I was messing around with the digamma function the other day, and I discovered this identity: $$\psi\left(\frac ab\right)=\sum_{\substack{\large\rho^b=1\\\large\rho\ne1}}(\rho^a-1)\ln(1-\bar\rho)-\gamma$$ when $0<\dfrac ab\le1$. It’s unusual in that it sums over the $b$-eth roots of unity (which I don’t see very often). (Note that $\bar\rho=\rho^{-1}$.) It also gives explicit values of the digamma function for all rational arguments, […]

I would like to obtain a closed form for the following limit: $$I_2=\lim_{k\to \infty} \left ( – (\ln k)^2/2 +\sum_{n=1}^{k} \psi(n) \, \psi'(n) \right)$$ Here $\psi(n)$ is digamma function. Using the method detailed in this answer, I was able to compute simpler, related series: $$\lim_{k\to \infty} \sum_{n=1}^{k} \left (\psi'(n) -1/n \right) =1 $$ $$\sum_{n=1}^{\infty} \psi”(n) […]

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 integers by letting $$\psi^{(n-1)}(x)=\int_a^x\psi^{(n)}(x)$$ Unfortunately, there is no fixed $a$ such that the above holds true for any $n$, but it does capture the general idea. Using fractional calculus, one can extend the polygamma […]

While trying to improve this interesting answer by @Anastasiya-Romanova I noticed that $$\psi^{(1)}\left(\frac 56\right)-\psi^{(1)}\left(\frac 16 \right)=5\left(\psi^{(1)}\left(\frac 23\right)-\psi^{(1)}\left(\frac 13\right)\right),$$ where $\psi^{(1)}$ is the trigamma function. I don’t know how to prove it. Can anybody help us to make her answer complete with the proof of this identity?

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: $$\begin{aligned} 1-\frac{1}{3^3}+\frac{1}{5^3}-\cdots &=\sum_{n=0}^{\infty}\frac{(-1)^n}{\left ( 2n+1 \right )^3} \\ &\overset{(*)}{=} \left ( 1+\frac{1}{5^3}+\frac{1}{9^3}+\cdots \right )-\left ( \frac{1}{3^3}+\frac{1}{7^3}+\frac{1}{11^3}+\cdots \right )\\ &=\sum_{n=0}^{\infty}\frac{1}{\left ( 4n+1 \right )^3} \; -\sum_{n=0}^{\infty}\frac{1}{\left […]

Let $\psi := \Gamma’/\Gamma$ denote the digamma function. Could you find, as $\alpha$ tends to $+\infty$, an equivalent term for the following series? $$ \sum_{n=1}^{\infty} \left( \psi (\alpha n) – \log (\alpha n) + \frac{1}{2\alpha n} \right) $$ Please I do have an answer, I’m curious about different approaches. Thanks.

There are few known closed form for values of the dilogarithm at specific points. Sometimes only the real part or only the imaginary part of the value is known, or a relation between several different values is known: [1][2][3][4][5][6]. Discovering a new identity of this sort is always of a great interest. I numerically discovered […]

I am looking for a book, paper, web site, etc. (or several ones) containing the most complete list of identities and special values for the polylogarithm $\operatorname{Li}_s(z)$ and polygamma $\psi^{(a)}(z)$ functions, including generalizations to negative, non-integer and complex orders and arguments. As far as I know, there is an ongoing research in this area, and […]

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 function. Does anyone know its Fourier Series representation?

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