# Questions tagged [digamma-function]

The digamma function, usually represented by the Greek letter psi or digamma, is the logarithmic derivative of the [tag:gamma-function]. It is the first of the polygamma functions.

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### Characterizing $\frac{m^k - e^{-x}(m+1)^k}{k!}$

I'm trying to characterize the behavior this function: $l_k(m,x)=\frac{m^k - e^{-x}(m+1)^k}{k!}$. I was wondering whether either of these functions are well-known in the probability theory and ...
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### On a log-gamma definite integral

A very famous log-gamma integral due to Raabe is $$\int_0^1 \log \Gamma (x) \, dx = \frac{1}{2} \log (2\pi).$$ Several proofs of this result can be found here. I would like to known about the ...
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### Understanding this property of the digamma function: $\pi(x + \frac{1}{2}) = -\gamma - 2\ln 2 + \sum\limits_{k=1}^n\frac{2}{2k-1}$

I am reading through the Wikipedia article on the digamma function where the digamma function is compared to the Harmonic Numbers defined as $H_n = \sum\limits_{k=1}^{n} \dfrac{1}{k}$. The first ...
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### How to prove that $3^{x^2+x} (x+1)^{-x} \Gamma (x+1)\ge 1$ for $x>0$?

Let $$f(x)=3^{x^2+x} (x+1)^{-x} \Gamma (x+1).$$ Drawing a picture with any computer algebra system, it is obviously that $f(x) \ge 1$ on $[0,\infty)$. But How can we prove this? If we take ...
Given two sequences $$a_n=\int_0^1 (1-x^2)^n dx$$ and $$b_n=\int_0^1 (1-x^3)^n dx$$ ,($n\in N$) then find the value of $$L=\lim_{n\to \infty} (10 \sqrt [n]{a_n} +5\sqrt [n]{b_n})$$ My try: a_n=\...