The powers of matrices with Stirling number-coefficients are investigated. It is revealed that the elements of these matrices have a number of properties of the ordinary Stirling numbers. Moreover, "higher order" Bell, Fubini and Eulerian numbers can be defined. Hence we give a new interpretation for E. T. Bell's iterated exponential integers. In addition, it is worth to note that these numbers appear in combinatorial physics, in the problem of the normal ordering of quantum field theoretical operators.
Given $R \subseteq \mathbb{N}$ let ${n \brace k}_R$, ${n \brack k}_R$, and $L(n,k)_R$ be the number of ways of partitioning the set $[n]$ into $k$ non-empty subsets, cycles and lists, respectively, with each block having cardinality in $R$. We refer to these as the $R$-restricted Stirling numbers of the second and first kind and the $R$-restricted Lah numbers, respectively. Note that the classical Stirling numbers of the second kind and first kind, and Lah numbers are ${n \brace k} = {n \brace k}_{\mathbb{N}}$, ${n \brack k} = {n \brack k}_{\mathbb{N}} $ and $L(n,k) = L(n,k)_{\mathbb{N}}$, res…
The $p$-th power of the logarithm of the Catalan generating function is computed using the Stirling cycle numbers. Instead of Stirling numbers, one may write this generating function in terms of higher order harmonic numbers.
The Jacobi-Stirling numbers were discovered as a result of a problem involving the spectral theory of powers of the classical second-order Jacobi differential expression. Specifically, these numbers are the coefficients of integral composite powers of the Jacobi expression in Lagrangian symmetric form. Quite remarkably, they share many properties with the classical Stirling numbers of the second kind which, as shown in LW, are the coefficients of integral powers of the Laguerre differential expression. In this paper, we establish several properties of the Jacobi-Stirling numbers and its compan…
We study set partitions with $r$ distinguished elements and block sizes found in an arbitrary index set $S$. The enumeration of these $(S,r)$-partitions leads to the introduction of $(S,r)$-Stirling numbers, an extremely wide-ranging generalization of the classical Stirling numbers and the $r$-Stirling numbers. We also introduce the associated $(S,r)$-Bell and $(S,r)$-factorial numbers. We study fundamental aspects of these numbers, including recurrence relations and determinantal expressions. For $S$ with some extra structure, we show that the inverse of the $(S,r)$-Stirling matrix encodes th…
We study matrices which transform the sequence of Fibonacci or Lucas polynomials with even index to those with odd index and vice versa. They turn out to be intimately related to generalized Stirling numbers and to Bernoulli, Genocchi and tangent numbers and give rise to various identities between these numbers. There is also a close connection with the Akiyama-Tanigawa algorithm.
The Legendre-Stirling numbers of the second kind were introduced by Everitt et al. in the spectral theory of powers of the Legendre differential expressions. In this paper, we provide a combinatorial code for Legendre-Stirling set partitions. As an application, we obtain combinatorial expansions of the Legendre-Stirling numbers of both kinds. Moreover, we present grammatical descriptions of the Jacobi-Stirling numbers of both kinds.
Several combinatorial identities are presented, involving Stirling functions of the second kind with a complex variable. The identities involve also Stirling numbers of the first kind, binomial coefficients and harmonic numbers.
Many combinatorial matrices --- such as those of binomial coefficients, Stirling numbers of both kinds, and Lah numbers --- are known to be totally non-negative, meaning that all minors (determinants of square submatrices) are non-negative. The examples noted above can be placed in a common framework: for each one there is a non-decreasing sequence $(a_1, a_2, \ldots)$, and a sequence $(e_1, e_2, \ldots)$, such that the $(m,k)$-entry of the matrix is the coefficient of the polynomial $(x-a_1)\cdots(x-a_k)$ in the expansion of $(x-e_1)\cdots(x-e_m)$ as a linear combination of the polynomials $1…
It is shown in this note that non-central Stirling numbers s(n,k,a) of the first kind naturally appear in the expansion of derivatives of the product of a power function and a logarithn function. We first obtain a recurrence relation for these numbers, and then, using Leibnitz rule we obtain an explicit formula for these numbers. We also obtain an explicit formula for s(n,1,a), and then derive several combinatorial identities related to these numbers.
In Combinatorics Stirling numbers may be defined in several ways. One such definition is given in [1], where an extensive consideration of Stirling numbers is presented. In this paper an alternative definition of Stirling numbers of both kind is given. Namely, Stirling numbers of the first kind appear in the closed formula for the n-th derivative of ln x. In the same way Stirling numbers of the second kind appear in the formula for the n-th derivative of f(e^x), where f(x) is an arbitrary smooth real function. This facts allow us to define Stirling numbers within the frame of differential calc…
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…
Several series expansions for real powers and several formulas for partial Bell polynomials of sinc and sinhc functions in terms of central factorial and Stirling numbers of second kind
本文利用 Faà di Bruno 公式、第二类中心阶乘数和第二类 Stirling 数，建立了 sinc 与 sinhc 函数实数幂的级数展开式，并给出相关偏 Bell 多项式闭式公式及组合数恒等式。
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 $k=5, 6,..., 20$.
Smith normal form evaluations found by Bessenrodt and Stanley for some Hankel matrices of q-Catalan numbers are proven in two ways. One argument generalizes the Bessenrodt-Stanley results for the Smith normal form of a certain multivariate matrix that refines one studied by Berlekamp, Carlitz, Roselle, and Scoville. The second argument, which uses orthogonal polynomials, generalizes to a number of other Hankel matrices, Toeplitz matrices, and Gram matrices. It gives new results for q-Catalan numbers, q-Motzkin numbers, q-Schröder numbers, q-Stirling numbers, q-matching numbers, q-factorials, q…
In the present article we introduce two new combinatorial interpretations of the $r$-Whitney numbers of the second kind obtained from the combinatorics of the differential operators associated to the grammar $G:=\{ y\rightarrow yx^{m}, x\rightarrow x\}$. By specializing $m=1$ we obtain also a new combinatorial interpretation of the $r$-Stirling numbers of the second kind. Again, by specializing to the case $r=0$ we introduce a new generalization of the Stirling number of the second kind and through them a binomial type family of polynomials that generalizes Touchard's. Moreover, we show severa…
We introduce, characterise and provide a combinatorial interpretation for the so-called $q$-Jacobi-Stirling numbers. This study is motivated by their key role in the (reciprocal) expansion of any power of a second order $q$-differential operator having the $q$-classical polynomials as eigenfunctions in terms of other even order operators, which we explicitly construct in this work. The results here obtained can be viewed as the $q$-version of those given by Everitt {\it et al.} and by the first author, whilst the combinatorics of this new set of numbers is a $q$-version of the Jacobi-Stirling …
Consider the Fibonacci numbers defined by setting $F_1=1=F_2$ and $F_n =F_{n-1}+F_{n-2}$ for $n \geq 3$. We let $n_F! = F_1 \cdots F_n$ and $\binom{n}{k}_F = \frac{n_F!}{k_F!(n-k)_F!}$. Let $(x)_{\downarrow_0} = (x)_{\uparrow_0} = 1$ and for $k \geq 1$, $(x)_{\downarrow_k} = x(x-1) \cdots (x-k+1)$ and $(x)_{\uparrow_k} = x(x+1) \cdots (x+k-1)$. Then the Stirling numbers of the first and second kind are the connections coefficients between the usual power basis $\{x^n:n \geq 0\}$ and the falling factorial basis $\{(x)_{\downarrow_n}:n \geq 0\}$ in the polynomial ring $\mathbb{Q}[x]$ and the Lah…
In this note, we derive an alternative recursive formula for the sums of powers of integers involving the Stirling numbers of the first kind. As a remarkable by-product, we provide a non-recursive definition of the Catalan numbers.
It is shown that for positive real numbers $ 0<λ_{1}<\dots<λ_{n}$, $\left[\frac{1}{β({λ_i}, {λ_j})}\right]$, where $ β(\cdot,\cdot)$ denotes the beta function, is infinitely divisible and totally positive. For $ \left[\frac{1}{β({i},{j})}\right]$, the Cholesky decomposition and successive elementary bidiagonal decomposition are computed. Let $\mathfrak w(n)$ be the $n$th Bell number. It is proved that $\left[\mathfrak w(i+j)\right]$ is a totally positive matrix but is infinitely divisible only upto order $4$. It is also shown that the symmetrized Stirling matrices are totally positive.
We define a generalized class of modified zeta series transformations generating the partial sums of the Hurwitz zeta function and series expansions of the Lerch transcendent function. The new transformation coefficients we define within the article satisfy expansions by generalized harmonic number sequences, or the partial sums of the Hurwitz zeta function, which are analogous to known properties for the Stirling numbers of the first kind and for the known transformation coefficients employed to enumerate variants of the polylogarithm function series. Applications of the new results we prove …
Umbral extensions of the stirling numbers of the second kind are considered and the resulting dobinski-like various formulas including new ones are presented. These extensions naturally encompass the two well known q-extensions. The further consecutive umbral extensions q-stirling numbers are therefore realized here in a two-fold way. The fact that the umbral q-extended dobinski formula may also be interpreted as the average of powers of random variable with the q-poisson distribution singles out the q-extensions which appear to be a kind of singular point in the domain of umbral extensions as…
In the paper, by virtue of expansions of two finite products of finitely many square sums, with the aid of series expansions of composite functions of (hyperbolic) sine and cosine functions with inverse sine and cosine functions, and in the light of properties of partial Bell polynomials, the author establishes Taylor's series expansions of real powers of two functions containing squares of inverse (hyperbolic) cosine functions in terms of the Stirling numbers of the first kind, presents an explicit formula of specific partial Bell polynomials at a sequence of derivatives of a function contain…
In this paper, we show that the infinite generalised Stirling matrices associated with boson strings with one annihilation operator are projective limits of approximate substitutions, the latter being characterised by a finite set of algebraic equations.
We obtain an explicit simple formula for the coefficients of the asymptotic expansion for the factorial of a natural number,in terms of derivatives of powers of an elementary function. The unique explicit expression for the coefficients that appears to be known is that in the book by L. Comtet, which is given in terms of sums of associated Stirling numbers of the first kind. By considering the bivariate generating function of the associated Stirling numbers of the second kind, another expression for the coefficients in terms of them follows also from our analysis. Comparison with Comtet's expr…
In the paper, the authors establish Maclaurin's series expansions and series identities for positive integer powers of the inverse sine function, for positive integer powers of the inverse hyperbolic sine function, for the composite of incomplete gamma functions with the inverse hyperbolic sine function, for positive integer powers of the inverse tangent function, and for positive integer powers of the inverse hyperbolic tangent function, in terms of the first kind Stirling numbers and binomial coefficients, apply the newly established Maclaurin's series expansion for positive integer powers o…
Congruences modulo prime powers involving generalized Harmonic numbers are known. While looking for similar congruences, we have encountered a curious triangular array of numbers indexed with positive integers $n,k$, involving the Bernoulli and cycle Stirling numbers. These numbers are all integers and they vanish when $n-k$ is odd. This triangle has many similarities with the Stirling triangle. In particular, we show how it can be extended to negative indices and how this extension produces a {\it second kind} of such integers which may be considered as a new generalization of the Genocchi nu…
It is shown that distance powers of an integral Cayley graph over an abelian group are again integral Cayley graphs over that group. Moreover, it is proved that distance matrices of integral Cayley graphs over abelian groups have integral spectrum.
Certain upper triangular matrices, termed as Parikh matrices, are often used in the combinatorial study of words. Given a word, the Parikh matrix of that word elegantly computes the number of occurrences of certain predefined subwords in that word. In this paper, we compute the Parikh matrix of any word raised to an arbitrary power. Furthermore, we propose canonical decompositions of both Parikh matrices and words into normal forms. Finally, given a Parikh matrix, the relation between its normal form and the normal forms of words in the corresponding M-equivalence class is established.
We derive explicit formulas for the normal ordering of powers of arbitrary monomials of boson operators. These formulas lead to generalisations of conventional Bell and Stirling numbers and to appropriate generalisations of the Dobinski relations. These new combinatorial numbers are shown to be coherent state matrix elements of powers of the monomials in question. It is further demonstrated that such Bell-type numbers, when considered as power moments, give rise to positive measures on the positive half-axis, which in many cases can be written in terms of known functions.
Certain triples of power series, considered by I. Macdonald, give a natural framework for many combinatorial and number theoretic sequences, such as the Stirling, Bernoulli and harmonic numbers and partitions of different kinds. The power series in such a triple are closely linked by identities coming from the theory of symmetric functions. We extend the work of Z-H. Sun, who developed similar ideas, and Macdonald, revealing more of the structure of these triples. De Moivre polynomials play a key role in this study.
Prime and Touchard Congruences of Mixed-Type Bell Numbers
Department of Computer science, University of Torbat-e Jam（托尔巴特-贾姆大学计算机科学系）；Department of Pure Mathematics, Faculty of Mathematical Sciences, Ferdowsi University of Mashhad（马什哈德费尔多西大学数学科学学院纯数学系）
本文建立混合型斯特林数与贝尔数的组合算术框架，连通分拆结构、Touchard 多项式与素幂同余式，结合 p 进赋值理论证明相关同余式，推广组合序列经典算术性质。
When doubly-affine matrices such as Latin and magic squares with a single non-zero eigenvalue are powered up they become constant matrices after a few steps. The process of compounding squares of orders m and n can then be used to generate an infinite series of such squares of orders mn. The Cayley-Hamilton theorem is used to understand this property, where their characteristic polynomials have just two terms.
Conventional Bell and Stirling numbers arise naturally in the normal ordering of simple monomials in boson operators. By extending this process we obtain generalizations of these combinatorial numbers, defined as coherent state matrix elements of arbitrary monomials, as well as the associated Dobinski relations. These Bell-type numbers may be considered as power moments and give rise to positive measures which allow the explicit construction of new classes of coherent states.
