Renaat Van Malderen
Bernoulli numbers are usually expressed in terms of their lower index numbers (recursive). This paper gives explicit formulas for Bernoulli numbers of even index. The formulas contain a remarkable sequence of determinants. The value of these determinants for variable dimension is investigated.
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In this paper, applying the Faà di Bruno formula and some properties of Bell polynomials, several closed formulas and determinantal expressions involving Stirling numbers of the second kind for higher-order Bernoulli and Euler polynomials are presented.
Hankel determinants of sequences related to Bernoulli and Euler numbers have been studied before, and numerous identities are known. However, when a sequence is shifted by one unit, the situation often changes significantly. In this paper we use classical orthogonal polynomials and related methods to prove a general result concerning Hankel determinants for shifted sequences. We then apply this result to obtain new Hankel determinant evaluations for a total of $13$ sequences related to Bernoulli and Euler numbers, one of which concerns Euler polynomials.
Bernoulli determinants and cuspidal subgroups
该研究针对任意整数\(N\)的模曲线\(X_1(N)\)，给出有理尖点类群阶数的显式公式，证明依赖斯特伦的结果及相关行列式计算，还定义了高权类似物并推测其阶数与特定行列式有关。
Explicit, recurrent, determinantal expressions of the $k$th power of formal power series and applications to the generalized Bernoulli numbers
本文给出形式幂级数第k次幂的闭式与递推行列式表达式，应用于广义伯努利数，推导新公式与递推关系，并推广欧拉恒等式，揭示与第二类斯特林数的联系。
This paper sets the groundwork for the consideration of families of recursively defined polynomials and rational functions capable of describing the Bernoulli numbers. These families of functions arise from various recursive definitions of the Bernoulli numbers. The derivation of these recursive definitions is shown here using the original application of the Bernoulli numbers: sums of powers of positive integers, i.e., Faulhaber's problem. Since two of the three recursive forms of the Bernoulli numbers shown here are readily available in literature and simple to derive, this paper focuses on t…
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…
On the restricted partition function via determinants with Bernoulli polynomials
本文利用由 Bernoulli 多项式值构成的行列式 $Δ_{r,D}$，证明了在行列式非零条件下，限制分拆函数 $p_{\mathbf a}(n)$ 可通过 Bernoulli 多项式值与 Bernoulli Barnes 数进行计算。
Determinants with Bernoulli polynomials and the restricted partition function
本文研究由伯努利多项式构造的两个 rD 阶行列式，并建立其与受限分拆函数之间的内在联系。
Bernoulli numbers are usually expressed in terms of their lower index numbers (recursive). This paper gives an explicit formula for Bernoulli numbers of even index. The formula contains a remarkable sequence of determinants.
In this paper, we introduce the concept of the (higher order) Appell-Carlitz numbers which unifies the definitions of several special numbers in positive characteristic, such as the Bernoulli-Carlitz numbers and the Cauchy-Carlitz numbers.Their generating function is usually named Hurwitz series in the function field arithmetic. By using Hasse-Teichmüller derivatives, we also obtain several properties of the (higher order) Appell-Carlitz numbers, including a recurrence formula, two closed forms expressions, and a determinant expression. The recurrence formula implies Carlitz's recurrence formu…
Using continued fraction expansions of certain polygamma functions as a main tool, we find orthogonal polynomials with respect to the odd-index Bernoulli polynomials $B_{2k+1}(x)$ and the Euler polynomials $E_{2k+ν}(x)$, for $ν=0, 1, 2$. In the process we also determine the corresponding Jacobi continued fractions (or J-fractions) and Hankel determinants. In all these cases the Hankel determinants are polynomials in $x$ which factor completely over the rationals.
On the restricted partition function via determinants with Bernoulli polynomials. II
本文在 D=1 或 D 为素数的条件下，将限制分拆函数的计算转化为系数为伯努利多项式与伯努利·巴恩斯数的线性方程组求解问题。
We introduce generalized hypergeometric Bernoulli numbers for Dirichlet characters. We study their properties, including relations, expressions and determinants. At the end in Appendix we derive first few expressions of these numbers.
We calculate the Hankel determinants of sequences of Bernoulli polynomials. This corresponding Hankel matrix comes from statistically estimating the variance in nonparametric regression. Besides its entries' natural and deep connection with Bernoulli polynomials, a special case of the matrix can be constructed from a corresponding Vandermonde matrix. As a result, instead of asymptotic analysis, we give a direct proof of calculating an entry of its inverse.
Hankel determinants and Jacobi continued fractions for $q$-Euler numbers
本文确定了$q$-Euler数的Hankel行列式，通过大$q$-Jacobi多项式的特化给出正交多项式，并利用Jacobi连分数表达式完成求值。
In this paper, we give the determinant expressions of the hypergeometric Bernoulli numbers, and some relations between the hypergeometric and the classical Bernoulli numbers which include Kummer's congruences. By applying Trudi's formula, we have some different expressions and inversion relations. We also determine explicit forms of convergents of the generating function of the hypergeometric Bernoulli numbers, from which several identities for hypergeometric Bernoulli numbers are given.
In this note, by using the Hasse-Teichmüller derivatives, we obtain two explicit expressions for the related numbers of higher order Appell polynomials. One of them presents a determinant expression for the related numbers of higher order Appell polynomials, which involves several determinant expressions of special numbers, such as the higher order generalized hypergeometric Bernoulli and Cauchy numbers, thus recovers the classical determinant expressions of Bernoulli and Cauchy numbers stated in an article by Glaisher in 1875.
Partial Results on Hankel Determinants and the Corresponding J-Fractions of a Sequence Related to Bernoulli Numbers
本文研究伯努利数相关序列的Hankel行列式，发现次高阶系数决定行列式，并证明其J-分数与Cao的快速收敛序列一致。
The Riemann zeta identity at even integers of Lettington, along with his other Bernoulli and zeta relations, are generalized. Other corresponding recurrences and determinant relations are illustrated. Another consequence is the application to sums of double zeta values. A set of identities for the Ramanujan and generalized Ramanujan polynomials is presented. An alternative proof of Lettington's identity is provided, together with its generalizations to the Hurwitz and Lerch zeta functions, hence to Dirichlet $L$ series, to Eisenstein series, and to general Mellin transforms. The Hurwitz number…
We study some properties of restricted and associated Fubini numbers. In particular, they have the natural extensions of the original Fubini numbers in the sense of determinants. We also introduce modified Bernoulli and Cauchy numbers and study characteristic properties.
Two Legendre-symbol determinants of Sun and Bernoulli numbers
本文给出Sun定义的两个Legendre符号行列式模p的显式公式（用Bernoulli数表示），并利用Reinhart的反例找到使行列式模p为零的素数，否定回答了Sun的一个问题。
Bernoulli numbers and the integer group determinants of $C_p \times C_p$
该研究针对素数$p\ge7$，结合伯努利数与圆单位，确定$C_p\times C_p$的整数群行列式集合的最小指数$e(p)$，给出其下界、等式成立条件及正则素数下的行列式性质。
Explicit expressions for restricted partition function $W(s,{\bf d}^m)$ and its quasiperiodic components $W_j(s,{\bf d}^m)$ (called {\em Sylvester waves}) for a set of positive integers ${\bf d}^m = \{d_1, d_2, ..., d_m\}$ are derived. The formulas are represented in a form of a finite sum over Bernoulli and Euler polynomials of higher order with periodic coefficients. A novel recursive relation for the Sylvester waves is established. Application to counting algebraically independent homogeneous polynomial invariants of the finite groups is discussed.
Carlitz has introduced q-analogues of the Bernoulli numbers around 1950. We obtain a representation of these q-Bernoulli numbers (and some shifted version) as moments of some orthogonal polynomials. This also gives factorisations of Hankel determinants of q-Bernoulli numbers, and continued fractions for their generating series. Some of these results are q-analogues of known results for Bernoulli numbers, but some are specific to the q-Bernoulli setting.
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.
Wolstenholme primes and group determinants of cyclic groups
本文给出了 Wolstenholme 素数的一个新等价定义，将其与循环群群行列式的项数以及特定受限分拆集合的基数联系起来。
We consider the numbers $\mathcal{B}_{r,s} = (\mathbf{B}+1)^r \mathbf{B}^s$ (in umbral notation $\mathbf{B}^n = \mathbf{B}_n$ with the Bernoulli numbers) that have a well-known reciprocity relation, which is frequently found in the literature and goes back to the 19th century. In a recent paper, self-reciprocal Bernoulli polynomials, whose coefficients are related to these numbers, appeared in the context of power sums and the so-called Faulhaber polynomials. The numbers $\mathcal{B}_{r,s}$ can be recursively expressed by iterated sums and differences, so it is not obvious that these numbers d…
For non-negative integers $r$ and $m$, let $S_m^{(r)}(n)$ denote the $r$-fold summation (or hyper-sum) over the first $n$ positive integers to the $m$th powers, with the initial condition $S_m^{(0)}(n) =n^m$. In this paper, we derive a new determinantal formula for $S_m^{(r)}(n)$. Specifically, we show that, for all integers $r\geq 0$ and $m \geq 1$, $S_m^{(r)}(n)$ is proportional to $S_1^{(r)}(n)$ times the determinant of a lower Hessenberg matrix of order $m-1$ involving the Bernoulli numbers and the variable $N_r = n + \frac{r}{2}$. Furthermore, whenever $r\geq 1$, evaluating this determina…
A Substitution to Bernoulli Numbers in easier computation of (ζ(2k))
提出一种无需伯努利数即可计算ζ(2k)的递归公式，提高了计算大值zeta函数的效率。
Closed-form formulas, determinantal expressions, recursive relations, power series, and special values of several functions used in Clark--Ismail's two conjectures
本文利用Faà di Bruno公式、偏Bell多项式恒等式及比值求导公式，系统建立了Clark--Ismail两个猜想中相关函数的闭式公式、行列式表达式、递推关系、幂级数与特殊值，并进一步得到Hessenberg矩阵行列式公式及相关数列的对数凸性。
Joint moments of higher order derivatives of CUE characteristic polynomials II: Structures, recursive relations, and applications
本文研究了 CUE 随机矩阵特征多项式导数联合矩首项阶系数的性质，建立了结构定理与递推关系，联系了 σ-Painlevé III' 方程解，并提出了适用于 Young diagrams 移位的一般 Hankel 矩阵行列式的方法。
Leading coefficient in the Hankel determinants related to binomial and $q$-binomial transforms
本文将二项式变换下 Hankel 行列式不变的经典结论推广至 $q$-二项式变换，研究其首项系数行为，并利用特殊二项式变换分析偶数下标 Bernoulli 多项式情形。
By studying Cameron's operator in terms of determinants, two kinds of "integer" sequences of incomplete numbers were introduced. One was the sequence of restricted numbers, including $s$-step Fibonacci sequences. Another was the sequence of associated numbers, including Lamé sequences of higher order. By the classical Trudi's formula and the inverse relation, more expressions were able to be obtained. These relations and identities can be extended to those of sequence of negative integers or rational numbers. As applications, we consider hypergeometric Bernoulli, Cauchy and Euler numbers with …
In this paper, we introduce the hypergeometric Euler number as an analogue of the hypergeometric Bernoulli number and the hypergeometric Cauchy number. We study several expressions and sums of products of hypergeometric Euler numbers. We also introduce complementary hypergeometric Euler numbers and give some characteristic properties. There are strong reasons why these hypergeometric numbers are important. The hypergeometric numbers have one of the advantages that yield the natural extensions of determinant expressions of the numbers, though many kinds of generalizations of the Euler numbers h…
Representations with k-generalized Fibonacci numbers
该研究探讨整数的k阶广义斐波那契数表示，给出系数为{-1,0,1}的零表示计数的递推关系，构造系数为{0,1}的 tribonacci 表示的二叉树模型，揭示其与自相似测度的联系。
Let $S_p(n)$ denote the sum of $p$th powers of the first $n$ positive integers $1^p + 2^p + \cdots + n^p$. In this paper, first we express $S_p(n)$ in the so-called Faulhaber form, namely, as an even or odd polynomial in $(n + 1/2)$, according as $p$ is odd or even. Then, using the relation $S_p(n) - S_p(n-1) = n^p$, we derive a recursive formula for the associated Faulhaber coefficients. Applying Cramer's rule to the corresponding system of equations, we obtain an explicit determinant formula for the said coefficients. Furthermore, we show how to convert the (even or odd) Faulhaber polynomial…
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