The r-central factorial numbers with even indices

Abstract

In this paper, we introduce the r-central factorial numbers with even indices of the first and second kind as extended versions of the central factorial numbers with even indices of both kinds. We obtain several fundamental properties and identities related to these numbers. The connections between the new numbers and the Stirling numbers are presented. In addition, we give the probability distribution of the unsigned r-central factorial numbers with even indices. Finally, we consider the r-central factorial matrices and study some of their properties.

Sums of Powers of Integers and Stirling Numbers

“…In what follows we denote these numbers by u(k, m), in accordance with the notation of Gelineau and Zeng [8]. The first property is the definition of u(k, m) through the generating function (see, e.g., [5,Equation (4.15)] and Equation (11) (with r = 0) of [15])…”

“…, where δ k,i denotes the Kronecker delta (see, e.g., [2, Proposition 2.2 (vii)] and Equation (13) (with r = 0) of [15]).…”

“…Remark 8. Knuth's formula (4) can be derived directly from (15) and the well-known recursive formula (see, e.g., [18, Proposition 1])…”

“…In the next proposition we show how the formulas (4) and (15) for S 2k (n) and S 2k+1 (n) can be quickly converted into the Faulhaber form by expressing the binomial coefficient n+m 2m in terms of the Legendre-Stirling numbers of the first kind. Proposition 11.…”

Laboratory Characterization of Asphalt Mixtures Containing Steel Slag

“…Recall that u(n, k) = t(2n, 2k) are called the central factorial numbers of the first kind with even indices, see, e.g., [14,23,31].…”