Stirling Numbers Of Second Kind Research Articles

Generating Function Of Extended Jacobi Polynomial Involving Stirling Number of Second Kind

In this paper we have obtained the generating function of extended Jacobi polynomial containing Stirling number of the second kind. Some particular cases are also obtained

On the generating functions and special functions associated with superoscillations

The aim of this paper is to study generating functions for the coefficients of the classical superoscillatory function associated with weak measurements. We also establish some new relations between the superoscillatory coefficients and many well-known families of special polynomials, numbers, and functions such as Bernstein basis functions, the Hermite polynomials, the Stirling numbers of second kind, and also the confluent hypergeometric functions. Moreover, by using generating functions, we are able to develop a recurrence relation and a derivative formula for the superoscillatory coefficients.

Relation Between Extended Stirling Numbers and q-B-splines

Stirling numbers of second kind S(n,k) denotes the number of ways partitioning a set of n elements into k nonempty sets. There are many types of stirling numbers which are studied up to now. In this study, we use extended stirling numbers of second kind which are defined for arbitrary reals. First, we define a relation between extended stirling numbers and q-B-splines by using the property that divided differences have a representation with q-B-splines. In addition, we derive identities on stirling numbers and q-integral of q-B-splines. Furthermore, we give q-generating functions of extended stirling numbers and define a q-difference equation for this function.

Generating functions for generalization Simsek numbers and their applications

Our purpose in this work is the complete the study of Simsek numbers. We give answer to some open problems concerning polynomial representations and associated generating function. At the end of the study we investigate a new generalization of these numbers and obtain useful identities which connect Simsek numbers and Stirling numbers of second kind.

On certain polynomial systems involving Stirling numbers of second kind

We solve a special type of linear systems with coefficients in multivariate polynomial rings. These systems arise in the computation of b-functions with respect to weights of certain hypergeometric ideals in the Weyl algebra.

Generalized Apostol Type Polynomials Based on Twin-Basic Numbers

In this work, we consider a class of new generating function for \((p,q)\)-analog of Apostol type polynomials of order \(\alpha\) including Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials of order \(\alpha\). By making use of their generating function, we derive some useful identities. We also introduce the generating functions of \((p,q)\)-analogues of the Stirling numbers of second kind of order \(\tau\) and the Bernstein polynomials by which we construct diverse correlations including aforementioned polynomials and the \((p,q)\)-gamma function.

Generalization of Stirling Number of the Second Kind and Combinatorial Identity

The Stirling numbers of second kind and related problems are widely used in combinatorial mathematics and number theory, and there are a lot of research results. This article discuss the function: ∑AC11 AC22 ···ACkk (C1+C2+···+Ck=N-K, Ci≥0), obtain its calculation formula and a series of conclusions, which generalize the results of existing literature, and further obtain the combinatorial identity: ∑(-1)K-i*C(K-1,K-i)C(A-1+i,N-1)=C(A,N-K).

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.

Generalized Positive Linear Operators Based on PED and IPED

The paper deals with generalized positive linear operators based on Polya–Eggenberger distribution (PED) as well as inverse Polya–Eggenberger distribution (IPED). Initially, we give the moments using Stirling numbers of second kind and then establish direct results for proposed operators.

Generalized Eulerian Numbers

We generalize the Eulerian numbers to sets of numbers Eμ(k,l), (μ=0,1,2,···) where the Eulerian numbers appear as the special case μ=1. This can be used for the evaluation of generalizations Eμ(k,Z) of the Geometric series G0(k;Z)=G1(0;Z) by splitting an essential part (1-Z)-(μK+1) where the numbers Eμ(k,l) are then the coefficients of the remainder polynomial. This can be extended for non-integer parameter k to the approximative evaluation of generalized Geometric series. The recurrence relations and for the Generalized Eulerian numbers E1(k,l) are derived. The Eulerian numbers are related to the Stirling numbers of second kind S(k,l) and we give proofs for the explicit relations of Eulerian to Stirling numbers of second kind in both directions. We discuss some ordering relations for differentiation and multiplication operators which play a role in our derivations and collect this in Appendices.

A NOTE ON q-ANALOGUE OF POLY-BERNOULLI NUMBERS AND POLYNOMIALS

In this paper, we define a q-analogue of the poly-Bernoulli numbers and polynomials which is generalization of the poly Bernoulli numbers and polynomials including q-polylogarithm function. We also give the relations between generalized poly-Bernoulli polynomials. We derive some relations that are connected with the Stirling numbers of second kind. By using special functions, we investigate some symmetric identities involving q-poly-Bernoulli polynomials.

A simple proof of convolution identities of Bernoulli numbers

T. Agoh and K. Dilcher proved convolution identities of Bernoulli numbers in 2007. Their proof was complicated calculations in more than 10 pages, which were based on the relation between the Stirling numbers of second kind and the Bernoulli numbers. In this short paper, we give a simple proof of it. Essentially, the proof is based on just one formula on a new kind of generating function.

About Extensions of Generalized Apostol-Type Polynomials

Under a slight modification on the parameters associated to the generalized Apostol-type polynomials and the use of the generating method, we obtain some new results concerning extensions of generalized Apostol-type polynomials. We state some algebraic and differential properties for a new class of extensions of generalized Apostol-type polynomials, as well as, some others identities which connect this polynomial class with the Stirling numbers of second kind, the Jacobi polynomials, the generalized Bernoulli polynomials, the Genocchi polynomials and the Apostol–Euler polynomials, respectively.

On a New Family of Generalized Stirling and Bell Numbers

A new family of generalized Stirling and Bell numbers is introduced by considering powers $(VU)^n$ of the noncommuting variables $U,V$ satisfying $UV=VU+hV^s$. The case $s=0$ (and $h=1$) corresponds to the conventional Stirling numbers of second kind and Bell numbers. For these generalized Stirling numbers, the recursion relation is given and explicit expressions are derived. Furthermore, they are shown to be connection coefficients and a combinatorial interpretation in terms of statistics is given. It is also shown that these Stirling numbers can be interpreted as $s$-rook numbers introduced by Goldman and Haglund. For the associated generalized Bell numbers, the recursion relation as well as a closed form for the exponential generating function is derived. Furthermore, an analogue of Dobinski's formula is given for these Bell numbers.

Generalized Stirling Functions of Second Kind and Representations of Fractional Order Differences via Derivatives

The paper is devoted to the study of generalized Stirling functions of second kind by using difference and differentiation operators of fractional order. The constructions under considerations give extensions of the classical Stirling numbers of second kind S(n, k) to functions S(n, β), whereby the second parameter k becomes any complex β, as well as to functions S(α, β), whereby also the first parameter n becomes any complex α. The main properties of these new functions, including recurrence relations are established. A chief application is a generalization of a basic formula of combinatorial and numerical analysis, namely expressing higher order differences in terms of derivatives, to the fractional instance: essentially a fractional order difference Δ β f(x) leads to S(n, β) in the above sum. Three concrete examples are presented.

The number of standard and of effective multiple alignments

We study the number of all possible alignments of N sequences, N ≥ 2, for two distinct alignment concepts proposed in the literature—standard alignments and effective alignments (consistent equivalence relations). Recursion formulae are developed to calculate these numbers. For standard alignments and for effective alignment of just two sequences, an explicit formula is also presented. The number of all effective alignments of a given site space is shown to be related to Stirling numbers of second kind.

Eulerian numbers with fractional order parameters

The aim of this paper is to generalize the well-known “Eulerian numbers”, defined by the recursion relationE(n, k) = (k + 1)E(n − 1, k) + (n − k)E(n − 1, k − 1), to the case thatn ∈ ℕ is replaced by α ∈ ℝ. It is shown that these “Eulerian functions”E(α, k), which can also be defined in terms of a generating function, can be represented as a certain sum, as a determinant, or as a fractional Weyl integral. TheE(α, k) satisfy recursion formulae, they are monotone ink and, as functions of α, are arbitrarily often differentiable. Further, connections with the fractional Stirling numbers of second kind, theS(α, k), α > 0, introduced by the authors (1989), are discussed. Finally, a certain counterpart of the famous Worpitzky formula is given; it is essentially an approximation ofx α in terms of a sum involving theE(α, k) and a hypergeometric function.