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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On the integral $\int_0^1\log(x!)dx$ revisited

I was interested in an integral that I known from , 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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Find limit of sequence defined by sum of previous terms and harmonics

I came across this sequence as part of my work. Could someone indicate me the methodology I should follow to solve it? I guess it involves harmonic numbers and/or the digamma function? I tried to ...
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 $g(x)$ ...