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The notion of generalized Bell numbers has appeared in several works but there is no systematic treatise on this topic. In this paper we fill this gap. We discuss the most important combinatorial, algebraic and analytic properties of these…
In the paper, the authors first inductively establish explicit formulas for derivatives of the arc sine function, then derive from these explicit formulas explicit expressions for a family of Bell polynomials related to the square function,…
Associated to each complex-valued random variable satisfying appropriate integrability conditions, we introduce a different generalization of the Stirling numbers of the second kind. Various equivalent definitions are provided. Attention,…
In the paper, the author finds an explicit formula for computing Bell numbers in terms of Lah numbers and Stirling numbers of the second kind.
In this paper we focus on r-geometric polynomials, r-exponential polynomials and their harmonic versions. It is shown that harmonic versions of these polynomials and their generalizations are useful to obtain closed forms of some series…
Several new estimates for the 2-adic valuations of Stirling numbers of the second kind are proved. These estimates, together with criteria for when they are sharp, lead to improvements in several known theorems and their proofs, as well as…
In this paper we study q-Bernoulli numbers and polynomials related to q-Stirling numbers. From thsese studying we investigate some interesting q-stirling numbers' identities related to q-Bernoulli numbers.
In this paper, we establish some expressions of Mneimneh-type binomial sums involving multiple harmonic-type sums in terms of finite sums of Stirling numbers, Bell numbers and some related variables. In particular, we present some new…
In the paper, the authors establish an explicit formula for computing Bernoulli polynomials at non-negative integer points in terms of $r$-Stirling numbers of the second kind.
In the paper, the authors provide four alternative proofs of an explicit formula for computing Bernoulli numbers in terms of Stirling numbers of the second kind.
In the paper, the authors find two closed forms involving the Stirling numbers of the second kind and in terms of a determinant of combinatorial numbers for the Bernoulli polynomials and numbers.
A combinatorial methods are used to investigate some properties of certain generalized Stirling numbers, including explicit formula and recurrence relations. Furthermore, an expression of these numbers with symmetric function is deduced.
In this study we introduce a second type of higher order generalised geometric polynomials. This we achieve by examining the generalised stirling numbers $S(n; k;\alpha;\beta;\gamma)$ [Hsu & Shiue,1998] for some negative arguments. We study…
We discuss a special function (polyexponential) that extends the natural exponential function and also the exponential integral. The basic properties of the polyexponential are listed and some applications are given. In particular, it is…
We prove characterizations of Appell polynomials by means of symmetric property. For these polynomials, we establish a simple linear expression in terms of Bernoulli and Euler polynomials. As applications, we give interesting examples. In…
Our paper deals about identities involving Bell polynomials. Some identities on Bell polynomials derived using generating function and successive derivatives of binomial type sequences. We give some relations between Bell polynomials and…
In the paper, the authors establish explicit formulas for the Dowling numbers and their generalizations in terms of generalizations of the Lah numbers and the Stirling numbers of the second kind. These results gen- eralize the Qi formula…
In the paper, the author finds an explicit formula for computing Bernoulli numbers of the second kind in terms of Stirling numbers of the first kind.
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…
In this paper we establish plenty of number theoretic and combinatoric identities involving generalized Bernoulli and Stirling numbers of both kinds. These formulas are deduced from Pascal type matrix representations of Bernoulli and…