Questions tagged [polygamma]

For questions about, or related to the polygamma function.

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Symbolic differentiation Gamma function (rings - algebra)

I was after an expression for the the nth derivative of the Gamma function and I managed to find this Symbolic differentiation Gamma which reads as \Gamma^{(n)}(z) = \Gamma(z) R(n,z) \...
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Is it possible to find the $n$th derivative of Gamma function?

By repeatedly differentiating $\Gamma(x)$, I noticed that $$\frac{d^{n}}{{dx}^{n}}\Gamma(x)=\sum_{k=0}^{n-1}\binom{n-1}{k}\psi^{(n-k-1)}(x)\,\frac{d^{k}}{{dx}^{k}}\Gamma(x),$$ where $\psi^{(a)}(x)$ is ...
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Integral representation of $\psi^{(\nu)}(z)$ for $\nu>0$ but $\nu\not\in\mathbb{N}$

I need the integral representation of the polygamma function for $n>0$ but $n\not\in\mathbb{N}$. I searched both among Wolfram functions and Digital Library of Mathematical Functions On Wolfram I ...
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Great difficulty in finding the residues of $\frac{\mathrm{Log}\Gamma\left(\frac{z+ai}{2\pi i}\right)}{\cosh(z)+1}$

$\newcommand{\log}{\operatorname{Log}}\newcommand{\res}{\operatorname{Res}}\newcommand{\d}{\mathrm{d}}$Let $\Lambda(z)=\log\Gamma(z)$, $a\gt0$, let $\psi$ denote digamma. It is written here, Page 49, ...
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1 vote
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Simplification of a difficult identity involving the digamma function

EDIT: The question here is not for the reader to laboriously scan all the working, but rather to suggest ways to continue the train of thought; for example, there is perhaps a closed form of the ...
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In an attempt to find $I = \int_0^\infty \frac{t}{e^t-1}dt$

I was trying to solve $I = \int_0^\infty \frac{t}{e^t-1}dt$ My approach I took the more general form of integral $f(s) = \int_0^{\infty}\frac{e^{-st}}{e^t-1}dt$ the same way as How to evaluate the ...
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Is there a decomposition for the digamma function as a sum of digamma functions?

Let $\psi(x)$ denote the digamma function $$\psi(x)=\Gamma(x)\frac{\partial}{\partial x} \Gamma(x).$$ Consider $x=x_1 +x_2+\dots +x_m$, where $x_j>0$, for $j=1, \ldots,m$. Is there any formula to ...
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What is the root of $\sum _{i=1}^{m-1} i^k=m^k$ given integer k?

Let $m'$ be the root of $\sum _{i=1}^{m-1} i^k=m^k$ given integer $k$ (solving for $m$). What is $m'$?
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I would like to know if there's a quicker way to verify: $$\partial_z^{n-1}\psi(tz)t^n = \partial_z^n\ln[\Gamma(tz)], \,\,\\ n \in \mathbb{N}^+, t \in \mathbb{C}_+\tag{1}\label{1}$$ That's true for $t=... • 516 1 vote 1 answer 214 views Upper bound for the nth derivative of$\Gamma(x)^n$I was trying to find an upper bound for $$\frac{d^n}{ds^n} \Gamma(s)^{n}|_{s=1}$$ yet, I only get the bound for the nth derivative of gamma, as follow: First, the integral of the nth derivative of ... 3 votes 2 answers 214 views Prove$\lim_{n\mapsto 0}[(\psi(n)+\gamma)\psi^{(1)}(n)-\frac12\psi^{(2)}(n)]=2\zeta(3)$How to prove that $$\lim_{n\mapsto 0}[(\psi(n)+\gamma)\psi^{(1)}(n)-\frac12\psi^{(2)}(n)]=2\zeta(3)\ ?$$ I encountered this limit while I was trying to solve$\int_0^1\frac{\ln x\ln(1-x)}{x(1-x)}dx\$ ...
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We can expression the first derivative of the gamma function as: $$\Gamma'(s) \sim -\frac{1}{s^2}+\frac{6\gamma^2+\pi^2}{12}+O(s)$$ but what about the second derivative? I do not know how to approach ...