# Questions tagged [polygamma]

For questions about, or related to the polygamma function.

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### Re-Expressing 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 ...
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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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### 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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### 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 $R(n,x)=\psi(x)\,R(-1+n,x)+R^{(0,1)}(-1+n,x)$ ...
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There are well known explicit formulas for negapolygamma expressions of the form $$\psi^{(-n)}(x)-\psi^{(-n)}(x-1)$$ for $n\in\mathbb{N}\gt1$ for example $$\psi^{(-2)}\left(x\right)-\psi^{(-2)}\left(... 1answer 136 views ### On the series expansion of \frac{\operatorname{Li}_3(-x)}{1+x} and its usage Recently I have asked about the evaluation of an integral involving a Trilogarithm (which can be found here). Pisco provided a quite elegant approach starting with a functional equation of the ... 2answers 180 views ### Relation between harmonic series H(m) and polygamma function? I have the following formula:$$h(x)=\sum_{R=1}^m \frac{1}{R+1}-1$$I have re-expressed this (correctly, I hope!) in terms of the harmonic number H(m)=\sum_{R=1}^m \frac{1}{R}:$$h(x)=H(m)+\frac{...
Is the reciprocal of polygamma functions of odd order convex on $\mathbb{R^+}$, while that of even order above 0 concave? Plotting the functions suggest so, but I've been trying for days to come up ...
### Bounds on the real and imaginary parts of the digamma function $\psi$
Let $\psi$ be the digamma function given by $$\psi (z)=\left.\frac {d}{dt}\log\Gamma (t)\right|_{t=z}.$$ I wonder does anyone know of any lower and/or upper bounds on the real and imaginary parts ...