Compositional Bernoulli numbers

Multiparameter poly-Cauchy and poly-Bernoulli numbers and polynomials

Recently, Komastu introduced the concept of poly-Cauchy numbers and polynomials which generalize Cauchy numbers and polynomials. In this paper, we introduce new generaliza- tion of poly-Cauchy and poly-Bernoulli numbers and polynomials. Also, we introduce new generalizations of Cauchy numbers and polynomials. Moreover, we derive some identities involving the new numbers and polynomials and some types of Stirling numbers. These gives generalization of some relations poly-Cauchy and poly-Bernoulli numbers and poly- nomials. Furthermore, we obtain some relations between the multiparameter poly-Ca…

On $q$-poly-Bernoulli numbers arising from combinatorial interpretations

In this paper we present several natural $q$-analogues of the poly-Bernoulli numbers arising in combinatorial contexts. We also recall some relating analytical results and ask for combinatorial interpretations.

An explicit Formula for Bernoulli Polynomials with a $q$ Parameter in Terms of $r$-Whitney Numbers

We define the Bernoulli polynomials with a $q$ parameter in terms of $r$-Whitney numbers of the second kind. Some algebraic properties and combinatorial identities of these polynomials are given. Also, we obtain several relations between the Cauchy and Bernoulli polynomials with a $q$ parameter in terms of $r$-Whitney numbers of both kinds.

Links Between Sums Over Paths in Bernoulli's Triangles and the Fibonacci Numbers

We investigate paths in Bernoulli's triangles, and derive several relations linking the partial sums of binomial coefficients to the Fibonacci numbers.

New Results and Matrix Representation for Daehee and Bernoulli Numbers and Polynomials

In this paper, we derive new matrix representation for Daehee numbers and polynomials, the lambda-Daehee numbers and polynomials and the twisted Daehee numbers and polynomials. This helps us to obtain simple and short proofs of many previous results on Daehee numbers and polynomials. Moreover, we obtained some new results for Daehee and Bernoulli numbers and polynomials.

Combinatorics of poly-Bernoulli numbers

The ${\mathbb B}_n^{(k)}$ poly-Bernoulli numbers --- a natural generalization of classical Bernoulli numbers ($B_n={\mathbb B}_n^{(1)}$) --- were introduced by Kaneko in 1997. When the parameter $k$ is negative then ${\mathbb B}_n^{(k)}$ is a nonnegative number. Brewbaker was the first to give combinatorial interpretation of these numbers. He proved that ${\mathbb B}_n^{(-k)}$ counts the so called lonesum $0\text{-}1$ matrices of size $n\times k$. Several other interpretations were pointed out. We survey these and give new ones. Our new interpretation, for example, gives a transparent, combina…

Cubic Harmonics and Bernoulli Numbers

The functions satisfying the mean value property for an n-dimensional cube are determined explicitly. This problem is related to invariant theory for a finite reflection group, especially to a system of invariant differential equations. Solving this problem is reduced to showing that a certain set of invariant polynomials forms an invariant basis. After establishing a certain summation formula over Young diagrams, the latter problem is settled by considering a recursion formula involving Bernoulli numbers. Keywords: polyhedral harmonics; cube; reflection groups; invariant theory; invariant dif…

Combinatorial properties of poly-Bernoulli relatives

In this note we augment the poly-Bernoulli family with two new combinatorial objects. We derive formulas for the relatives of the poly-Bernoulli numbers using the appropriate variations of combinatorial interpretations. Our goal is to show connections between the different areas where poly-Bernoulli numbers and their relatives appear and give examples how the combinatorial methods can be used for deriving formulas between integer arrays.

Poly-Bernoulli numbers and lonesum matrices

A lonesum matrix is a matrix that can be uniquely reconstructed from its row and column sums. Kaneko defined the poly-Bernoulli numbers $B_m^{(n)}$ by a generating function, and Brewbaker computed the number of binary lonesum $m\times n$-matrices and showed that this number coincides with the poly-Bernoulli number $B_m^{(-n)}$. We compute the number of $q$-ary lonesum $m\times n$-matrices, and then provide generalized Kaneko's formulas by using the generating function for the number of $q$-ary lonesum $m\times n$-matrices. In addition, we define two types of $q$-ary lonesum matrices that are c…

New Results on Higher-Order Daehee and Bernoulli Numbers and Polynomials

We derive new matrix representation for higher order Daehee numbers and polynomials, the higher order lambda-Daehee numbers and polynomials and the twisted lambda-Daehee numbers and polynomials of order k. This helps us to obtain simple and short proofs of many previous results on higher order Daehee numbers and polynomials. Moreover, we obtained recurrence relation, explicit formulas and some new results for these numbers and polynomials. Furthermore, we investigated the relation between these numbers and polynomials and Stirling numbers, Norlund and Bernoulli numbers of higher order. The res…

Additional Fibonacci-Bernoulli relations

We continue our study on relationships between Fibonacci (Lucas) numbers and Bernoulli numbers and polynomials. The derivations of our results are based on functional equations for the respective generating functions, which in our case are combinations of hyperbolic functions. Special cases and some corollaries will highlight interesting aspects of our findings.

Symmetrized poly-Bernoulli numbers and combinatorics

Poly-Bernoulli numbers are one of generalizations of the classical Bernoulli numbers. Since a negative index poly-Bernoulli number is an integer, it is an interesting problem to study this number from combinatorial viewpoint. In this short article, we give a new combinatorial relation between symmetrized poly-Bernoulli numbers and the Dumont-Foata polynomials.

伯努利数的新表达式及其应用

A New Expression for the Bernoulli Numbers and its Applications

本文提出伯努利数的新表达式，并通过应用该表达式重新证明了线性递推关系，同时获得了新的递推关系及同余关系。

Stirling numbers with level $2$ and poly-Bernoulli numbers with level $2$

In this paper, we introduce poly-Bernoulli numbers with level $2$, related to the Stirling numbers of the second kind with level $2$, and study several properties of poly-Bernoulli numbers with level $2$ from their expressions, relations, and congruences. Poly-Bernoulli numbers with level $2$ have strong connections with poly-Cauchy numbers with level $2$. In a special case, we can determine the denominators of Bernoulli numbers with level $2$ by showing a von Staudt-Clausen like theorem.

On the combinatorics of symmetrized poly-Bernoulli numbers

In this paper we introduce three combinatorial models for symmetrized poly-Bernoulli numbers. Based on our models we derive generalizations of some identities for poly-Bernoulli numbers. Finally, we set open questions and directions of further studies.

涉及两类Bernoulli数和Stirling数的新公式

New formulas involving Bernoulli and Stirling numbers of both kinds

本文提出了几个涉及两类Bernoulli数和Stirling数的新公式。

A closed formula for the generating function of $p$-Bernoulli numbers: An elementary proof

For a two parameter family of Bernoulli numbers $B_{n, p}$ the exponential generating function is derived by elementary methods.

A congruence concerning a convolution involving weighted Bernoulli numbers

Given a prime $p\geq 5$, we reduce modulo p a convolution of order p-1 of powers of two weighted Bernoulli numbers with Bernoulli numbers in terms of harmonic numbers and generalized harmonic numbers. Our proof is based on studying the p-adic expansion of the number of permutations of Sym(p-2) with an even number of ascents, up to the modulus $p^2$.

A new explicit formula for Bernoulli and Genocchi numbers in terms of Stirling numbers

In the paper, the authors review some explicit formulas and establish a new explicit formula for Bernoulli and Genocchi numbers in terms of Stirling numbers of the second kind.

Combinatorial aspects of poly-Bernoulli polynomials and poly-Euler numbers

In this article, we introduce combinatorial models for poly-Bernoulli polynomials and poly-Euler numbers of both kinds. As their applications, we provide combinatorial proofs of some identities involving poly-Bernoulli polynomials.

Fibonacci polynomials, generalized Stirling numbers, and Bernoulli, Genocchi and tangent numbers

We study matrices which transform the sequence of Fibonacci or Lucas polynomials with even index to those with odd index and vice versa. They turn out to be intimately related to generalized Stirling numbers and to Bernoulli, Genocchi and tangent numbers and give rise to various identities between these numbers. There is also a close connection with the Akiyama-Tanigawa algorithm.

Hurwitz型多重伯努利数与多项式的取值递推关系

Recurrences for values of the Hurwitz type poly-Bernoulli numbers and polynomials

该研究针对一类新的广义Hurwitz型多重伯努利数与多项式，通过引入新的第二类Stirling数推广形式，推导其计算算法与负上指标组合公式，并建立该推广Stirling数与图论的联系。

Tangent and Bernoulli numbers related to Motzkin and Catalan numbers by means of numerical triangles

It is shown that Bernoulli numbers and tangent numbers (the derivatives of the tangent function at zero) can be obtained by means of easily defined triangles of numbers in several ways, some of them very similar to the Catalan triangle and a Motzkin-like triangle. Our starting point in order to show this is a new expression of Zeta(2n) involving Motzkin paths.

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 generali…

Exotic series with Bernoulli, harmonic, Catalan, and Stirling numbers

In this paper, we present a formula for generating various exotic series in the spirit of Ovidiu Furdui and Alina Sintamarian. Our new series (evaluated in closed form) involve Bernoulli, harmonic, and Catalan numbers. Also Stirling numbers of the second kind, other special numbers, and exponential polynomials. The results include series identities with Laguerre polynomials and derangement polynomials.

自然数连续幂和的q-模拟与扩展Carlitz q-Bernoulli数及多项式

$q$-analogues of sums of consecutive powers of natural numbers and extended Carlitz $q$-Bernoulli numbers and polynomials

本文通过构造经典Appell序列的q-模拟框架，扩展了Carlitz q-Bernoulli数与多项式，并利用第二类q-Stirling数给出其显式级数表示和新连接公式。

用第二类Stirling数表示Bernoulli数的一个显式公式

An explicit formula for Bernoulli numbers in terms of Stirling numbers of the second kind

该研究发现了一个以第二类Stirling数表示Bernoulli数的显式计算公式，为数论相关计算提供了新的显式表达工具。

On a Generalization of Bernoulli and Euler Numbers

We introduce a series of numbers which serve as a generalization of Bernoulli, Euler numbers and binomial coefficients. Their properties are applied to solve a probability problem and suggest a statistical test for independence and identical distribution of random variables.

An integral representation and properties of Bernoulli numbers of the second kind

In the paper, the author establishes an integral representation and properties of Bernoulli numbers of the second kind and reveals that the generating function of Bernoulli numbers of the second kind is a Bernstein function on $(0,\infty)$.

Three pearls of Bernoulli numbers

The Bernoulli numbers are fascinating and ubiquitous numbers, they occur in several domains of Mathematics like Number theory (FLT), Group theory, Calculus and even in Physics. Since Bernoulli's work, they are yet studied to understand their deep nature and particularly to find relationships between them. In this paper, we give, firstly, a short response to a problem stated, in 1971, by Carlitz and studied by many authors like Prodinger \cite{PRO}, the second pearl is an answer to a question raised, in 2008, by Tom Apostol .The third pearl is another proof of a relationship already given in 20…

Alternative proofs of a formula for Bernoulli numbers in terms of Stirling numbers

In the paper, the authors provide four alternative proofs of an explicit formula for computing Bernoulli numbers in terms of Stirling numbers of the second kind.

Powers of two weighted sum of the first p divided Bernoulli numbers modulo p

We show that, modulo some odd prime p, the powers of two weighted sum of the first p-2 divided Bernoulli numbers equals the Agoh-Giuga quotient plus twice the number of permutations on p-2 letters with an even number of ascents and distinct from the identity. We provide a combinatorial characterization of Wieferich primes, as well as of primes p for which p^2 divides the Fermat quotient q_p(2).

Bernoulli-Fibonacci Polynomials

By using definition of Golden derivative, corresponding Golden exponential function and Fibonomial coefficients, we introduce generating functions for Bernoulli-Fibonacci polynomials and related numbers. Properties of these polynomials and numbers are studied in parallel with usual Bernoulli counterparts. Fibonacci numbers and Golden ratio are intrinsically involved in formulas obtained.

An explicit formula for Bernoulli polynomials in terms of $r$-Stirling numbers of the second kind

In the paper, the authors establish an explicit formula for computing Bernoulli polynomials at non-negative integer points in terms of $r$-Stirling numbers of the second kind.

An explicit formula for computing Bernoulli numbers of the second kind in terms of Stirling numbers of the first kind

In the paper, the author finds an explicit formula for computing Bernoulli numbers of the second kind in terms of Stirling numbers of the first kind.

Remarkable relations between the central binomial series, Eulerian polynomials, and poly-Bernoulli numbers

The central binomial series at negative integers are expressed as a linear combination of values of certain two polynomials. We show that one of the polynomials is a special value of the bivariate Eulerian polynomial and the other polynomial is related to the antidiagonal sum of poly-Bernoulli numbers. As an application, we prove Stephan's observation from 2004.

An integral representation, some inequalities, and complete monotonicity of Bernoulli numbers of the second kind

In the paper, the authors discover an integral representation, some inequalities, and complete monotonicity of Bernoulli numbers of the second kind.

Two closed forms for the Bernoulli polynomials

In the paper, the authors find two closed forms involving the Stirling numbers of the second kind and in terms of a determinant of combinatorial numbers for the Bernoulli polynomials and numbers.

Unimodality of the Expectation of Betti Numbers for Bernoulli Random Quota Complexes

We study certain random simplicial complexes, called random quota complexes. A quota complex on $N+1$ weighted vertices is constructed by adding an $n$-simplex if the sum of the weights of the vertices is below a given quota, $q$. In this paper, the weights of the vertices are chosen i.i.d. with a Bernoulli distribution. The main result of this paper is that the expectation of the $m^{\textrm{th}}$ Betti number, i.e., the dimension of the $m^{\textrm{th}}$ homology group, is unimodal in $m$.

Sums of Products of Bernoulli numbers of the second kind

The Bernoulli numbers b_0,b_1,b_2,.... of the second kind are defined by \sum_{n=0}^\infty b_nt^n=\frac{t}{\log(1+t)}. In this paper, we give an explicit formula for the sum \sum_{j_1+j_2+...+j_N=n, j_1,j_2,...,j_N>=0}b_{j_1}b_{j_2}...b_{j_N}. We also establish a q-analogue for \sum_{k=0}^n b_kb_{n-k}=-(n-1)b_n-(n-2)b_{n-1}.

Hankel Determinants of sequences related to Bernoulli and Euler Polynomials

We evaluate the Hankel determinants of various sequences related to Bernoulli and Euler numbers and special values of the corresponding polynomials. Some of these results arise as special cases of Hankel determinants of certain sums and differences of Bernoulli and Euler polynomials, while others are consequences of a method that uses the derivatives of Bernoulli and Euler polynomials. We also obtain Hankel determinants for sequences of sums and differences of powers and for generalized Bernoulli polynomials belonging to certain Dirichlet characters with small conductors. Finally, we collect a…

Explicit formulas for computing Bernoulli numbers of the second kind and Stirling numbers of the first kind

In the paper, by establishing a new and explicit formula for computing the $n$-th derivative of the reciprocal of the logarithmic function, the author presents new and explicit formulas for calculating Bernoulli numbers of the second kind and Stirling numbers of the first kind. As consequences of these formulas, a recursion for Stirling numbers of the first kind and a new representation of the reciprocal of the factorial $n!$ are derived. Finally, the author finds several identities and integral representations relating to Stirling numbers of the first kind.

Asymptotics of signed Bernoulli convolutions scaled by multinacci numbers

We study the signed Bernoulli convolution $$ν_β^{(n)}=*_{j=1}^n \left (\frac12δ_{β^{-j}}-\frac12δ_{-β^{-j}}\right ),\ n\ge 1$$ where $β>1$ satisfies $$β^m=β^{m-1}+\cdots+β+1$$ for some integer $m\ge 2$. When $m$ is odd, we show that the variation $|ν_β^{(n)}|$ coincides the unsigned Bernoulli convolution $$μ_β^{(n)}=*_{j=1}^n \left (\frac12δ_{β^{-j}}+\frac12δ_{-β^{-j}}\right ).$$ When $m$ is even, we obtain the exact asymptotic of the total variation $\|ν_β^{(n)}\|$ as $n\rightarrow\infty$.

Bernoulli 划分

Bernoulli Partitions

本文提出基于尺度不变散射的 Bernoulli 数 B_{2n} 划分方法，将其表示为同号单调有理数的有限和，并探讨这些有理数在大 n 极限下的性质。

伯努利分拆与欧拉分拆

Bernoulli and Euler Partitions

该研究提出了伯努利数与欧拉数的精确有理分拆形式，这类分拆是涉及Faulhaber系数和Salié系数的新型求和表达式。

Fast computation of Bernoulli, Tangent and Secant numbers

We consider the computation of Bernoulli, Tangent (zag), and Secant (zig or Euler) numbers. In particular, we give asymptotically fast algorithms for computing the first n such numbers in O(n^2.(log n)^(2+o(1))) bit-operations. We also give very short in-place algorithms for computing the first n Tangent or Secant numbers in O(n^2) integer operations. These algorithms are extremely simple, and fast for moderate values of n. They are faster and use less space than the algorithms of Atkinson (for Tangent and Secant numbers) and Akiyama and Tanigawa (for Bernoulli numbers).