# Questions tagged [poly-gamma-function]

The polygamma function may be represented as \begin{align} \psi^{(m)}(z)&= (-1)^{m+1}\int_0^\infty\frac{t^m e^{-zt}} {1-e^{-t}}\ dt\\ &=-\int_0^1\frac{t^{z-1}}{1-t}\ln^mt\ dt \end{align} which holds for $Re z >0$ and $m > 0$. For $m = 0$ see the digamma function definition.

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### How to prove negativity of a $3\times3$ determinant whose elements involve trigamma, tetragamma, and pentagamma functions?

The classical Euler gamma function can be defined by the integral \begin{equation*} \Gamma(z)=\int_0^{\infty}t^{z-1}\operatorname{e}^{-t}\operatorname{d}t, \quad \Re(z)>0. \end{equation*} Its ...
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### How to extend this sum involving generalized harmonic numbers?

It is well-known since Euler that the Generalized harmonic numbers, defined for $n\in\mathbb N$ by $$H_n^{(r)}=\sum_{k=1}^n\frac1{k^r},$$ can be naturally extended for non integer $n$ in terms of ...
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### Polygamma function in mathematical physics

Are there situations in which the polygamma pops up naturally in a mathematical physics context? In particular: are there examples of potentials having some interest for which the dependence on the ...
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### How to prove the convexity of a simple function involving a ratio of two polygamma functions?

Let \begin{equation*} \Gamma(z)=\int_0^{\infty}t^{z-1}\textrm{e}^{-t}\textrm{d}t, \quad \Re(z)>0 \end{equation*} and $$\psi(z)=[\ln\Gamma(z)]'=\frac{\Gamma'(z)}{\Gamma(z)}.$$ In the literature, ...
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### Class of differentiated Gamma functions: are there any algebras where they are elementary?

There is a class of differentiated gamma functions (DGFs) that basically can be expressed via derivatives of Gamma function. They include the Gamma function, Polygamma function, and Hurwitz Zeta ...
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### On the integral $\int_0^1\log(x!)dx$ revisited

I was interested in an integral that I known from [1], it is $$\int_0^1 \log(x!)dx.$$ I tried to get such closed-form using myself ideas and symbolic calculations, also with the help of Wolfram ...
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### A challenging inequality that involves the digamma function and polygamma functions

Let $f(x)=x \psi(x+1)$, where $\psi$ is the digamma function. Define $$g(x)=(f(ax)+f((1-a)x)-f(x))-(f(ax+by)+f((1-a)x+(1-b)y)-f(x+y)),$$ where $0\le a,b\le 1$ and $x,y\ge 0$. How to show that $g(x)$ ...
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