Let $n, k$ and $a$ be positive integers. The Stirling numbers of the first kind, denoted by $s(n,k)$, count the number of permutations of $n$ elements with $k$ disjoint cycles. Let $p$ be a prime. In recent years, Lengyel, Komatsu and Young, Leonetti and Sanna, Adelberg, Hong and Qiu made some progress in the study of the $p$-adic valuations of $s(n,k)$. In this paper, by using Washington's congruence on the generalized harmonic number and the $n$-th Bernoulli number $B_n$ and the properties of $m$-th Stirling numbers of the first kind obtained recently by the authors, we arrive at an exact expression or a lower bound of $v_p(s(ap, k))$ with $a$ and $k$ being integers such that $1\le a\le p-1$ and $1\le k\le ap$. This infers that for any regular prime $p\ge 7$ and for arbitrary integers $a$ and $k$ with $5\le a\le p-1$ and $a-2\le k\le ap-1$, one has $v_p(H(ap-1,k))<-\frac{\log{(ap-1)}}{2\log p}$ with $H(ap-1, k)$ being the $k$-th elementary symmetric function of $1, \frac{1}{2}, ..., \frac{1}{ap-1}$. This gives a partial support to a conjecture of Leonetti and Sanna raised in 2017. We also present results on $v_p(s(ap^n,ap^n-k))$ from which one can derive that under certain condition, for any prime $p\ge 5$, any odd number $k\ge 3$ and any sufficiently large integer $n$, if $(a,p)=1$, then $v_p(s(ap^{n+1},ap^{n+1}-))=v_p(s(ap^n,ap^n-k))+2$. It confirms partially Lengyel's conjecture proposed in 2015.
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.
The aim of this paper is to prove conjectures concerning $p$-adic valuations of Stirling numbers of the second kind $S(n,k)$, $n,k\in\mathbb{N}_+$, stated by Amdeberhan, Manna and Moll and Berrizbeitia et al., where $p$ is a prime number. The proof is based on elementary facts from $p$-adic analysis.
Let $m, n, k$ and $c$ be positive integers. Let $ν_2(k)$ be the 2-adic valuation of $k$. By $S(n,k)$ we denote the Stirling numbers of the second kind. In this paper, we first establish a convolution identity of the Stirling numbers of the second kind and provide a detailed 2-adic analysis to the Stirling numbers of the second kind. Consequently, we show that if $2\le m\le n$ and $c$ is odd, then $ν_2(S(c2^{n+1},2^m-1)-S(c2^n, 2^m-1))=n+1$ except when $n=m=2$ and $c=1$, in which case $ν_2(S(8,3)-S(4,3))=6$. This solves a conjecture of Lengyel proposed in 2009.
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.
In this paper, we use our previous study of the higher order Bernoulli numbers $B_n^{(l)}$ to investigate the $p$-adic properties of the Stirling numbers of the second kind $S(n,k)$. For example, we give a new, greatly simplified proof of the formula $ν_2(S(2^h,k))=d_2(k)-1$ if $1\le k \le 2^h$, and generalize this result to arbitrary primes $p$. We also consider the Stirling numbers of the first kind $s(n,k)$, with new results analogous to those for the Stirling numbers of the second kind. New mod $p$ congruences for Stirling numbers of both kinds are also given.
In this paper, we investigate the 2-adic valuations of the Stirling numbers $S(n, k)$ of the second kind. We show that $v_2(S(4i, 5))=v_2(S(4i+3, 5))$ if and only if $i\not\equiv 7\pmod {32}$. This confirms a conjecture of Amdeberhan, Manna and Moll raised in 2008. We show also that $v_2(S(2^n+1, k+1))= s_2(n)-1$ for any positive integer $n$, where $s_2(n)$ is the sum of binary digits of $n$. It proves another conjecture of Amdeberhan, Manna and Moll.
We generalize results on the $p$-adic valuations of $S(n,k)$, the Stirling number of the second kind and $s(n,k)$ the Stirling number of the first kind. We have several new estimates for these valuations, along with criteria for when the estimates are sharp. The primary foci are the explicit evaluation of $ν_2(S(n,k))$ with $n=c2^h$, $k=b2^h+a$, $a, b, c, h, k, n \in Z^+$, and $1\le a \le 2^{h-1}$, and $ν_p(S(n,k))$ when $n=cp^h$ for an odd prime $p$. We have strong new results, which generalize and strengthen previous results, for all primes. We also have some new results on the $p$-adic valu…
Let $n$ and $k$ be positive integers. We denote by $v_2(n)$ the 2-adic valuation of $n$. The Stirling numbers of the first kind, denoted by $s(n,k)$, counts the number of permutations of $n$ elements with $k$ disjoint cycles. In recent years, Lengyel, Komatsu and Young, Leonetti and Sanna, and Adelberg made some progress on the $p$-adic valuations of $s(n,k)$. In this paper, by introducing the concept of $m$-th Stirling numbers of the first kind and providing a detailed 2-adic analysis, we show an explicit formula on the 2-adic valuation of $s(2^n, k)$. We also prove that $v_2(s(2^n+1,k+1))=v_…
Let S(n,k) denote the Stirling numbers of the second kind. We prove that the p-adic limit of S(p^e a + c, p^e b + d) as e goes to infinity exists for all integers a, b, c, and d. We call the limiting p-adic integer S(p^\infty a + c, p^\infty b + d). When a equiv b mod (p-1) or d \le 0, we express them in terms of p-adic binomial coefficients introduced in a recent paper.
In this paper, we study $λ$-analogues of the r-Stirling numbers of the first kind which have close connections with the r-Stirling numbers of the first kind and $λ$-Stirling numbers of the first kind. Specifically, we give the recurrence relations for these numbers and show their connections with the $λ$-Stirling numbers of the first kind and higher-order Daehee polynomials.
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 analyze congruence classes of $S(n,k)$, the Stirling numbers of the second kind, modulo powers of 2. This analysis provides insight into a conjecture posed by Amdeberhan, Manna and Moll, which those authors established for $k\le5$. We provide a framework that can be used to justify the conjecture by computational means, which we then complete for $k=5, 6,..., 20$.
Recently, the degenerate Stirling numbers of the first kind were introduced. In this paper, we give some formulas for the degenerate Stirling numbers of the first kind in the terms of the complete Bell polynomials with higher-order harmonic number arguments. Also, we derive an identity connecting the degenerate Stirling numbers of the first kind and the degenerate derangement numbers by using probabilistic method.
Recently, several authors have studied the Stirling numbers of the second kind and Bell polynomials. In this paper, we study the extended Stirling polynomials of the second kind and the extended Bell polynomials associated with the Stirling numbers of the second kind. In addition, we note that the extended Bell polynomials can be expressed in terms of the moments of the Poisson random variable with parameter
In a recent work, the degenerate Stirling polynomials of the second kind were studied by T. Kim. In this paper, we investigate the extended degenerate Stirling numbers of the second kind and the extended degenerate Bell polynomials associated with them. As results, we give some expressions, identities and properties about the extended degener- ate Stirling numbers of the second kind and the extended degenerate Bell polynomials.
Let $P_n(x)=\frac1{n!}\sum\binom n{2i+1}(2i+1)^x$. This extends to a continuous function on the 2-adic integers, the $n$th 2-adic partial Stirling function. We show that $(-1)^{n+1}P_n$ is the only 2-adically continuous approximation to $S(x,n)$, the Stirling number of the second kind. We present extensive information about the zeros of $P_n$, for which there are many interesting patterns. We prove that if $e\ge2$ and $2^e+1\le n\le 2^e+4$, then $P_n$ has exactly $2^{e-1}$ zeros, one in each mod $2^{e-1}$ congruence. We study the relationship between the zeros of $P_{2^e+Δ}$ and $P_Δ$, for $1\…
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…
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.
In this paper, we consider the degenerate Stirling polynomials of the second kind which are derived from the generating function. In addition, we give some new identities for these polynomials.
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.
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, 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.
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…
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
Associated to each random variable $Y$ having a finite moment generating function, we introduce a different generalization of the Stirling numbers of the second kind. Some characterizations and specific examples of such generalized numbers are provided. As far as their applications are concerned, attention is focused in extending in various ways the classical formula for sums of powers on arithmetic progressions. Illustrations involving rising factorials, Bell polynomials, polylogarithms, and a certain class of Appell polynomials, in connection with appropriate random variables $Y$ in each cas…
Recently, Kargin et al. (arXiv:2008.00284 [math.NT]) obtained (among many other things) the following formula for the hyper-sums of powers of integers $S_k^{(m)}(n)$ \begin{equation*} S_k^{(m)}(n) = \frac{1}{m!} \sum_{i=0}^{m} (-1)^i \genfrac{[}{]}{0pt}{}{m+n+1}{i+n+1}_{n+1} S_{k+i}(n), \end{equation*} where $S_k^{(0)}(n) \equiv S_k(n)$ is the ordinary power sum $1^k + 2^k + \cdots + n^k$. In this note we point out that a formula equivalent to the preceding one was already established in a different form, namely, a form in which $\genfrac{[}{]}{0pt}{}{m+n+1}{i+n+1}_{n+1}$ is given explicitly a…
In this paper, we exploit the r-Stirling numbers of both kinds in order to give explicit formulae for the values of the high order Bernoulli numbers and polynomials of both kinds at integers. We give also some identities linked the r-Stirling numbers and binomial coefficients.
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.
In this paper we give simple expressions, involving binomials coefficients, for the value of $c(n,k)$ modulo $p^{v_p(n)}$, when $v_p(n) > 0$. Here $c(n,k)$ denotes a Stirling number of the first kind, and $v_p(n)$ is the highest power of $p$ dividing $n$. As an application, we compute the Chern classes of permutation representations of cyclic groups.
For all integers $n \geq k \geq 1$, define $H(n,k) := \sum 1 / (i_1 \cdots i_k)$, where the sum is extended over all positive integers $i_1 < \cdots < i_k \leq n$. These quantities are closely related to the Stirling numbers of the first kind by the identity $H(n,k) = s(n + 1, k + 1) / n!$. Motivated by the works of Erdős-Niven and Chen-Tang, we study the $p$-adic valuation of $H(n,k)$. In particular, for any prime number $p$, integer $k \geq 2$, and $x \geq (k-1)p$, we prove that $ν_p(H(n,k)) < -(k - 1)(\log_p(n/(k - 1)) - 1)$ for all positive integers $n \in [(k-1)p, x]$ whose base $p$ repre…
The partial Stirling numbers T_n(k) used here are defined as the sum over odd values of i of (n choose i) i^k. Their 2-exponents nu(T_n(k)) are important in algebraic topology. We provide many specific results, applying to all values of n, stating that, for all k in a certain congruence class mod 2^t, nu(T_n(k)) = nu(k - k0) + c0, where k0 is a 2-adic integer and c0 a positive integer. Our analysis involves several new general results for nu(sum (n choose 2i+1) i^j), the proofs of which involve a new family of polynomials. Following Clarke, we interpret T_n as a function on the 2-adic integers…
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.
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.
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.
The purpose of this paper is to present a systemic study of some families of q-Euler numbers and polynomials of Norlund's type by using multivariate fermionic p-adic integral on Zp. Moreover, the study of these higher-order q-Euler numbers and polynomials implies some interesting q-analogue of stirling numbers identities.
In this paper we consider carlitz q-Bernoulli numbers and q-stirling numbers of the first and the second kind. From these numbers we derive many interesting formulae associated with q-Bernoulli numbers.
By applying the Newton-Gregory expansion to the polynomial associated with the sum of powers of integers $S_k(n) = 1^k + 2^k + \cdots + n^k$, we derive a couple of infinite families of explicit formulas for $S_k(n)$. One of the families involves the $r$-Stirling numbers of the second kind $\genfrac{\{}{\}}{0pt}{}{k}{j}_r$, $j=0,1,\ldots,k$, while the other involves their duals $\genfrac{\{}{\}}{0pt}{}{k}{j}_{-r}$, with both families of formulas being indexed by the non-negative integer $r$. As a by-product, we obtain three additional formulas for $S_k(n)$ involving the numbers $\genfrac{\{}{\}…
Recently, introduced are the generalized Euler-Genocchi and generalized degenerate Euler-Genocchi polynomials. The aim of this note is to study the multi-Euler-Genocchi and degenerate multi-Euler-Genocchi polynomials which are defined by means of the multiple logarithm and generalize respectively the generalized Euler-Genocchi and generalized degenerate Euler-Genocchi polynomials. Especially, we express the former by the generalized Euler-Genocchi polynomials, the multi-Stirling numbers of the first kind and Stirling numbers of the second kind, and the latter by the generalized degenerate Eule…
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…
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 the paper, by establishing a new and explicit formula for computing the $n$-th derivative of the reciprocal of the logarithmic function, the author presents new and explicit formulas for calculating Bernoulli numbers of the second kind and Stirling numbers of the first kind. As consequences of these formulas, a recursion for Stirling numbers of the first kind and a new representation of the reciprocal of the factorial $n!$ are derived. Finally, the author finds several identities and integral representations relating to Stirling numbers of the first kind.
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 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 th…
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.
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…
In this paper we establish plenty of number theoretic and combinatoric identities involving generalized Bernoulli and Stirling numbers of both kinds. These formulas are deduced from Pascal type matrix representations of Bernoulli and Stirling numbers. For this we define and factorize a modified Pascal matrix corresponding to Bernoulli and Stirling cases.
The first aim of this paper is to construct new generating functions for the generalized λ-Stirling type numbers of the second kind, generalized array type polynomials and generalized Eulerian type polynomials and numbers, attached to Dirichlet character. We derive various functional equations and differential equations using these generating functions. The second aim is provide a novel approach to deriving identities including multiplication formulas and recurrence relations for these numbers and polynomials using these functional equations and differential equations. Furthermore, by applying…
In this paper, we exhibit two matrix representations of the rational roots of generalized Fibonacci polynomials (GFPs) under convolution product, in terms of determinants and permanents, respectively. The underlying root formulas for GFPs and for weighted isobaric polynomials (WIPs), which appeared in an earlier paper by MacHenry and Tudose, make use of two types of operators. These operators are derived from the generating functions for Stirling numbers of the first kind and second kind. Hence we call them Stirling operators. To construct matrix representations of the roots of GFPs, we use th…
