广义幂级数的参数变化

Zeta Series Generating Function Transformations Related to Generalized Stirling Numbers and Partial Sums of the Hurwitz Zeta Function

We define a generalized class of modified zeta series transformations generating the partial sums of the Hurwitz zeta function and series expansions of the Lerch transcendent function. The new transformation coefficients we define within the article satisfy expansions by generalized harmonic number sequences, or the partial sums of the Hurwitz zeta function, which are analogous to known properties for the Stirling numbers of the first kind and for the known transformation coefficients employed to enumerate variants of the polylogarithm function series. Applications of the new results we prove …

关于广义双重级数的一个注记

A note on a generalized double series

本文利用围道积分推导出涉及 Hurwitz-Lerch zeta 函数的广义双重有限级数的闭式公式，并求出其用特殊函数表示的特例。

Expansions of generalized Euler's constants into the series of polynomials in $π^{-2}$ and into the formal enveloping series with rational coefficients only

In this work, two new series expansions for generalized Euler's constants (Stieltjes constants) $γ_m$ are obtained. The first expansion involves Stirling numbers of the first kind, contains polynomials in $π^{-2}$ with rational coefficients and converges slightly better than Euler's series $\sum n^{-2}$. The second expansion is a semi-convergent series with rational coefficients only. This expansion is particularly simple and involves Bernoulli numbers with a non-linear combination of generalized harmonic numbers. It also permits to derive an interesting estimation for generalized Euler's cons…

Three Series for the Generalized Golden Mean

As is well-known, the ratio of adjacent Fibonacci numbers tends to phi = (1 + sqrt(5))/2, and the ratio of adjacent Tribonacci numbers (where each term is the sum of the three preceding numbers) tends to the real root eta of X^3 - X^2 - X - 1 = 0. Letting alpha(n) denote the corresponding ratio for the generalized Fibonacci numbers, where each term is the sum of the n preceding, we obtain rapidly converging series for alpha(n), 1/alpha(n), and 1/(2-alpha(n)).

Generalized $k$-regular sequences III: Arithmetical properties of generalized $k$-regular series

Let $F(z)$ be a $k$-regular series in $\mathbb{Z}[[z]]$ and $b$ be an integer with $b\ge2$. Bell, Bugeaud and Coons [BelBC] proved that $F(\frac{1}{b})$ is either rational or transcendental. In [Mi], we introduce a generalized $k$-regular sequence as a unification of several kinds of important sequences including $k$-regular, $k$-additive and $k$-multiplicative sequences. In this paper, we give a generalization of the result of Bell, Bugeaud and Coons for certain generalized $k$-regular series. Especially, we show that the values of irrational generating functions of certain sum of $k$-additiv…

Some Generalized Harmonic Number Identities

Summation by parts is used to find the sum of a finite series of generalized harmonic numbers involving a specific polynomial or rational function. The Euler-Maclaurin formula for sums of powers is used to find the sums of some finite series of generalized harmonic numbers involving nonnegative integer powers, which can be used to evaluate the sums of the finite series of generalized harmonic numbers involving polynomials. Many examples and a computer program are provided.

On some $q$-series identities related to a generalized divisor function and their implications

In this article, a $q$-series examined by Kluyver and Uchimura is generalized. This allows us to find generalization of the identities in the random acyclic digraph studied by Simon, Crippa, and Collenberg in 1993. As one of the corollaries of our main theorem, we get results of Dilcher and Andrews, Crippa, and Simon. This main theorem involves a surprising new generalization of the divisor function $σ_s(n)$, which we denote by $σ_{s,z}(n)$. Analytic properties of $σ_{s,z}(n)$ are also studied. As a special case of one of our theorem we obtain result from a recent paper of Bringmann and Jennin…

Exact counting of Euler Tours for generalized series-parallel graphs

We give a simple polynomial-time algorithm to exactly count the number of Euler Tours (ETs) of any Eulerian generalized series-parallel graph, and show how to adapt this algorithm to exactly sample a random ET of the given generalized series-parallel graph. Note that the class of generalized seriesparallel graphs includes all outerplanar graphs. We can perform the counting in time $O(mΔ^3)$, where $Δ$ is the maximum degree of the graph with $m$ edges. We use $O(mΔ^2 \log Δ)$ bits to store intermediate values during our computations. To date, these are the first known polynomial-time algorithms…

Generalized Dirichlet series of n variables associated with automatic sequences

This article consists to give a necessary and sufficient condition of the meromorphic continuity of Dirichlet series defined as $\sum_{x\in \mathbf{N}^n} \frac{a_{x}}{P(x)^s}$, Where $a_{x}$ is a $q$-automatic sequence of $n$ parameters and $P: \mathbf{C}^n \to \mathbf{C}$ a polynomial, such that $P$ does not have zeros on $\mathbf{Q}^{n}_{+}$. And some specific cases of $n=1$ will also be studied in this article as examples to show the possibility to have an holomorphic continuity on the whole complex plane. Some equivalences between infinite products are also built as consequences of these r…

广义$n$-级数与de Rham复形

Generalized $n$-series and de Rham complexes

本文研究了交换环上广义$n$-级数的代数与组合性质，证明了经典组合恒等式的$s$-类似物，并定义了导数的$s$-类似物以证明仿射直线上的Poincaré引理与Cartier同构。