Kunle Adegoke, Sourangshu Ghosh
We derive new infinite series involving Fibonacci numbers and Riemann zeta numbers. The calculations are facilitated by evaluating linear combinations of polygamma functions of the same order at certain arguments.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Following the Mellin and inverse Mellin transform techniques presented in our paper arXiv:1606.02150 (NT), we have established close forms of Laurent series expansions of products of bi- and trigamma functions /psi(z)*/psi(-z) and /psi_(1)(z)*/psi_(1)(-z). These series were used to find summation rules which include generalized harmonic numbers of first, second and third powers and values of the Riemann zeta-functions at integers / Bernoulli numbers, for example 2*Sum_(k-1)^(infinity)(H_(k)^((2))/k^3)=6*/zeta(2)*/zeta(3)-9*/zeta(5). Some of these rules were tested numerically.
In a recent work, Gun and co-workers have proposed that $\,\sum_{n=-\infty}^{\infty}{(n+α)^{-k}}\,$ is a transcendental number for all integer $\,k$, $k > 1$, and $\,α\in \mathbb{Q} \backslash \mathbb{Z}$. Here in this work, this proposition is shown to be \emph{false} whenever $\,k\,$ is odd and $\,α\,$ is a half-integer. It is also shown that these are the only counterexamples, which allows for a correct reformulation of the original proposition. This leads to a theorem yielding a closed-form expression for the summation of that series, which determines its arithmetic nature. The result is t…
The purpose of this article is to present closed forms for various types of infinite series involving Fibonacci (Lucas) numbers and the Riemann zeta function at integer arguments.
A new definition for the Dirichlet beta function for positive integer arguments is discovered and presented for the first time. This redefinition of the Dirichlet beta function, based on the polygamma function for some special values, provides a general method for obtaining all special constants associated with Dirichlet beta function. We also show various new and fundamental relations between the polygamma function, Riemann zeta, the even-indexed euler numbers, the Dirichlet beta functions in a way never seen or imagined before.
On the linear independence of $p$-adic polygamma values
本文提出$p$-进多伽马函数值的新线性无关判别准则，推导$p$-进Hurwitz zeta函数多族平移值的线性无关性，改进推广已有结果，核心是利用正交多项式Rodrigues公式的差分类似构造第二类Padé型逼近。
Analytic continuation of $\ell$-generalized Fibonacci zeta function
本文定义ℓ-广义斐波那契zeta函数，研究其到全复平面的解析延拓，计算奇点与留数，并证明其在负整数处取有理数值。
The Fibonacci Zeta Function and Modular Forms
本文证明了一族推广 Fibonacci zeta 函数的 Dirichlet 级数可借助二面体 GL(2) Maass 形式实现亚纯延拓。
A New Representation of the Riemann Zeta Function
本文提出 Riemann Zeta 函数的新表示，通过两个结构化二重和的极限差导出含最大值函数的新恒等式，连接经典分析与现代求和技术。
The Fibonacci Zeta Function and Continuation
本文引入一类推广普通斐波那契泽塔函数的狄利克雷级数，给出将其亚纯延拓至复数域的三种方法，包括两种经典解析方法与一种基于移位卷积和模形式的方法。
In this paper, we study some Euler-Apéry-type series which involve central binomial coefficients and (generalized) harmonic numbers. In particular, we establish elegant explicit formulas of some series by iterated integrals and alternating multiple zeta values. Based on these formulas, we further show that some other series are reducible to ln(2), zeta values, and alternating multiple zeta values by considering the contour integrals related to gamma functions, polygamma functions and trigonometric functions. The evaluations of a large number of special Euler-Apéry-type series are presented as …
We derive two new analogues of a transformation formula of Ramanujan involving the Gamma and Riemann zeta functions present in the Lost Notebook. Both involve infinite series consisting of Hurwitz zeta functions and yield modular relations. As a special case of the first formula, we obtain an identity involving polygamma functions given by Andrew P. Guinand and as a limiting case of the second formula, we derive the transformation formula of Ramanujan.
Cancellation profiles of Fibonacci-Lucas zeta sums
研究Fibonacci-Lucas zeta和的项量级分布，证明归一化绝对项和的指数展开及收敛性，并确定最大项索引与条件数增长。
The Right Edge of the Zero Set of the Fibonacci Zeta Function
该研究确定了斐波那契ζ函数零点实部闭包的精确右边界，证明了相关零点性质，推导了部分和边界的渐近关系，还推广到卢卡斯ζ函数并构造了完备化整函数。
We develop series representations for the Hurwitz and Riemann zeta functions in terms of generalized Bernoulli numbers (Nörlund polynomials), that give the analytic continuation of these functions to the entire complex plane. Special cases yield series representations of a wide variety of special functions and numbers, including log Gamma, the digamma, and polygamma functions. A further byproduct is that $ζ(n)$ values emerge as nonlinear Euler sums in terms of generalized harmonic numbers. We additionally obtain series and integral representations of the first Stieltjes constant $γ_1(a)$. The …
Curious multisection identities by index factorization
本研究提出可等价表示为无穷二重乘积/级数的通用多截恒等式，推导出多种特殊函数相关新恒等式，并证明其含参数形式的特例对应标准多截恒等式。
Simplicial $d$-Polytopic Numbers Defined on Generalized Fibonacci Polynomials
本文定义了广义Fibonacci多项式上的单纯形$d$-多胞形数，建立了基本恒等式与$q$-恒等式，求出了生成函数及倒数之和，并引入了相应的Zeta函数。
Let $F_n$ and $L_n$ be the Fibonacci and Lucas numbers, respectively. Four corresponding zeta functions in $s$ are defined by \[ζ_F(s) \,:=\, \sum_{n=1}^{\infty} \frac{1}{F_n^s}\,,\quad ζ_F^*(s) \,:=\,\sum_{n=1}^{\infty} \frac{{(-1)}^{n+1}}{F_n^s}\,,\quad ζ_L(s) \,:=\, \sum_{n=1}^{\infty} \frac{1}{L_n^s}\,,\quad ζ_L^*(s) \,:=\, \sum_{n=1}^{\infty} \frac{{(-1)}^{n+1}}{L_n^s} \,.\] For positive integers $s$ the transcendence of these values is known as well as algebraic independence or dependence results for sets of these numbers. In this paper, we investigate linear forms in the above zeta func…
We present explicit formulas for the following family of parametric binomial sums involving harmonic numbers for $p=0,1,2$ and $|t|\leq1$. $$ \sum_{k=1}^{\infty}\frac{H_{k-1}t^k}{k^p\binom{n+k}{k}}\quad \mbox{and}\quad \sum_{k=1}^{\infty}\frac{t^k}{k^p\binom{n+k}{k}}. $$ We also generalize the following relation between the Stirling numbers of the first kind and the Riemann zeta function to polygamma function and give some applications. $$ ζ(n+1)=\sum_{k=n}^{\infty}\frac{s(k,n)}{kk!}, \quad n=1,2,3,... . $$ As examples, \begin{equation*} ζ(3)=\frac{1}{7}\sum_{k=1}^{\infty}\frac{H_{k-1}4^k}{k^2…
We introduce an algorithm to compute the functions belonging to a suitable set ${\mathscr F}$ defined as follows: $f\in {\mathscr F}$ means that $f(s,x)$, $s\in A\subset {\mathbb R}$ being fixed and $x>0$, has a power series expansion centred at $x_0=1$ with convergence radius greater or equal than $1$; moreover, it satisfies a functional equation of step $1$ and the Euler-Maclaurin summation formula can be applied to $f$. Denoting the Euler gamma-function as $Γ$, we will show that, for $x>0$, $\log Γ(x)$, the digamma function $ψ(x)$, the polygamma functions $ψ^{(w)}(x)$, $w\in {\mathbb N}$, $…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。