Sums of products of Cauchy numbers

Abstract

a b s t r a c tIn this paper, we consider a kind of sums involving Cauchy numbers, which have not been studied in the literature. By means of the method of coefficients, we give some properties of the sums. We further derive some recurrence relations and establish a series of identities involving the sums, Stirling numbers, generalized Bernoulli numbers, generalized Euler numbers, Lah numbers, and harmonic numbers. In particular, we generalize some relations between two kinds of Cauchy numbers and some identities for Cauchy numbers and Stirling numbers.

Explicit Expressions for Higher Order Binomial Convolutions of Numerical Sequences

“…thus showing (11). Finally, (12) is an immediate consequence of (7), (9), and (11). ⊓ ⊔ Lemma 2 tells us that in order to compute the right-hand side in (12), we only need to look at the nth coefficient in the expansion of G (µ) (u (1) × · · · × u (m) , z).…”

Riordan arrays and harmonic number identities

“…By the method of coefficients, particularly, the Riordan array method, many properties of the Cauchy numbers, the Cauchy polynomials and the sums of products of Cauchy numbers are established in [21,24,26], including some identities which relate the Cauchy numbers (polynomials) to the Stirling, Bernoulli and harmonic numbers. Following the notation of [21], we define (1 + t) log(1 + t) , (5.12) where C k andĈ k are the Cauchy numbers of the first and second kinds.…”

Shifted poly-Cauchy numbers

“…are sometimes called the Bernoulli numbers of the second kind (see, e.g., [1,17]). Such numbers have been studied by several authors (see [4,14,15,16,18]) because they are related to various special combinatorial numbers, including Stirling numbers of both kinds, Bernoulli numbers, and harmonic numbers. The poly-Cauchy numbers c …”