poly-Bernoulli Numbers Research Articles

Bijective enumerations of Γ-free 0–1 matrices

We construct a new bijection between the set of n×k 0–1 matrices with no three 1's forming a Γ configuration and the set of (n,k)-Callan sequences, a simple structure counted by poly-Bernoulli numbers. We give two applications of this result: We derive the generating function of Γ-free matrices, and we give a new bijective proof for an elegant result of Aval et al. that states that the number of complete non-ambiguous forests with n leaves is equal to the number of pairs of permutations of {1,…,n} with no common rise.

On Arakawa–Kaneko zeta-functions associated with $GL_2(\mathbb{C})$ and their functional relations

We construct a certain class of Arakawa–Kaneko zeta-functions associated with $GL_2(\mathbb{C})$, which includes the ordinary Arakawa–Kaneko zeta-function. We also define poly-Bernoulli polynomials associated with $GL_2(\mathbb{C})$ which appear in their special values of these zeta-functions. We prove some functional relations for these zeta-functions, which are regarded as interpolation formulas of various relations among poly-Bernoulli numbers. Considering their special values, we prove difference relations and duality relations for poly-Bernoulli polynomials associated with $GL_2(\mathbb{C})$.

Restricted lonesum matrices

Lonesum matrices are matrices that are uniquely reconstructible from their row and column sum vectors. These matrices are enumerated by the poly-Bernoulli numbers; a sequence related to the multiple zeta values with a rich literature in number theory. Lonesum matrices are in bijection with many other combinatorial objects: several permutation classes, matrix classes, acyclic orientations in complete bipartite graphs etc. Motivated of these facts, we study in this paper lonesum matrices with restriction on the number of columns and rows of the same type. Keywords: enumeration, poly-Bernoulli numbers, restricted Stirling numbers, lonesum matrices MSC: 05A05, 05A15, 05A18, 05A19

On a duality formula for certain sums of values of poly-Bernoulli polynomials and its application

We prove a duality formula for certain sums of values of poly-Bernoulli polynomials which generalizes dualities for poly-Bernoulli numbers. We first compute two types of generating functions for these sums, from which the duality formula is apparent. Secondly we give an analytic proof of the duality from the viewpoint of our previous study of zeta-functions of Arakawa-Kaneko type. As an application, we give a formula that relates poly-Bernoulli numbers to the Genocchi numbers.

Generalized poly-Cauchy and poly-Bernoulli numbers by using incomplete $${\varvec{r}}$$ r -Stirling numbers

In this paper we introduce restricted r-Stirling numbers of the first kind. Together with restricted r-Stirling numbers of the second kind and the associated r-Stirling numbers of both kinds, by giving more arithmetical and combinatorial properties, we introduce a new generalization of incomplete poly-Cauchy numbers of both kinds and incomplete poly-Bernoulli numbers.

A NOTE ON q-ANALOGUE OF POLY-BERNOULLI NUMBERS AND POLYNOMIALS

In this paper, we define a q-analogue of the poly-Bernoulli numbers and polynomials which is generalization of the poly Bernoulli numbers and polynomials including q-polylogarithm function. We also give the relations between generalized poly-Bernoulli polynomials. We derive some relations that are connected with the Stirling numbers of second kind. By using special functions, we investigate some symmetric identities involving q-poly-Bernoulli polynomials.

Multi-poly-Bernoulli numbers and polynomials with a q parameter

We introduce themulti-poly-Bernoulli numbers and polynomialswith a q parameter, which are generalizations of the poly-Bernoulli numbers and polynomials with a q parameter, respectively.We give several combinatorial identities and properties of these new numbers and polynomials.

Some applications of degenerate poly-Bernoulli numbers and polynomials

Abstract In this paper, we consider degenerate poly-Bernoulli numbers and polynomials associated with a polylogarithmic function and a p-adic invariant integral on ℤ p {\mathbb{Z}_{p}} . By using umbral calculus, we derive some identities of those numbers and polynomials.

MULTI-POLY-BERNOULLI NUMBERS AND RELATED ZETA FUNCTIONS

We construct and study a certain zeta function which interpolates multi-poly-Bernoulli numbers at nonpositive integers and whose values at positive integers are linear combinations of multiple zeta values. This function can be regarded as the one to be paired up with the $\unicode[STIX]{x1D709}$-function defined by Arakawa and Kaneko. We show that both are closely related to the multiple zeta functions. Further we define multi-indexed poly-Bernoulli numbers, and generalize the duality formulas for poly-Bernoulli numbers by introducing more general zeta functions.

Incomplete Poly-Bernoulli Numbers and Incomplete Poly-Cauchy Numbers Associated to the q-Hurwitz–Lerch Zeta Function

In this paper we introduce a q-analogue of the incomplete poly-Bernoulli numbers and incomplete poly-Cauchy numbers by using the q-Hurwitz–Lerch zeta Function. Then we study several combinatorial properties of these new sequences. Moreover, we give some relations between the q-Hurwitz type incomplete poly-Bernoulli numbers, the q-Hurwitz type incomplete poly-Cauchy numbers and the incomplete Stirling numbers of both kinds.

Incomplete Multi-Poly-Bernoulli Numbers and Multiple Zeta Values

Using the restricted and associated Stirling numbers of the second kind, we define the incomplete multi-poly-Bernoulli numbers which generalize poly-Bernoulli numbers. We study their analytic and combinatorial properties. As an application, we present a new infinite series representation of the multiple zeta values via the Lambert W-function. We also give similar results for incomplete Bernoulli numbers of Hurwitz type and incomplete multi-poly-Bernoulli-star numbers.

Generalized incomplete poly-Bernoulli and poly-Cauchy numbers

By using the restricted and associated Stirling numbers of the first kind and by generalizing the (unsigned) Stirling numbers of the first kind, we define the generalized incomplete poly-Cauchy numbers by combining the generalized and the incomplete poly-Cauchy numbers, and study their arithmetical and combinatorial properties. We also study the corresponding generalized incomplete poly-Bernoulli numbers.

ON POLY-EULER NUMBERS

Poly-Euler numbers are introduced as a generalization of the Euler numbers in a manner similar to the introduction of the poly-Bernoulli numbers. In this paper, some number-theoretic properties of poly-Euler numbers, for example, explicit formulas, a Clausen–von Staudt type formula, congruence relations and duality formulas, are given together with their combinatorial properties.

Incomplete poly-Bernoulli numbers associated with incomplete Stirling numbers

By using the associated and restricted Stirling numbers of the second kind, we give some generalizations of the poly-Bernoulli numbers. We also study their arithmetical and combinatorial properties. As an application, at the end of the paper we present a new infinite series representation of the Riemann zeta function via the Lambert $W$.

A Note on Hermite poly-Bernoulli Numbers and Polynomials of the Second Kind

In the paper, we introduce a new concept of poly-Bernoulli numbers and polynomials of the second kind which is called Hermite poly-Bernoulli numbers and polynomials of the second kind. We also investigate and analyse its applications in number theory, combinatorics and other fields of mathematics. The results derived here are a generalization of some known summation formulae earlier studied by Jolany et al. [17,18], Dattoli et al [14] and Pathan et al [29,30].

Fully degenerate poly-Bernoulli numbers and polynomials

Abstract In this paper, we introduce the new fully degenerate poly-Bernoulli numbers and polynomials and inverstigate some properties of these polynomials and numbers. From our properties, we derive some identities for the fully degenerate poly-Bernoulli numbers and polynomials.

Combinatorics of poly-Bernoulli numbers

The Bn(k) poly-Bernoulli numbers — a natural generalization of classical Bernoulli numbers (Bn = Bn(1)) — were introduced by Kaneko in 1997. When the parameter k is negative then Bn(k) is a nonnegative number. Brewbaker was the first to give combinatorial interpretation of these numbers. He proved that Bn(−k) counts the so called lonesum 0–1 matrices of size n × k. Several other interpretations were pointed out. We survey these and give new ones. Our new interpretation, for example, gives a transparent, combinatorial explanation of Kaneko’s recursive formula for poly-Bernoulli numbers.

Formula omitted]-poly-Bernoulli numbers and [formula omitted]-poly-Cauchy numbers with a parameter by Jackson’s integrals

We define q-poly-Bernoulli polynomials Bn,ρ,q(k)(z) with a parameter ρ, q-poly-Cauchy polynomials of the first kind cn,ρ,q(k)(z) and of the second kind ĉn,ρ,q(k)(z) with a parameter ρ by Jackson’s integrals, which generalize the previously known numbers and polynomials, including poly-Bernoulli numbers Bn(k) and the poly-Cauchy numbers of the first kind cn(k) and of the second kind ĉn(k). We investigate their properties connected with usual Stirling numbers and weighted Stirling numbers. We also give the relations between generalized poly-Bernoulli polynomials and two kinds of generalized poly-Cauchy polynomials.

Poly-Bernoulli numbers and polynomials with a q parameter

We define poly-Bernoulli numbers and polynomials with a q parameter. These numbers and polynomials show a correspondence to poly-Cauchy numbers and polynomials with a q parameter recently defined by the second author. We study some arithmetical and number theoretical properties of these numbers and polynomials as well.

Explicit formula for generalization of poly-Bernoulli numbers and polynomials with a,b,c parameters

In this paper we investigate special generalized Bernoulli polynomials with a,b,c parameters that generalize classical Bernoulli numbers and polynomials. The present paper deals with some recurrence formulae for the generalization of poly-Bernoulli numbers and polynomials with a,b,c parameters. Poly-Bernoulli numbers satisfy certain recurrence relationships which are used in many computations involving poly-Bernoulli numbers. Obtaining a closed formula for generalization of poly-Bernoulli numbers with a,b,c paramerers therefore seems to be a natural and important problem. By using the generalization of poly-Bernoulli polynomials with a,b,c parameters of negative index we define symmetrized generalization of poly-Bernoulli polynomials with a,b parameters of two variables and we prove duality property for them. Also by stirling numbers of the second kind we will find a closed formula for them. Furthermore we generalize the Arakawa-Kaneko Zeta functions and by using the Laplace-Mellin integral, we define generalization of Arakawa-Kaneko Zeta functions with a,b parameters and we obtain an interpolation formula for the generalization of poly-Bernoulli numbers and polynomials with a,b parameters. Furthermore we present a link between this type of Zeta functions and Dirichlet series. By our interpolation formula, we will interpolate the generalization of Arakawa-Kaneko Zeta functions with a,b parameters.