# Questions tagged [polygamma]

For questions about, or related to the polygamma function.

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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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### Expressions of G-BARNES

All people know expressions G-BARNES FUNCTION for example G(1/2), G(3/2) etc ... or G(1/4), G(3/4). But someone know G(1/8), G(3/8), G(5/8) or G(7/8) in terms of Psi(1,1/8) ? Thanks.
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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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### 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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### Is this similarity just a coincidence?

Here is a graph of the function $y=-1/x$: If we add infinitely many similar functions with a shift of $\pi/2$ each in both directions, we get $\tan x$. But if we do the same only in one direction, we ...
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### Derivative of polygamma function

I am working on my Matlab homework and I have to make a derivative of function $f(x)=\psi (x)\cdot \sin (x)$ , where $\psi(x)$ is polygamma function. What the derivative of $\psi(x)$ will be?
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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. ...
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 ...
Does this reduce down to the PolyGamma function? $H_n=$ \lim_{s\to 0} \, \left(-\frac{\left(\frac{1}{s}+1\right)^n (s+1)^{-n} \left(\sum _{k=0}^{\infty } \frac{\left(-\frac{1}{s}\right)^k \left(\...