Terms Of Stirling Numbers Research Articles

An Application of Pascal Distribution Series on Certain Analytic Functions Associated with Stirling Numbers and Sălăgean Operator

In the present paper, we will observe that the Sălăgean differential operator can be written in terms of Stirling numbers. Furthermore, we find a necessary and sufficient condition and inclusion relation for Pascal distribution series to be in the class ℙ k λ , α of analytic functions with negative coefficients defined by the Sălăgean differential operator. Also, we consider an integral operator related to Pascal distribution series. Several corollaries and consequences of the main results are also considered.

Closed formulas and determinantal expressions for higher-order Bernoulli and Euler polynomials in terms of Stirling numbers

In this paper, applying the Fa\`a 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.

Explicit Expressions for Higher Order Binomial Convolutions of Numerical Sequences

We give explicit expressions for higher order binomial convolutions of sequences of numbers having a finite exponential generating function. Illustrations involving Cauchy, Bernoulli, and Apostol–Euler numbers are presented. In these cases, we obtain formulas easy to compute in terms of Stirling numbers.

Some notes on the [formula omitted]-Stirling numbers

The orthogonality of the (q,t)-version of the Stirling numbers has recently been proved by Cai and Readdy using a bijective argument. In this paper, we introduce new recurrences for the (q,t)-Stirling numbers and provide a (q,t)-analogue for sums of powers. Specializations of these results are given in terms of Stirling numbers or q-Stirling numbers.

Asymptotic expansions for certain mathematical constants and special functions

For fixed real b > 1 and ? > 0, let S[?]b (n) = ?n,k=1 bkk-?. Abel proved that S[?]b(n) ~ bn ??,k =0 ckn-(k+?)(n ? ?), and gave an explicit formula for determining the coefficients ck ? ck(b,?) in terms of Stirling numbers of the second kind. We here provide a recurrence relation for determining the coefficients ck, without Stirling numbers. We also consider asymptotic expansions concerning Somos' quadratic recurrence constant, Glaisher-Kinkelin constant, Choi-Srivastava constants, and the Barnes G-function.

On a generalization of a waiting time problem and some combinatorial identities

In this paper we give an extension of the results of the generalized waiting time problem given by El-Desouky and Hussen (1990). An urn contains m types of balls of unequal numbers, and balls are drawn with replacement until first duplication. In the case of finite memory of order k, let ni be the number of type i, i = 1, 2, …, m. The probability of success pi = ni/N, i = 1, 2, …, m, where ni is a positive integer and Let Ym,k be the number of drawings required until first duplication. We obtain some new expressions of the probability function, in terms of Stirling numbers, symmetric polynomials, and generalized harmonic numbers. Moreover, some special cases are investigated. Finally, some important new combinatorial identities are obtained.

On a generalization of a waiting time problem and some combinatorial identities

In this paper we give an extension of the results of the generalized waiting time problem given by El-Desouky and Hussen (1990). An urn contains m types of balls of unequal numbers, and balls are drawn with replacement until first duplication. In the case of finite memory of order k, let ni be the number of type i, i = 1, 2, …, m. The probability of success pi = ni/N, i = 1, 2, …, m, where ni is a positive integer and Let Ym,k be the number of drawings required until first duplication. We obtain some new expressions of the probability function, in terms of Stirling numbers, symmetric polynomials, and generalized harmonic numbers. Moreover, some special cases are investigated. Finally, some important new combinatorial identities are obtained.

An explicit formula for Bell numbers in terms of Stirling numbers and hypergeometric functions

In the paper, the author finds an explicit formula for computing Bell numbers in terms of Kummer confluent hypergeometric functions and Stirling numbers of the second kind.

Alternative proofs of a formula for Bernoulli numbers in terms of Stirling numbers

Alternative proofs of a formula for Bernoulli numbers in terms of Stirling numbers

Cumulants of multinomial and negative multinomial distributions

Abstract For the first time, explicit expressions are given for the cumulants of the multinomial and the negative multinomial distributions. These expressions are given in terms of Stirling numbers. Computational efficiency of the expressions is illustrated.

Enumerating set partitions according to the number of descents of size d or more

Let P(n, k) denote the set of partitions of {1, 2, ..., n} having exactly k blocks. In this paper, we find the generating function which counts the members of P(n, k) according to the number of descents of size d or more, where d ≥ 1 is fixed. An explicit expression in terms of Stirling numbers of the second kind may be given for the total number of such descents in all the members of P(n, k). We also compute the generating function for the statistics recording the number of ascents of size d or more and show that it has the same distribution on P(n, k) as the prior statistics for descents when d ≥ 2, by both algebraic and combinatorial arguments.

Codimension Growth of Strong Lie Nilpotent Associative Algebras

We study Lie nilpotent varieties of associative algebras. We explicitly compute the codimension growth for the variety of strong Lie nilpotent associative algebras. The codimension growth is polynomial and found in terms of Stirling numbers of the first kind. To achieve the result we take the free Lie algebra of countable rank L(X), consider its filtration by the lower central series and shift it. Next we apply generating functions of special type to the induced filtration of the universal enveloping algebra U(L(X)) = A(X).

Distributional analysis of a generalization of the Polya process

A nonhomogeneous birth process generalizing the Polya process is analyzed, and the distribution of the transition probabilities is shown to be the convolution of a negative binomial distribution and a compound Poisson distribution, whose secondary distribution is a mixture of zero-truncated geometric distributions. A simplified form of the secondary distribution is obtained when the transition intensities have a particular structure, and may sometimes be expressed in terms of Stirling numbers and special functions such as the incomplete gamma function, the incomplete beta function, and the exponential integral. Conditions under which the compound Poisson form of the marginal distributions may be improved to a geometric mixture are also given.

Exponential Polynomials, Stirling Numbers, and Evaluation of Some Gamma Integrals

This article is a short elementary review of the exponential polynomials (also called single-variable Bell polynomials) from the point of view of analysis. Some new properties are included, and several analysis-related applications are mentioned. At the end of the paper one application is described in details—certain Fourier integrals involving and are evaluated in terms of Stirling numbers.

Arithmetical properties of a sequence arising from an arctangent sum

The sequence { x n } defined by x n = ( n + x n − 1 ) / ( 1 − n x n − 1 ) , with x 1 = 1 , appeared in the context of some arctangent sums. We establish the fact that x n ≠ 0 for n ⩾ 4 and conjecture that x n is not an integer for n ⩾ 5 . This conjecture is given a combinatorial interpretation in terms of Stirling numbers via the elementary symmetric functions. The problem features linkage with a well-known conjecture on the existence of infinitely many primes of the form n 2 + 1 , as well as our conjecture that ( 1 + 1 2 ) ( 1 + 2 2 ) ⋯ ( 1 + n 2 ) is not a square for n > 3 . We present an algorithm that verifies the latter for n ⩽ 10 3200 .

On the Representation of Birth-Death Processes with Polynomial Transition Rates

Birth-death processes with polynomial transition rates are considered. It is shown that a deterministic population model with a polynomial growth rate corresponds to the expected population size of a birth-death process with polynomial transitions. A representation in terms of Stirling numbers, for the partial differential equation for the probability generating function of a birth-death process with polynomial transition rates is derived.

On k-th record times, record values and their moments

We give new formula for moments of k-th record values in terms of Stirling numbers of the first kind. In particular, the formulae allow to derive the explicit formulae for moments of k-th lower record values from exponential distribution which have not been known yet. Moreover, some interesting identities involving harmonic numbers are also obtained as corollaries to presented results.

Ordering of boson operator functions by the Hausdorff similarity transform

A generating function technique making use of the Hausdorff similarity transform is used to give arbitrary products of bosons in normal or antinormal ordered form. Operator commutator formulas and ordering in terms of Stirling numbers are also discussed.

Search trees and Stirling numbers

Search trees, specially binary search trees, are very important data structures thatcontributed immensely to improved performance of different search algorithms. In this paper, we express certain parameters of search trees in terms of Stirling numbers. We also introduce two new inversion formulas relating Stirling numbers of the first and second kinds.

Asymptotic approximation with Kantorovich polynomial

We present the complete asymptotic expansion for the Kantorovich polynomials Kn as n→∞. The result is in a form convenient for applications. All coefficients of n−k (k=1,2,...) are calculated explicitly in terms of Stirling numbers of the first and second kind.