Stirling Numbers Research Articles

A distribution function from population genetics statistics using Stirling numbers of the first kind: Asymptotics, inversion and numerical evaluation

Stirling numbers of the first kind are common in number theory and combinatorics; through Ewen’s sampling formula, these numbers enter into the calculation of several population genetics statistics, such as Fu’s F s F_s . In previous papers we have considered an asymptotic estimator for a finite sum of Stirling numbers, which enables rapid and accurate calculation of Fu’s F s F_s . These sums can also be viewed as cumulative distribution functions, leading directly to the possibility of an inversion function, where, given a value for Fu’s F s F_s , the goal is to solve for one of the input parameters. We solve this inversion using Newton iteration for small parameters. For large parameters, we have to extend our earlier obtained asymptotic results to solve the inversion problem asymptotically. Numerical experiments are given to show the efficiency of both solving the inversion problem and the expanded asymptotic estimator for sums of Stirling numbers.

A formula relating Bell polynomials and Stirling numbers of the first kind

A formula relating Bell polynomials and Stirling numbers of the first kind

Lerch's $\Phi$ and the Polylogarithm at the Negative Integers

At the negative integers, there is a simple relation between the Lerch $\Phi$ function and the polylogarithm. The literature has a formula for the polylogarithm at the negative integers, which utilizes the Stirling numbers of the second kind. Starting from that formula, we can deduce a simple closed formula for the Lerch $\Phi$ function at the negative integers, where the Stirling numbers are not needed. Leveraging that finding, we also produce alternative formulae for the $k$-th derivatives of the cotangent and cosecant (ditto, tangent and secant), as simple functions of the negative polylogarithm and Lerch $\Phi$, respectively, which is evidence of the importance of these functions (they are less exotic than they seem). Lastly, we present a new formula for the Hurwitz zeta function at the positive integers using this novelty.

Polynomials, numerical triangles and sequences associated with the probability distribution of the hyperbolic cosine type for odd values of the parameter

The article introduces a new class of polynomials that first appeared in the probability distribution density function of the hyperbolic cosine type. With an integer change in one of the parameters of this distribution, polynomials in the form of a product of positive factors are written out with an increasing degree. Earlier, the author found a connection between the distribution of the hyperbolic cosine type and numerical sets, in particular, in the simplest cases with the triangle of coefficients of Bessel polynomials, the triangle of Stirling numbers, sequences of coefficients in the expansion of various functions, etc. Also from the distribution formed numerous numerical sequences, both new and widely known. Consideration of polynomials separately from the density function made it possible to reconstruct numerical sets of coefficients, ordered in the form of numerical triangles and numerical sequences. The connections between the elements of the sets are established. Among the sequences obtained, in the simplest cases, there are those known from others, for example, physical problems. However, the overwhelming majority of the found number sets have not been encountered earlier in the literature. The obvious applications of this research are number theory and algebra. And the interdisciplinarity of the results indicates the possibility of applications and enhances their practical significance in other areas of knowledge.

Polynomials and number sets associated with the probability distribution of the hyperbolic cosine type for even values of the parameter

The polynomials used in the formation of the probability distribution density function of the hyperbolic cosine type are investigated. Earlier, on the basis of a hyperbolic cosine distribution, the author obtained numerical sets, among which not only new ones, but also, for example, the triangle of Stirling numbers, the triangle of the coefficients of Bessel polynomials, sequences of coefficients in the expansion of various functions, etc. In this paper, depending on the natural parameter m and the real distribution parameter β , a new class of polynomials is obtained. For even and odd m , the polynomials are constructed using similar, but different formulas. The article presents polynomials for even values m . Structurally, polynomials consist of quadratic factors. The coefficients of the polynomials, ordered by m , form numerical triangles depending on β . Some relations are found between the coefficients. From the numerical triangles, a set of numerical sequences is obtained, which for integers β are integers. Also, polynomials with respect to x turn out to be polynomials with respect to β . With this interpretation the variable acts as a parameter. New numerical triangles and sequences for different x were found. The overwhelming majority of the obtained numerical sequences are new. The class of polynomials arising from problems of probability theory indicates the possibility of applying the results.

New Properties on Degenerate Bell Polynomials

The aim of this paper is to study the degenerate Bell numbers and polynomials which are degenerate versions of the Bell numbers and polynomials. We derive some new identities and properties of those numbers and polynomials that are associated with the degenerate Stirling numbers of both kinds.

Strict monotonic trees arising from evolutionary processes: Combinatorial and probabilistic study

In this paper we study two models of labelled random trees that generalise the original unlabelled Schröder tree. Our new models can be seen as models for phylogenetic trees in which nodes represent species and labels encode the order of appearance of these species, and thus the chronology of evolution. One important feature of our trees is that they can be generated efficiently thanks to a dynamical, recursive construction. Our first model is an increasing tree in the classical sense (labels increase along each branch of the tree and each label appears only once). To better model phylogenetic trees, we relax the rules of labelling by allowing repetitions in the second model.For each of the two models, we provide asymptotic theorems for different characteristics of the tree (e.g. degree of the root, degree distribution, height, etc.), thus giving extensive information about the typical shapes of these trees. We also provide efficient algorithms to generate large trees efficiently in the two models. The proofs are based on a combination of analytic combinatorics, probabilistic methods, and bijective methods (we exhibit bijections between our models and well-known models of the literature such as permutations and Stirling numbers of both kinds).It turns out that even though our models are labelled, they can be specified simply in the world of ordinary generating functions. However, the resulting generating functions will be formal. Then, by applying Borel transforms the models will be amenable to techniques of analytic combinatorics.

Stirling operators in spatial combinatorics

We define and study a spatial (infinite-dimensional) counterpart of Stirling numbers. In classical combinatorics, the Pochhammer symbol (m)n can be extended from a natural number m∈N to the falling factorials (z)n=z(z−1)⋯(z−n+1) of an argument z from F=R or C, and Stirling numbers of the first and second kinds are the coefficients of the expansions of (z)n through zk, k≤n and vice versa. When taking into account spatial positions of elements in a locally compact Polish space X, we replace N by the space of configurations—discrete Radon measures γ=∑iδxi on X, where δxi is the Dirac measure with mass at xi. The spatial falling factorials (γ)n:=∑i1∑i2≠i1⋯∑in≠i1,…,in≠in−1δ(xi1,xi2,…,xin) can be naturally extended to mappings M(1)(X)∋ω↦(ω)n∈M(n)(X), where M(n)(X) denotes the space of F-valued, symmetric (for n≥2) Radon measures on Xn. There is a natural duality between M(n)(X) and the space CF(n)(X) of F-valued, symmetric continuous functions on Xn with compact support. The Stirling operators of the first and second kind, s(n,k) and S(n,k), are linear operators, acting between spaces CF(n)(X) and CF(k)(X) such that their dual operators, acting from M(k)(X) into M(n)(X), satisfy (ω)n=∑k=1ns(n,k)⁎ω⊗k and ω⊗n=∑k=1nS(n,k)⁎(ω)k, respectively. In the case where X has only a single point, the Stirling operators can be identified with Stirling numbers. We derive combinatorial properties of the Stirling operators, present their connections with a generalization of the Poisson point process and with the Wick ordering under the canonical commutation relations.

Angle Sums of Schläfli Orthoschemes

We consider the simplices KnA={x∈Rn+1:x1≥x2≥⋯≥xn+1,x1-xn+1≤1,x1+⋯+xn+1=0}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} K_n^A=\\{x\\in {\\mathbb {R}}^{n+1}:x_1\\ge x_2\\ge \\cdots \\ge x_{n+1},x_1-x_{n+1}\\le 1,\\,x_1+\\cdots +x_{n+1}=0\\} \\end{aligned}$$\\end{document}and KnB={x∈Rn:1≥x1≥x2≥⋯≥xn≥0},\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} K_n^B=\\{x\\in {\\mathbb {R}}^n:1\\ge x_1\\ge x_2\\ge \\cdots \\ge x_n\\ge 0\\}, \\end{aligned}$$\\end{document}which are called the Schläfli orthoschemes of types A and B, respectively. We describe the tangent cones at their j-faces and compute explicitly the sums of the conic intrinsic volumes of these tangent cones at all j-faces of K_n^A and K_n^B. This setting contains sums of external and internal angles of K_n^A and K_n^B as special cases. The sums are evaluated in terms of Stirling numbers of both kinds. We generalize these results to finite products of Schläfli orthoschemes of type A and B and, as a probabilistic consequence, derive formulas for the expected number of j-faces of the Minkowski sums of the convex hulls of a finite number of Gaussian random walks and random bridges. Furthermore, we evaluate the analogous angle sums for the tangent cones of Weyl chambers of types A and B and finite products thereof.

Reduction Formulas for Generalized Hypergeometric Series Associated with New Sequences and Applications

In this paper, by introducing two sequences of new numbers and their derivatives, which are closely related to the Stirling numbers of the first kind, and choosing to employ six known generalized Kummer’s summation formulas for 2F1(−1) and 2F1(1/2), we establish six classes of generalized summation formulas for p+2Fp+1 with arguments −1 and 1/2 for any positive integer p. Next, by differentiating both sides of six chosen formulas presented here with respect to a specific parameter, among numerous ones, we demonstrate six identities in connection with finite sums of 4F3(−1) and 4F3(1/2). Further, we choose to give simple particular identities of some formulas presented here. We conclude this paper by highlighting a potential use of the newly presented numbers and posing some problems.

A Combinatorial Approach to the Stirling Numbers of the First Kind with Higher Level

In this paper, we investigate a generalization of the classical Stirling numbers of the first kind by considering permutations over tuples with an extra condition on the minimal elements of the cycles. The main focus of this work is the analysis of combinatorial properties of these new objects. We give general combinatorial identities and some recurrence relations. We also show some connections with other sequences such as poly-Cauchy numbers with higher level and central factorial numbers. To obtain our results, we use pure combinatorial arguments and classical manipulations of formal power series.

A Family of the r-Associated Stirling Numbers of the Second Kind and Generalized Bernoulli Polynomials

In this article, we derive representation formulas for a class of r-associated Stirling numbers of the second kind and examine their connections with a class of generalized Bernoulli polynomials. Herein, we use the Blissard umbral approach and the familiar Bell polynomials. Links with available literature on this subject are also pointed out. The extension to the bivariate case is discussed.

Factorization of Hermite subdivision operators from polynomial over-reproduction

We study the case of Hermite subdivision operators satisfying a spectral condition of order greater than their size. We show that this can be characterized by operator factorizations involving Taylor operators and difference factorizations of a rank one vector scheme. Giving explicit expressions for the factorization operators, we put into evidence that the factorization only depends on the order of the spectral condition but not on the polynomials that define it. We further show that the derivation of these operators is based on an interplay between Stirling numbers and p-Cauchy numbers (or generalized Gregory coefficients).

Construction on the Degenerate Poly-Frobenius-Euler Polynomials of Complex Variable

In this paper, we introduce degenerate poly-Frobenius-Euler polynomials and derive some identities of these polynomials. We give some relationships between degenerate poly-Frobenius-Euler polynomials and degenerate Whitney numbers and Stirling numbers of the first kind. Moreover, we define degenerate poly-Frobenius-Euler polynomials of complex variables and then we derive several properties and relations.

New Families of Special Polynomial Identities Based upon Combinatorial Sums Related to p-Adic Integrals

The aim of this paper is to study and investigate generating-type functions, which have been recently constructed by the author, with the aid of the Euler’s identity, combinatorial sums, and p-adic integrals. Using these generating functions with their functional equation, we derive various interesting combinatorial sums and identities including new families of numbers and polynomials, the Bernoulli numbers, the Euler numbers, the Stirling numbers, the Daehee numbers, the Changhee numbers, and other numbers and polynomials. Moreover, we present some revealing remarks and comments on the results of this paper.

On New Formulas of Fibonacci and Lucas Numbers Involving Golden Ratio Associated with Atomic Structure in Chemistry

The main purpose of this paper is to give many new formulas involving the Fibonacci numbers, the golden ratio, the Lucas numbers, and other special numbers. By using generating functions for the special numbers with their functional equations method, we also give many new relations among the Fibonacci numbers, the Lucas numbers, the golden ratio, the Stirling numbers, and other special numbers. Moreover, some applications of the Fibonacci numbers and the golden ratio in chemistry are given.

Implementation of computation formulas for certain classes of Apostol-type polynomials and some properties associated with these polynomials

The main purpose of this paper is to present various identities and computation formulas for certain classes of Apostol-type numbers and polynomials. The results of this paper contain not only the $\lambda$-Apostol-Daehee numbers and polynomials, but also Simsek numbers and polynomials, the Stirling numbers of the first kind, the Daehee numbers, and the Chu-Vandermonde identity. Furthermore, we derive an infinite series representation for the $\lambda$-Apostol-Daehee polynomials. By using functional equations containing the generating functions for the Cauchy numbers and the Riemann integrals of the generating functions for the $\lambda$-Apostol-Daehee numbers and polynomials, we also derive some identities and formulas for these numbers and polynomials. Moreover, we give implementation of a computation formula for the $\lambda$-Apostol-Daehee polynomials in Mathematica by Wolfram language. By this implementation, we also present some plots of these polynomials in order to investigate their behaviour some randomly selected special cases of its parameters. Finally, we conclude the paper with some comments and observations on our results.

A unified family of multivariable Legendre poly-Genocchi polynomials

In this paper, we introduce a new class of Legendre poly-Genocchi polynomials and give some identities of these polynomials related to the Stirling numbers of the second kind. The concept of poly-Bernoulli numbers $B_{n}^{(k)}(a,b)$, poly-Bernoulli polynomials $B_{n}^{(k)}(x,a,b)$ of Jolany et al., Hermite-Bernoulli polynomials ${}_{H}B_{n}(x,y)$ of Dattoli et al., ${}_{H}B_{n}^{(\alpha)}(x,y)$ of Pathan et al. and ${}_{H}G_{n}^{(k)}(x,y)$ of Khan are generalized to the one $_{S}G_{n}^{(k)}(x,y,z)$. Some implicit summation formulae and general symmetry identities are derived by using different analytical means and applying generating function. These results extended some known summation and identities of Hermite poly-Genocchi numbers and polynomials.

Degenerate Derangement Polynomials and Numbers

In this paper, we consider a new type of degenerate derangement polynomial and number, which shall be called the degenerate derangement polynomials and numbers of the second kind. These concepts are motivated by Kim et al.’s work on degenerate derangement polynomials and numbers. We investigate some properties of these new degenerate derangement polynomials and numbers and explore their connections with the degenerate gamma distributions for the case λ∈(−1,0). In more detail, we derive their explicit expressions, recurrence relations, and some identities involving our degenerate derangement polynomials and numbers and other special polynomials and numbers, which include the fully degenerate Bell polynomials, the degenerate Fubini polynomials, and the degenerate Stirling numbers of the first and the second kinds. We also show that those polynomials and numbers are connected with the moments of some variants of the degenerate gamma distributions. Moreover, we compare the degenerate derangement polynomials and numbers of the second kind to those of Kim et al.

Sums of powers of integers and hyperharmonic numbers

In this paper, we obtain a new formula for the sums of k-th powers of the first n positive integers, Sk(n), that involves the hyperharmonic numbers and the Stirling numbers of the second kind. Then, using an explicit representation for the hyperharmonic numbers, we generalize this formula to the sums of powers of an arbitrary arithmetic progression. Furthermore, we express the Bernoulli polynomials in terms of hyperharmonic polynomials and Stirling numbers of the second kind. Finally, we extend the obtained formula for Sk(n) to negative values of n.