# Questions tagged [polygamma]

For questions about, or related to the polygamma function.

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### Computing solutions to polygamma function of order 1?

I want to preface this with the fact that I am WAY out of my depth with my mathematical familiarity with these topics. While trying to figure out how to compute the solution to the polygamma function ...
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### How do you solve for x in this equation which includes the polygamma function.

How do you solve for x given: $$0 = 2x+18 - \int_{0}^{\infty}\left(t^x * e^\left(-t\right) * ln(t)\right)$$ The background is I have this function: $f\left(x\right)\ =x^{2}+\ 19x\ -\ x!$ I took the ...
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### Polygamma sum problem ...

Hello guys i have a problem evaluating the following sum $$\sum_{n=1}^{+\infty}\frac{n(n+1)}{2}\frac{4x(3\pi ^{2}(n+1)^{2}+x^{2})}{(x^{2}-\pi ^{2}(n+1)^{2})^{3}}$$ It is obviously of the polygamma ...
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### Polygamma sum problem

I have a problem evaluating the following sum, $$\sum_{n=1}^{+\infty}\frac{4nx(3\pi ^{2}(n+1)^{2}+x^{2})}{(x^{2}-\pi ^{2}(n+1)^{2})^{3}}$$ The sum obviously is of the form of a polygamma function. ...
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### Ask for a proof of logarithmically complete monotonicity of a power-exponential function involving the difference of the psi and logarithmic functions

It is common knowledge that the classical Euler gamma function $\Gamma(z)$ can defined by \begin{equation*} \Gamma(z)=\int^\infty_0t^{z-1} e^{-t}\textrm{d}t, \quad \Re(z)>0 \end{equation*} and the ...
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### What is the general formula of the sum $\sum_{k=0}^{n}(-1)^{k} \binom{n}{k}\binom{k/2}{m}$ for $m,n\in\mathbb{N}$?

The classical Euler's gamma function $\Gamma(z)$ can be defined by \Gamma(z)=\lim_{n\to\infty}\frac{n!n^z}{\prod_{k=0}^n(z+k)}, \quad z\in\mathbb{C}\setminus\{0,-1,-2,\dotsc\}. \end{...
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### correct property of a Gamma function?

On Wikipedia, https://en.wikipedia.org/wiki/Gamma_function, I read that the following hold true for any positive integer $n$: \begin{aligned}\Gamma \left({\tfrac {1}{2}}+n\right)&={(2n)! \over 4^{...
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### The limit of the ratio of polygamma functions

I want to calculate this quantity: $$\lim_{x \rightarrow \infty}\frac{\Psi_1 (x)}{\Psi_1 (x + y)}$$ where $$\Psi_1 (x)=\frac{d^2}{dx^2}\log \Gamma (x)=\sum_{k=0}^{\infty}\frac{1}{(x+k)^2}.$$ I ...
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