# Questions tagged [polygamma]

For questions about, or related to the polygamma function.

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### Conjecture $\int_0^1\frac{\ln^2\left(1+x+x^2\right)}x dx\stackrel?=\frac{2\pi}{9\sqrt3}\psi^{(1)}(\tfrac13)-\frac{4\pi^3}{27\sqrt3}-\frac23\zeta(3)$

I'm interested in the following definite integral: $$I=\int_0^1\frac{\ln^2\!\left(1+x+x^2\right)}x\,dx.\tag1$$ The corresponding antiderivative can be evaluated with Mathematica, but even after ...
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### Closed form for $\sum_{n=1}^{\infty} \frac{1}{1+n+n^2+\cdots+n^a}$

Does anyone know if there happens to exist a closed form solution for this sum: $$\sum_{n=1}^{\infty} \frac{1}{1+n+n^2+\cdots+n^a}$$ For low values of $a$, Wolfram Alpha gives a closed form in terms ...
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### Various evaluations of the series $\sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)^3}$

I recently ran into this series: $$\sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)^3}$$ Of course this is just a special case of the Beta Dirichlet Function , for $s=3$. I had given the following solution:...
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### Evaluating $\psi^{(1/2)}(x)$ of the extended polygamma function

The polygamma function is generally given by $$\psi^{(n)}(x)=\frac{d^{n+1}}{dx^{n+1}}\ln(\Gamma(x)),~n\in\mathbb N_{\ge0}$$ where $\Gamma$ is the gamma function. This can be extended to negative ...
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### Residue of $\Gamma^{2}$ and $\Gamma^{3}$

Based on wiki, the residues of $\Gamma$ at non positive integers are given by: $$\text{Res}\left ( \Gamma(z),z=-n \right )=\frac{(-1)^{n}}{n!}.$$ I have been trying to find residue for $\Gamma^{2}$ ...
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### Asymptotic behavior of Harmonic-like series $\sum_{n=1}^{k} \psi(n) \psi'(n) - (\ln k)^2/2$ as $k \to \infty$

I would like to obtain a closed form for the following limit: $$I_2=\lim_{k\to \infty} \left ( - (\ln k)^2/2 +\sum_{n=1}^{k} \psi(n) \, \psi'(n) \right)$$ Here $\psi(n)$ is digamma function. Using ...
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### Seeking methods to solve: $\int_0^\infty \frac{\ln(t)}{t^n + 1}\:dt$

Seeking Methods to solve the following two definite integrals: \begin{equation} I_(n) = \int_0^\infty \frac{\ln(t)}{t^n + 1}\:dt \qquad J(n) = \int_0^\infty \frac{\ln^2(t)}{t^n + 1}\:dt \end{equation}...
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### On the series expansion of $\frac{\operatorname{Li}_3(-x)}{1+x}$ and its usage

Recently I have asked about the evaluation of an integral involving a Trilogarithm $($which can be found here$)$. Pisco provided a quite elegant approach starting with a functional equation of the ...
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### Looking for an identity connecting polylogarithm and polygamma functions of arguments $\frac14$ and $\frac34$

I have a recollection of seeing an identity connecting polylogarithm and polygamma functions of arguments $\frac14$ and $\frac34$. But I don't remember details, and searching my books and the Internet ...
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### Logarithmic Integral I

Consider the integral \begin{align} I = \int_{0}^{1} \frac{ \ln^{2}x}{(x^{2} - x + 1)^{2}} \, dx. \end{align} It is speculated that the value is \begin{align} I = \frac{10 \, \pi^{3}}{3^{5} \, \sqrt{3}...
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### Is it possible to simplify $\psi^{(2)}(\frac18)$ or $\psi^{(2)}(\frac pq)$?

Is it possible to simplify $\psi^{(2)}(\frac18)$, where $\psi$ denotes the polygamma function? Or more generalized, $\psi^{(2)}(\frac pq)$ and $\psi^{(2n)}(\frac pq)$? Background Noticing there is ...
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### Generalized hypergeometric function at unity

I wonder if there is a closed formula for the following generalized hypergeometric function at $z=1$: $${}_4F_3\left(a,a,a,a;a+1,a+1,2a;1\right)$$ Specifically, I would like to have a formula in ...
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### Path containing zeros of all derivatives

Following @mercio's comments, I've rewritten my question in terms of zeros instead of saddles. Also, after more careful consideration, I've decided that perhaps the path I seek might not depend on the ...
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### Relation between the derivative of hurwitz zeta and trigamma

How to prove that $$\zeta '(-1, 1/3)- \zeta '(-1, 2/3) = \frac{\psi ' (1/3)}{6 \sqrt{3} \pi}-\frac{\pi}{9 \sqrt{3}}$$ Does there exist a general formula for $$\zeta '(-1, x)- \zeta '(-1, 1-x)$$
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### Is there a nice solution to the equation $\Psi(x)=\ln(\pi)$ with a positive real $x$?

I tried to find a nice solution to the following equation: $$\Psi(x)=\ln(\pi)$$ with $x\in\Bbb R_{\ge0}$ and where $\Psi(x)=\frac{\Gamma'(x)}{\Gamma(x)}$. Is there a nice expression for x satisfying ...
### I ran $\frac{d^n}{dx^n}[(x!)!]$ through the calculator but don’t understand the $R^{(0,1)}$ and $R(n,x)$ in the output
I ran $\frac{d^n}{dx^n}[(x!)!]$ through Wolfram|Alpha, which returned $$\frac{\partial^n(x!)!}{\partial x^n} = \Gamma(1+x!)\,R(n,1+x!)$$ for $R(n,x)=\psi(x)\,R(-1+n,x)+R^{(0,1)}(-1+n,x)$ ...