Identities on products of Genocchi numbers

Abstract

In this paper, we discuss the properties of a class of Genocchi numbers by generating functions and Riordan arrays, we establish some identities involving Genocchi numbers, the Stirling numbers, the generalized Stirling numbers, the higher order Bernoulli numbers and Cauchy numbers. Further, we get asymptotic value of some sums relating the Genocchi numbers.

Explicit expressions for a certain class of Appell polynomials. A probabilistic approach

“…Different generalizations of these numbers have been considered in the literature (see, for instance, Hsu and Shiue [12], Mező [13], Luo and Srivastava [9] and Wuyungaowa [10]). Here, we consider the following probabilistic generalization…”

“…which, thanks to (6), implies that B(t, m; x) ∈ E t (Y ) with associated random variable Y = β m . By (10) and (23), the jth moment of the random variable Y = β m is given by…”

A probabilistic generalization of the Stirling numbers of the second kind

“…We keep here the same notations used in Sections 1 and 2, particularly, the random variables Y , (Y j ) j≥1 and (S k ) k≥0 given in ( 6) and ( 7), and the coefficients c n,N (k) defined in (14). Example 4.1.…”

“…Motivated by various specific problems, different generalizations of the Stirling numbers S(n, m) have been considered in the literature (see, for instance, Comtet [9], Hsu and Shine [10], Mező [11], Luo and Srivastava [12], Mihoubi and Maamra [13], Wuyungaowa [14], Cakić et al [15], Mihoubi and Tiachachat [16], and El-Desouky et al [17], among many others). As far as the applications of the polynomials S Y (n, m; x) are concerned, attention is focused in showing the following extension of formulas (3) and ( 4)…”