Stirling Numbers Research Articles

A new explicit formula for Bernoulli numbers involving the Euler number

We derive a new explicit formula for Bernoulli numbers in terms of the Stirling numbers of the second kind and the Euler numbers. As a corollary of our result, we obtain an explicit formula for the even Euler numbers in terms of the Stirling numbers of the second kind.

New definitions of the generalized Stirling numbers

We introduce new combinatorial and algebraic definitions for the generalized Stirling numbers of Hsu and Shiue, and we use the interplay between these definitions to extend existing identities and discover new ones. We also explore how some of these results specialize to new identities of r-Whitney numbers.

Notes on q-Hermite based unified Apostol type polynomials

In this article, a new class of q-Hermite based unified Apostol type polynomials is introduced by means of generating function and series representation. Several important formulas and recurrence relations for these polynomials are derived via different generating methods. We also introduce q-analog of Stirling numbers of second kind of order v by which we construct a relation including aforementioned polynomials.

2-Adic valuations of Stirling numbers of the first kind

Let [Formula: see text] and [Formula: see text] be positive integers. We denote by [Formula: see text] the 2-adic valuation of [Formula: see text]. The Stirling numbers of the first kind, denoted by [Formula: see text], count the number of permutations of [Formula: see text] elements with [Formula: see text] disjoint cycles. In recent years, Lengyel, Komatsu and Young, Leonetti and Sanna, and Adelberg made some progress on the study of the [Formula: see text]-adic valuations of [Formula: see text]. In this paper, by introducing the concept of [Formula: see text]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 [Formula: see text]. We also prove that [Formula: see text] holds for all integers [Formula: see text] between 1 and [Formula: see text]. As a corollary, we show that [Formula: see text] if [Formula: see text] is odd and [Formula: see text]. This confirms partially a conjecture of Lengyel raised in 2015. Furthermore, we show that if [Formula: see text], then [Formula: see text] and [Formula: see text], where [Formula: see text] stands for the [Formula: see text]th elementary symmetric functions of [Formula: see text]. The latter one supports the conjecture of Leonetti and Sanna suggested in 2017.

An approach to negative hypergeometric distribution by generating function for special numbers and polynomials

Recently, the numbers $Y_{n}(\lambda )$ and the polynomials $Y_{n}(x,\lambda)$ have been introduced by the second author [22]. The purpose of this paper is to construct higher-order of these numbers and polynomials with their generating functions. By using these generating functions with their functional equations and derivative equations, we derive various identities and relations including two recurrence relations, Vandermonde type convolution formula, combinatorial sums, the Bernstein basis functions, and also some well known families of special numbers and their interpolation functions such as the Apostol--Bernoulli numbers, the Apostol--Euler numbers, the Stirling numbers of the first kind, and the zeta type function. Finally, by using Stirling's approximation for factorials, we investigate some approximation values of the special case of the numbers $Y_{n}\left( \lambda \right) $.

Explicit expressions and integral representations for the Stirling numbers. A probabilistic approach

We show that the Stirling numbers of the first and second kind can be represented in terms of moments of appropriate random variables. This allows us to obtain, in a unified way, a variety of old and new explicit expressions and integral representations of such numbers.

Two Parametric Kinds of Eulerian-Type Polynomials Associated with Euler’s Formula

The purpose of this article is to construct generating functions for new families of special polynomials including two parametric kinds of Eulerian-type polynomials. Some fundamental properties of these functions are given. By using these generating functions and the Euler’s formula, some identities and relations among trigonometric functions, two parametric kinds of Eulerian-type polynomials, Apostol-type polynomials, the Stirling numbers and Fubini-type polynomials are presented. Computational formulae for these polynomials are obtained. Applying a partial derivative operator to these generating functions, some derivative formulae and finite combinatorial sums involving the aforementioned polynomials and numbers are also obtained. In addition, some remarks and observations on these polynomials are given.

Local and global degree profiles of randomly grown self-similar hooking networks under uniform and preferential attachment

We investigate node degrees in a network grown from a seed by hooking self-similar components under two models of randomness: a uniform attachment model and a model based on preferential attachment. We study two degree profiles: a local profile tracking the evolution of the degree of a particular node over time, and a global profile concerned about counts of the number of nodes of a particular degree.For the local profile, under uniform attachment growth, we have the exact mean, variance and probability distribution in terms of standard combinatorial numbers like generalized harmonic numbers and Stirling numbers of the first kind. Asymptotically, we observe phases: The early nodes have an asymptotically normal distribution, intermediate nodes have a Poisson distribution and the late nodes have a degenerate distribution. In contrast, under preferential attachment, the moments of the degree of a node contain Stirling numbers of the second kind and (under appropriate scaling) the profile has a gamma-type limit law.As for the global profile, we use Pólya urns to derive strong laws. Four regimes arise according to the structure of the seed. Within these regimes, we identify a few degenerate cases. Barring these degenerate cases, we uncover an asymptotically normal joint multivariate distribution for nodes of very small degrees.

Peters type polynomials and numbers and their generating functions: Approach with p‐adic integral method

The aim of this paper is to introduce and investigate some of the primary generalizations and unifications of the Peters polynomials and numbers by means of convenient generating functions and p‐adic integrals method. Various fundamental properties of these polynomials and numbers involving some explicit series and integral representations in terms of the generalized Stirling numbers, generalized harmonic sums, and some well‐known special numbers and polynomials are presented. By using p‐adic integrals, we construct generating functions for Peters type polynomials and numbers (Apostol‐type Peters numbers and polynomials). By using these functions with their partial derivative eqautions and functional equations, we derive many properties, relations, explicit formulas, and identities including the Apostol‐Bernoulli polynomials, the Apostol‐Euler polynomials, the Boole polynomials, the Bernoulli polynomials, and numbers of the second kind, generalized harmonic sums. A brief revealing and historical information for the Peters type polynomials are given. Some of the formulas given in this article are given critiques and comments between previously well‐known formulas. Finally, two open problems for interpolation functions for Apostol‐type Peters numbers and polynomials are revealed.

On the performance of transmit antenna selection in multiuser asymmetric RF/FSO system under generalized order user scheduling

In this paper, the performance analysis of transmit antenna selection on a multiuser asymmetric RF/FSO system under generalized order user scheduling is investigated. A dual-hop system with one source, one relay and multiple FSO users at destination is considered without direct link between the source and destination. The RF link is subjected to Nakagami-m fading and FSO links follow Exponential Weibull (EW) distribution. Using the amplify-and-forward (AF) channel-state-information (CSI) based gain relay protocol, the equivalent cumulative distribution function (CDF) end-to-end SNR at the destination is derived and used to obtain the analytical expression for the system outage probability. By utilizing the derived statistics, the closed-form expression for the system average bit error rate (ABER) is then derived through the Gauss–Laguerre quadrature rule and second kind Stirling number. Moreover, the results demonstrate that the atmospheric turbulence significantly degrades the system performance and increase in the number of source transmit antenna offers the system better performance. Also, the system performance is deteriorated by increasing the orders N or decreasing the number of user K. The accuracy of the derived expressions is validated through Montel-Carlo Simulations.

Some Theorems on Tauber's Generalized Stirling, Lah and Bell Numbers

In this paper, some properties for Tauber's generalized Stirling and Lah numbers are obtained including other forms of recurrence relations, orthogonality and inverse relations, rational generating function and explicit formual in symmetric function form. Moreover, a new explicit formula is derived, which is analogous to the Qi formula.

Layer regrouping optimization for multilayer heterogeneous reservoirs at a high water cut stage

Due to the vertical heterogeneity of the multilayer reservoirs, which results in a great interlayer interference, the recovery of each layer is significantly different when all layers are perforated and produced simultaneously. The layer with poor reservoir properties has a smaller recovery than that with good reservoir properties. Layer regrouping is an effective method to reduce interlayer interference and improve the recovery of the layer with poor reservoir properties. However, not all layer regrouping programs are reasonable when considering the relationship of investment and income. We must find the optimal layer regrouping program. In this paper, we generate all possible layer regrouping programs by using the enumeration method with the second kind Stirling numbers. The optimal layer regrouping program is obtained with considering the effect of technique and economic factors. Based on the theory of Buckley-Leveret displacement, the cumulative oil production of each program is obtained. The more oil produced, the more favorable the program is. The feasibility of layer regrouping program is seriously depended on economic benefits, which can be calculated by net present value method. The feasibility of layer regrouping program under different drilling well number and crude oil price is analyzed. Finally, a case for layer regrouping optimization is introduced. From the case, we can conclude that when oil price is $70 per barrel and drilling well number is 2, the 237th program is the optimal layer regrouping program, whose net present value is 422.44 × 104 Chinese yuan.

Some Identities Involving Second Kind Stirling Numbers of Types $B$ and $D$

Using Reiner's definition of Stirling numbers of the second kind in types $B$ and $D$, we generalize two well-known identities concerning the classical Stirling numbers of the second kind. The first identity relates them with Eulerian numbers and the second identity interprets them as entries in a transition matrix between the elements of two standard bases of the polynomial ring $\mathbb{R}[x]$. Finally, we generalize these identities to the group of colored permutations $G_{m,n}$.

Degenerate central factorial numbers of the second kind

In this paper, we introduce the degenerate central factorial polynomials and numbers of the second kind which are degenerate versions of the central factorial polynomials and numbers of the second kind. We derive some properties and identities for those polynomials and numbers. We obtain, among other things, recursive formulas for the degenerate central factorial polynomials and numbers of the second kind. Recently, Dolgy and Kim (Proc Jangjeon Math Soc 21(2):309–317, 2018) obtained some explicit formulas of degenerate Stirling numbers associated with the degenerate special numbers and polynomials. This paper motivated our to do this research.

The r-Bell numbers and matrices containing non-central Stirling and Lah numbers

The r-Bell numbers and matrices containing non-central Stirling and Lah numbers

Truncated Fubini Polynomials

In this paper, we introduce the two-variable truncated Fubini polynomials and numbers and then investigate many relations and formulas for these polynomials and numbers, including summation formulas, recurrence relations, and the derivative property. We also give some formulas related to the truncated Stirling numbers of the second kind and Apostol-type Stirling numbers of the second kind. Moreover, we derive multifarious correlations associated with the truncated Euler polynomials and truncated Bernoulli polynomials.

Generating functions for finite sums involving higher powers of binomial coefficients: Analysis of hypergeometric functions including new families of polynomials and numbers

The aim of this paper is to construct and investigate some of the fundamental generalizations and unifications of new families of polynomials and numbers involving finite sums of higher powers of binomial coefficients and the Franel numbers by means of suitable generating functions and hypergeometric function. We derive several fundamental properties involving the generating functions, formulas, recurrence relations for these polynomials and numbers. These new families of polynomials and numbers are shown to generalize some known special polynomials such as the Legendre polynomials, the Michael Vowe polynomials, the Mirimanoff polynomials, Golombek type polynomials and also the Franel numbers. We give relations between the generalized Franel numbers, the Legendre polynomials, the Bernoulli numbers, the Stirling numbers, the Catalan numbers and other special numbers. Using the Riemann integral and p-adic integrals representations of these new polynomials, we derive combinatorial sums and identities related to sums of powers of binomial coefficients. Moreover, we introduce some revealing and historical remarks and observations on the finite sums of powers of binomial coefficients, special polynomials and numbers. Appropriate connections of identities, formulas, relations and results given in this paper with those in earlier and future studies are pointed out in detail. Special values of explicit formulas of our new numbers give solutions of the open problem 1 raised by Srivastava [58, p. 416, Open Problem 1]. Finally, we pose two open questions related to ordinary generating functions for these new numbers.

Combinatorial Identities with Generalized Higher-order Genocchi Sequences

In this paper, we make use of the probabilistic method to calculate the moment representation of generalized higher-order Genocchi polynomials. We obtain the moment expression of the generalized higher-order Genocchi numbers with a and b parameters. Some characteriza tions and identities of generalized higher-order Genocchi polynomials are given by the proof of the moment expression. As far as properties given by predecessors are concerned, we prove them by the probabilistic method. Finally, new identities of relationships involving generalized higher-order Genocchi numbers and harmonic numbers, derangement numbers, Fibonacci numbers, Bell numbers, Bernoulli numbers, Euler numbers, Cauchy numbers and Stirling numbers of the second kind are established.Â

A Geometric Interpretation of the Intertwining Number

We exhibit a connection between two statistics on set partitions, the intertwining number and the depth-index. In particular, results link the intertwining number to the algebraic geometry of Borel orbits. Furthermore, by studying the generating polynomials of our statistics, we determine the $q=-1$ specialization of a $q$-analogue of the Bell numbers. Finally, by using Renner's $H$-polynomial of an algebraic monoid, we introduce and study a $t$-analog of $q$-Stirling numbers.

Hurwitz-Lerch Type Multi-Poly-Cauchy Numbers

In this paper, we define Hurwitz–Lerch multi-poly-Cauchy numbers using the multiple polylogarithm factorial function. Furthermore, we establish properties of these types of numbers and obtain two different forms of the explicit formula using Stirling numbers of the first kind.