Questions tagged [polygamma]

For questions about, or related to the polygamma function.

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2
votes
1answer
255 views

Re-Expressing the Digamma

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 ...
6
votes
6answers
344 views

Various evaluations of the series $\sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)^3}$

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:...
2
votes
1answer
87 views

The meaning and definition of $\psi^{(-2)}(x)$, and the convergence of some related series involving the Möbius function

While I was playing with a CAS I find that makes sense the function $$\psi^{(-k)}(x),$$ for example $\psi^{(-2)}(x)$, where $\psi^{(n)}(x)$ denotes the $n$th derivative of the digamma function, see ...
14
votes
2answers
528 views

Conjecture $\int_0^1\frac{\ln^2\left(1+x+x^2\right)}x dx\stackrel?=\frac{2\pi}{9\sqrt3}\psi^{(1)}(\tfrac13)-\frac{4\pi^3}{27\sqrt3}-\frac23\zeta(3)$

I'm interested in the following definite integral: $$I=\int_0^1\frac{\ln^2\!\left(1+x+x^2\right)}x\,dx.\tag1$$ The corresponding antiderivative can be evaluated with Mathematica, but even after ...
12
votes
1answer
332 views

Generalized FoxTrot Series $F(a,b,q,x) = \sum_{k=q}^{\infty} \dfrac {(-1)^{k+1} k^a}{k^b+x}$

The FoxTrot Series is defined as: $$F = \sum_{k=1}^{\infty} \dfrac {(-1)^{k+1} k^2}{k^3+1}.$$ Using partial fraction decomposition we can show that $$F = \frac 13 \left[ 1 - \ln2 + \pi\operatorname{...
5
votes
2answers
200 views

Asymptotic behavior of Harmonic-like series $\sum_{n=1}^{k} \psi(n) \psi'(n) - (\ln k)^2/2$ as $k \to \infty$

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 ...
1
vote
1answer
402 views

Show some properties of the Digamma Function

Let $\psi(z)$ denote the Digamma function, $\psi(z)=\frac{d}{dz}\ln \Gamma(z)=\frac{\Gamma'(z)}{\Gamma(z)}$. I am meant to show the following properties of $\psi$: $\psi$ is meromorphic in $\mathbb{C}...
13
votes
2answers
2k views

The most complete reference for identities and special values for polylogarithm and polygamma functions

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^{(...
16
votes
2answers
594 views

Conjectured closed form for $\operatorname{Li}_2\!\left(\sqrt{2-\sqrt3}\cdot e^{i\pi/12}\right)$

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 ...
7
votes
2answers
232 views

How is $ \sum_{n=1}^{\infty}\left(\psi(\alpha n)-\log(\alpha n)+\frac{1}{2\alpha n}\right)$ when $\alpha$ is great?

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 ...
6
votes
1answer
180 views

Evaluating $\psi^{(1/2)}(x)$ of the extended polygamma function

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 ...
3
votes
1answer
595 views

Prove $\psi^{(1)}\left(\frac 56\right)-\psi^{(1)}\left(\frac 16 \right)=5\left(\psi^{(1)}\left(\frac 23\right)-\psi^{(1)}\left(\frac 13\right)\right)$

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^{(...
2
votes
1answer
102 views

Expressions of G-BARNES

All people know expressions G-BARNES FUNCTION for example G(1/2), G(3/2) etc ... or G(1/4), G(3/4). But someone know G(1/8), G(3/8), G(5/8) or G(7/8) in terms of Psi(1,1/8) ? Thanks.
5
votes
1answer
153 views

Digamma equation identification

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)-\...
1
vote
2answers
60 views

I ran $\frac{d^n}{dx^n}[(x!)!]$ through the calculator but don’t understand the $R^{(0,1)}$ and $R(n,x)$ in the output

I ran $\frac{d^n}{dx^n}[(x!)!]$ through Wolfram|Alpha, which returned $$\frac{\partial^n(x!)!}{\partial x^n} = \Gamma(1+x!)\,R(n,1+x!)$$ for $R(n,x)=\psi(x)\,R(-1+n,x)+R^{(0,1)}(-1+n,x)$ ...
0
votes
1answer
33 views

Explicit forms of negapolygamma difference with arguments that differ by a half?

There are well known explicit formulas for negapolygamma expressions of the form $$\psi^{(-n)}(x)-\psi^{(-n)}(x-1)$$ for $n\in\mathbb{N}\gt1$ for example $$\psi^{(-2)}\left(x\right)-\psi^{(-2)}\left(...
5
votes
2answers
153 views

On the series expansion of $\frac{\operatorname{Li}_3(-x)}{1+x}$ and its usage

Recently I have asked about the evaluation of an integral involving a Trilogarithm $($which can be found here$)$. Pisco provided a quite elegant approach starting with a functional equation of the ...
2
votes
2answers
189 views

Relation between harmonic series $H(m)$ and polygamma function?

I have the following formula: $$h(x)=\sum_{R=1}^m \frac{1}{R+1}-1$$ I have re-expressed this (correctly, I hope!) in terms of the harmonic number $H(m)=\sum_{R=1}^m \frac{1}{R}$: $$h(x)=H(m)+\frac{...
2
votes
2answers
361 views

Asymptotic expansion of Polygamma functions

According to Wikipedia, the log-Gamma and Polygamma functions have the following asymptotic behaviour on the real line for $x\to\infty$: $$\ln\Gamma(x) = (x - \tfrac{1}{2}) \ln(x) - x + \tfrac{1}{2}\...
1
vote
0answers
104 views

Convexity of reciprocal polygamma

Is the reciprocal of polygamma functions of odd order convex on $ \mathbb{R^+}$, while that of even order above 0 concave? Plotting the functions suggest so, but I've been trying for days to come up ...
0
votes
1answer
378 views

Bounds on the real and imaginary parts of the digamma function $\psi $

Let $\psi $ be the digamma function given by $$\psi (z)=\left.\frac {d}{dt}\log\Gamma (t)\right|_{t=z}. $$ I wonder does anyone know of any lower and/or upper bounds on the real and imaginary parts ...