# Questions tagged [polygamma]

For questions about, or related to the polygamma function.

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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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### Integral $\int_0^\infty\Big[\log\left(1+x^2\right)-\psi\left(1+x^2\right)\Big]dx$

I found this intriguing integral: $$\int_0^\infty\Big[\log\left(1+x^2\right)-\psi\left(1+x^2\right)\Big]dx\approx0.84767315533332877726581...$$ where $\psi(z)=\partial_z\log\Gamma(z)$ is the digamma. ...
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### Closed Form for the Integral: $\int_0^1 t^n\log\Gamma(t+a)dt$

I am wondering if someone could tell me whether or not the following integral has a closed form representation: $$\int_0^1 t^n\log\Gamma(t+a)dt$$ In Srivastava's and Choi's wonderful book Zeta and q-...
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Let $f(r)$ be a function defined as follows \begin{align} f(x)=A^{\frac{tx-1}{x-1}} \left( c^x \frac{\Gamma(0.5+x)}{\sqrt{\pi}} \right)^{\frac{1-t}{x-1}} \end{align} where $0 < A,c$ and $t\in (0,... 1answer 192 views ### Closed form for$\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 ... 2answers 594 views ### Conjectured closed form for$\operatorname{Li}_2\!\left(\sqrt{2-\sqrt3}\cdot e^{i\pi/12}\right)$There are few known closed form for values of the dilogarithm at specific points. Sometimes only the real part or only the imaginary part of the value is known, or a relation between several different ... 2answers 752 views ### Integral${\large\int}_0^1\ln^3\!\left(1+x+x^2\right)dxI'm interested in this integral: $$I=\int_0^1\ln^3\!\left(1+x+x^2\right)dx.\tag1$$ Can we prove that \begin{align}I&\stackrel{\color{gray}?}=\frac32\ln^33-9\ln^23+36\ln3+2\pi^2\ln3-\frac{4\pi^2}... 2answers 528 views ### 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$$The corresponding antiderivative can be evaluated with Mathematica, but even after ... 0answers 220 views ### Asymptotic behavior of the generalized polygamma function The generalized polygamma function^{[1]}$$\!^{[2]} is defined as $$\psi^{(\nu)}(z)=e^{-\gamma\!\;\nu}\;\partial_\nu\!\left(\frac{e^{\gamma\!\;\nu}\;\zeta(\nu+1,z)}{\Gamma(-\nu)}\right),\tag1$$ where ...
I was reading some articles on the digamma function, and I was wondering if anyone knows how to express the digamma function $\psi^{(0)}(n)$ in terms of a trigonometric function or a logarithmic ...