Extended Bernoulli and Stirling matrices and related combinatorial identities

Abstract

In this paper we establish plenty of number theoretic and combinatoric identities involving generalized Bernoulli and Stirling numbers of both kinds. These formulas are deduced from Pascal type matrix representations of Bernoulli and Stirling numbers. For this we define and factorize a modified Pascal matrix corresponding to Bernoulli and Stirling cases.

Polynomials with r-Lah coefficient and hyperharmonic numbers

“…Additionally, we show that Bernoulli polynomials can be described as a sum of the products of hyperharmonic numbers and r-Stirling numbers of the second kind (see Theorem 5). This formula leads a generalization of some identities, obtained by Pascal matrix method, presented in [14]. Finally, we examine a polynomial which we call harmonic geometric r-Lah polynomial (see Section 5).…”

Harmonic number identities via polynomials with r-Lah coefficients

“…Utilizing (15) and the generating function of r -Stirling numbers of the second kind [14, Theorem 16]…”

“…There exist many elegant identities involving hyperharmonic numbers. Some of these identities are exhibited by using combinatorial technique [4], Euler-Siedel matrix [24,39], derivative and difference operators [22,23,26,38], Pascal type matrix [15]. Whether the properties of harmonic numbers are provided by hyperharmonic numbers are actively studied.…”

A closed formula for the generating function of p-Bernoulli numbers

“…The Bernoulli numbers are connected with some well known special numbers [7,8,18,19,20,21]. Rahmani [23] also gave explicit formulas involving different kind of special numbers.…”