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In this paper, we introduce generalizations of rising factorials and falling factorials, respectively, and study their relations with the well-known Stirling numbers, Lah numbers, and so on. The first stage is to define poly-falling factorial sequences in terms of the polyexponential functions, reducing them to falling factorials if k=1k=1, necessitating a demonstration of the relations: between poly-falling factorial sequences and the Stirling numbers of the first and second kind, respectively; between poly-falling factorial sequences and the poly-Bell polynomials; between poly-falling factorial sequences and the poly-Bernoulli numbers; between poly-falling factorial sequences and poly-Genocchi numbers; and recurrence formula of these sequences. The later part of the paper deals with poly-rising factorial sequences in terms of the polyexponential functions, reducing them to rising factorial if k=1k=1. We study some relations: between poly-falling factorial sequences and poly-rising factorial sequences; between poly-rising factorial sequences and the Stirling numbers of the first kind and the power of xx; and between poly-rising factorial sequences and Lah numbers and the poly-falling factorial sequences. We also derive recurrence formula of these sequences and reciprocal formula of the poly-falling factorial sequences.

Abstract In this paper, we introduce poly-central factorial sequences and poly-central Bell polynomials arising from the polyexponential functions, reducing them to central factorials and central Bell polynomials of the second kind respectively when k = 1 $k = 1$ . We also show some relations: between poly-central factorial sequences and power of x; between poly-central Bell polynomials and power of x; between poly-central Bell polynomials and the poly-Bell polynomials; between poly-central Bell polynomials and higher order type 2 Bernoulli polynomials of second kind; recurrence formula of poly-central Bell polynomials.

In this paper, we introduce poly-central factorial sequences and poly-central Bell polynomials arising from the polyexponential functions, reducing them to central factorials and central Bell polynomials of the second kind respectively when k = 1 . We also show some relations: between poly-central factorial sequences and power of x; between poly-central Bell polynomials and power of x; between poly-central Bell polynomials and the poly-Bell polynomials; between poly-central Bell polynomials and higher order type 2 Bernoulli polynomials of second kind; recurrence formula of poly-central Bell polynomials. [ABSTRACT FROM AUTHOR]

In this paper, using geometric polynomials, we obtain a generating function of p-Bernoulli numbers in terms of harmonic numbers. As consequences of this generating function, we derive closed formulas for the finite summation of Bernoulli and harmonic numbers involving Stirling numbers of the second kind. We also give a relationship between the p-Bernoulli numbers and the generalized Bernoulli polynomials. [ABSTRACT FROM AUTHOR]

In this paper, we present a systematic and unified investigation for the Apostol-Bernoulli polynomials, the Apostol-Euler polynomials and the Apostol-Genocchi polynomials. By applying the generating-function methods and summation-transform techniques, we establish some higher-order convolutions for the Apostol-Bernoulli polynomials, the Apostol-Euler polynomials and the Apostol-Genocchi polynomials. Some results presented here are the corresponding extensions of several known formulas.

Abstract: The aim of this paper is to introduce and investigate several new identities related to a unification and generalization of the three families of generalized Apostol type polynomials such as the Apostol–Bernoulli polynomials, the Apostol–Euler polynomials and the Apostol–Genocchi polynomials. The results presented here are based upon the theory of the Umbral Calculus and the Umbral Algebra. We also introduce some operators. By using a unified generating function for these Apostol type polynomials, which was constructed recently by Özden et al. (2010) [42], we derive many new properties of these polynomials. Moreover, we give relations between these polynomials and the Stirling numbers of the first and second kind. [Copyright &y& Elsevier]

In a manner analogous to the definition of Stirling numbers of the first and second kind we define a set of numbers for restricted random walks of finite memory. These two sequences of counting numbers should be useful in the exact enumeration of properties of simple and restricted random walks. [ABSTRACT FROM AUTHOR]

The concern of this paper is a beta type operator \(L_n\) recently introduced by D.D. Stancu. We present the complete asymptotic expansion for \(L_n\) as \(n\) tends to infinity. All coefficients of \(n^{-k} \ (k=1,2, \ldots)\) are calculated explicitly in terms of Stirling numbers of the first and second kind. Moreover, we give an asymptotic expansion for \(L_n\) into a series of reciprocal factorials.

In this paper, we present a systematic and unified investigation for the Apostol-Bernoulli polynomials, the Apostol-Euler polynomials and the Apostol-Genocchi polynomials. By applying the generating-function methods and summation-transform techniques, we establish some higher-order convolutions for the Apostol-Bernoulli polynomials, the Apostol-Euler polynomials and the Apostol-Genocchi polynomials. Some results presented here are the corresponding extensions of several known formulas. [ABSTRACT FROM AUTHOR]