in the Weyl algebra, while the unsigned Stirling numbers of the first kind are related to normal orderings in the shift algebra. Kim-Kim introduced a λ-analogue of the unsigned Stirling numbers of the first kind and that of the r-Stirling numbers of the first kind. In this paper, we introduce a λ-analogue of the shift algebra (called λ-shift algebra) and investigate normal orderings in the λ-shift algebra. From the normal orderings in the λ-shift algebra, we derive some identities about the λ-analogue of the unsigned Stirl…
Several new estimates for the 2-adic valuations of Stirling numbers of the second kind are proved. These estimates, together with criteria for when they are sharp, lead to improvements in several known theorems and their proofs, as well as to new theorems. The estimates and criteria all depend on our previous analysis of powers of 2 in the denominators of coefficients of higher order Bernoulli polynomials. The corresponding estimates for Stirling numbers of the first kind are also proved. Some attention is given to asymptotic cases, which will be further explored in subsequent publications.
This paper presents a number of identities for Dirichlet series and series with Stirling numbers of the first kind. As coefficients for the Dirichlet series we use Cauchy numbers of the first and second kinds, hyperharmonic numbers, derangement numbers, binomial coefficients, central binomial coefficients, and Catalan numbers.
Let P be the set of the sequence of polynomials of degree n. The aim of this paper is to study the Stirling numbers of the second kind associated with P and of the first kind associated with P, in a unified and systematic way with the help of umbral calculus technique. Our results are illustrated with many examples which give rise to interesting inverse relations in each case.
In this paper, we present a formula for generating various exotic series in the spirit of Ovidiu Furdui and Alina Sintamarian. Our new series (evaluated in closed form) involve Bernoulli, harmonic, and Catalan numbers. Also Stirling numbers of the second kind, other special numbers, and exponential polynomials. The results include series identities with Laguerre polynomials and derangement polynomials.
Broder introduced the r-Stirling numbers of the first kind and of the second kind which enumerate restricted permutations and respectively restricted partitions, the restriction being that the first r elements must be in distinct cycles and respectively in distinct subsets. Kim-Kim-Lee-Park constructed the degenerate r-Stirling numbers of both kinds as degenerate versions of them. The aim of this paper is to derive some identities and recurrence relations for the degenerate r-Stirling numbers of the first kind and of the second kind via boson operators. In particular, we obtain the normal orde…
