Generalizations of poly-Bernoulli and poly-Cauchy numbers

Abstract

In this paper we consider some generalizations of poly-Bernoulli and poly-Cauchy numbers. The first is by means of the Hurwitz-Lerch zeta function. The second generalization is via weighted Stirling numbers. The third one is given with the help of degenerate Stirling numbers. All these generalizations lead to symmetries between various types of Stirling numbers, and enable us to investigate and expand algebraic properties of poly-Bernoulli and poly-Cauchy numbers. We also combine these generalizations and derive numerous combinatorial and arithmetical identities.

Hurwitz-Lerch Type Multi-Poly-Cauchy Numbers

“…Parallel to the results of Cencki and Young [7], relations between Hurwitz-Lerch-type multi-poly-Cauchy numbers and Hurwitz-Lerch-type multi-poly-Bernoulli numbers can be shown using the orthogonality and inverse relations for Stirling numbers [12]. By the orthogonality relations:…”

“…To obtain a kind of generalization of the results in [7] (Theorem 2.7), we introduce modified Hurwitz-Lerch-type multi-poly-Bernoulli numbers, denoted byB…”

“…Certain generalizations of poly-Cauchy numbers of the first and second kind were introduced by Cenkci and Young [7]. This generalization was motivated by the concept of the Hurwitz-Lerch factorial zeta function defined by:…”

Multi Poly-Bernoulli and Multi Poly-Euler Polynomials

“…The numbers B (k) n,a are called the Hurwitz type poly Bernoulli numbers. These numbers have been shown Theorem 2.1 of [8] to have explicit formula…”

“…Recently, these numbers have been interpreted as the number of binary lonesum matrices of size n × k where the binary lonesum matrix is a binary matrix which can be reconstructed from its row and column sums [5]. It is worth-mentioning that the above generalization of Kaneko has been generalized further by Cenkci and Young [8] using the concept of Hurwitz-Lerch zeta function Φ(z, s, a) as follows…”