Questions tagged [polygamma]

For questions about, or related to the polygamma function.

Filter by
Sorted by
Tagged with
39
votes
1answer
819 views

A closed form of $\sum_{k=1}^\infty \psi^{(1)} (k+a)\psi^{(1)} (k+b)$?

The following result $$ \sum_{k=1}^\infty\left(\psi^{(1)} (k)\right)^2 = 3\zeta(3) $$ where $\psi^{(1)}$ is the polygamma function makes me think there is a nice sum for the series $$ \sum_{k=1}^\...
24
votes
2answers
737 views

Integral ${\large\int}_0^1\ln^3\!\left(1+x+x^2\right)dx$

I'm interested in this integral: $$I=\int_0^1\ln^3\!\left(1+x+x^2\right)dx.\tag1$$ Can we prove that $$\begin{align}I&\stackrel{\color{gray}?}=\frac32\ln^33-9\ln^23+36\ln3+2\pi^2\ln3-\frac{4\pi^2}...
18
votes
1answer
743 views

Evaluating $\int_0^{\Large\frac{\pi}{2}}\left(\frac{1}{\log(\tan x)}+\frac{1}{1-\tan(x)}\right)^3dx$

Using the method shown here, I have found the following closed form. $$ \int_0^{\!\Large \frac{\pi}{2}}\!\!\left(\frac{1}{\log(\tan x)}+\frac{1}{1-\tan x}\right)^2\! \mathrm dx= 3\ln2-\frac{4}{\pi}...
16
votes
2answers
585 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 ...
15
votes
0answers
390 views

Is this similarity just a coincidence?

Here is a graph of the function $y=-1/x$: If we add infinitely many similar functions with a shift of $\pi/2$ each in both directions, we get $\tan x$. But if we do the same only in one direction, we ...
14
votes
1answer
485 views

Closed form for $\sum_{n=0}^\infty\frac{\Gamma\left(n+\tfrac14\right)}{2^n\,(4n+1)^2\,n!}$

I was experimenting with hypergeometric-like series and discovered the following conjecture (so far confirmed by more than $5000$ decimal digits): $$\sum_{n=0}^\infty\frac{\Gamma\!\left(n+\tfrac14\...
14
votes
2answers
518 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 ...
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^{(...
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{...
11
votes
2answers
317 views

Trigamma identity $4\,\psi_1\!\left(\frac15\right)+\psi_1\!\left(\frac25\right)-\psi_1\!\left(\frac1{10}\right)=\frac{4\pi^2}{\phi\,\sqrt5}.$

I heuristically discovered the following identity for the trigamma function, that I could not find in any tables or papers or infer from existing formulae (e.g. [1], [2], [3], [4], [5], [6]): $$4\,\...
10
votes
0answers
216 views

Asymptotic behavior of the generalized polygamma function

The generalized polygamma function$^{[1]}$$\!^{[2]}$ is defined as $$\psi^{(\nu)}(z)=e^{-\gamma\!\;\nu}\;\partial_\nu\!\left(\frac{e^{\gamma\!\;\nu}\;\zeta(\nu+1,z)}{\Gamma(-\nu)}\right),\tag1$$ where ...
9
votes
2answers
388 views

Integral $\int_0^\infty\Big[\log\left(1+x^2\right)-\psi\left(1+x^2\right)\Big]dx$

I found this intriguing integral: $$\int_0^\infty\Big[\log\left(1+x^2\right)-\psi\left(1+x^2\right)\Big]dx\approx0.84767315533332877726581...$$ where $\psi(z)=\partial_z\log\Gamma(z)$ is the digamma. ...
7
votes
2answers
229 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 ...
7
votes
1answer
190 views

Closed form for $\sum_{n=1}^{\infty} \frac{1}{1+n+n^2+\cdots+n^a}$

Does anyone know if there happens to exist a closed form solution for this sum: $$\sum_{n=1}^{\infty} \frac{1}{1+n+n^2+\cdots+n^a}$$ For low values of $a$, Wolfram Alpha gives a closed form in terms ...
6
votes
6answers
335 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:...
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 ...
5
votes
3answers
350 views

Residue of $\Gamma^{2}$ and $\Gamma^{3}$

Based on wiki, the residues of $\Gamma$ at non positive integers are given by: $$\text{Res}\left ( \Gamma(z),z=-n \right )=\frac{(-1)^{n}}{n!}.$$ I have been trying to find residue for $\Gamma^{2}$ ...
5
votes
2answers
199 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 ...
5
votes
2answers
174 views

Seeking methods to solve: $\int_0^\infty \frac{\ln(t)}{t^n + 1}\:dt$

Seeking Methods to solve the following two definite integrals: \begin{equation} I_(n) = \int_0^\infty \frac{\ln(t)}{t^n + 1}\:dt \qquad J(n) = \int_0^\infty \frac{\ln^2(t)}{t^n + 1}\:dt \end{equation}...
5
votes
1answer
136 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 ...
5
votes
1answer
149 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)-\...
5
votes
0answers
120 views

Closed form for $\sum_{k\geq 1}\frac{1}{(2^k-1)^a}$

Is there a closed form for $$f(a)=\sum_{k=1}^\infty\frac{1}{(2^k-1)^a},$$ where $0<a\in\mathbb{R}$. My attempts so far have considered $a\in\mathbb{N}$, which appears to give finite sums of the q-...
4
votes
4answers
180 views

Value of $\psi\left(\frac{1}{2}\right)$

I apologise if this is a dumb question, but I have trouble deriving $\displaystyle\psi\left(\frac{1}{2}\right)=-\gamma-2\ln{2}$. I have tried the following. \begin{align} \psi\left(\frac{1}{2}\right) &...
4
votes
0answers
164 views

Recurrence relation for polygamma reflection polynomials

In the reflection formula for the polygamma function $$ \psi^{(n)}(1-z) + (-1)^{n+1}\psi^{(n)}(z) % = (-1)^{n} \pi \frac{d^{n}}{d z^{n}} \cot (\pi z) $$ the right hand side is a polynomial $(-1)^{n}\...
3
votes
2answers
127 views

Alternative proof for $\zeta\left(2,\frac14\right)=\psi^{(1)}\left(\frac14\right)=\pi^2+8G$

On the German Wikipedia page of the Hurwitz Zeta Function I have come across the following formula $$\zeta\left(2,\frac14\right)~=~\pi^2+8G\tag1$$ Where $G$ is Catalan's Constant. Even though I ...
3
votes
2answers
173 views

Special values $\psi \left(\frac12\right)$ and $\psi \left(\frac13\right)$

I wonder if it is easy to prove that $$ \begin{align} \psi \left(\frac12\right) & = -\gamma - 2\ln 2, \\ \psi \left(\frac13\right) & = -\gamma + \frac\pi6\sqrt{3}- \frac32\ln 3, \end{align} $$...
3
votes
1answer
591 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^{(...
3
votes
1answer
54 views

Solving an integral with limits of polygamma functions

I've been trying to solve the following integral: $$I_n=\int_0^1\frac1x\ln^n(x)\ln^{8-n}(1-x)~\mathrm dx$$ for $n\in[2,6]$. It can be computed as a limit of derivatives of the Beta function: $$I_n=\...
3
votes
1answer
472 views

Asymptotic behavior of the zeros of the digamma function

The gamma function has just one extremum on each interval $(k,k+1)$, where $k$ is a negative integer. These extrema occur at the zeros of the derivative of the gamma function. Let $z_n$ denote the $n$-...
3
votes
1answer
1k views

Derivative of Binomial Coefficient $\binom{2N}{N-x}$ with respect to $x$

I've got $\binom{2N}{N-x}$ and I'd like to take the derivative with respect to $x$. I know that I can take the derivative of $\binom{n}{k}$ w.r.t. n using logarithmic differentiation, but that's not ...
2
votes
1answer
177 views

Sum of complex digamma functions

It seems that the sum of the digamma function of $z$ and the digamma function of its conjugate $z^*$ is always real-valued. $$\psi(z)+\psi(z^*)=\frac{\Gamma'(z)}{\Gamma(z)}+\frac{\Gamma'(z^*)}{\Gamma(...
2
votes
1answer
291 views

Looking for an identity connecting polylogarithm and polygamma functions of arguments $\frac14$ and $\frac34$

I have a recollection of seeing an identity connecting polylogarithm and polygamma functions of arguments $\frac14$ and $\frac34$. But I don't remember details, and searching my books and the Internet ...
2
votes
1answer
87 views

Logarithmic Integral I

Consider the integral \begin{align} I = \int_{0}^{1} \frac{ \ln^{2}x}{(x^{2} - x + 1)^{2}} \, dx. \end{align} It is speculated that the value is \begin{align} I = \frac{10 \, \pi^{3}}{3^{5} \, \sqrt{3}...
2
votes
2answers
179 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
1answer
136 views

Minimizing $f(x)=A^{\frac{tx-1}{x-1}} \left( c^x \frac{\Gamma(0.5+x)}{\sqrt{\pi}} \right)^{\frac{1-t}{x-1}}$ subject to the constraint

Let $f(r)$ be a function defined as follows \begin{align} f(x)=A^{\frac{tx-1}{x-1}} \left( c^x \frac{\Gamma(0.5+x)}{\sqrt{\pi}} \right)^{\frac{1-t}{x-1}} \end{align} where $0 < A,c$ and $ t\in (0,...
2
votes
1answer
322 views

Poles of hypergeometric function $_2F_1$

We consider the hypergeometric function $ _2F_1 [\dfrac{1}{2}(1+k+l+\omega), \dfrac{1}{2}(1+k-l+\omega), 1+k, -r^2]$, and use its expansion as given in [1] in terms of the rising Pochhammer symbols. ...
2
votes
2answers
350 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}\...
2
votes
1answer
72 views

Prove or disprove: Is $\psi(x) - \psi(x + y)$ an increasing function of $x > 0$, for all $y > 0$?

The title says it all. Let $y$ be a fixed positive real value. For any such $y$, is $\psi(x) - \psi(x + y)$ an increasing function of $x > 0$? Here $\psi(x)$ is the digamma function, defined by: $...
2
votes
1answer
156 views

Closed Form for the Integral: $\int_0^1 t^n\log\Gamma(t+a)dt$

I am wondering if someone could tell me whether or not the following integral has a closed form representation: $$\int_0^1 t^n\log\Gamma(t+a)dt$$ In Srivastava's and Choi's wonderful book Zeta and q-...
2
votes
1answer
250 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 ...
2
votes
1answer
114 views

Bounding infinite series derived from polygamma functions

Let $f(x) = 2 \psi^{(1)}(x+1) + x \psi^{(2)}(x+1) $ for $ x > 0 $, where $\psi^{(i)}(x)$ is the $i^{th}$ derivative of the digamma function $\psi(x)$. The goal is to prove that $ f(x) < \frac{...
2
votes
1answer
141 views

Is it possible to simplify $\psi^{(2)}(\frac18)$ or $\psi^{(2)}(\frac pq)$?

Is it possible to simplify $\psi^{(2)}(\frac18)$, where $\psi$ denotes the polygamma function? Or more generalized, $\psi^{(2)}(\frac pq)$ and $\psi^{(2n)}(\frac pq)$? Background Noticing there is ...
2
votes
0answers
61 views

Generalized hypergeometric function at unity

I wonder if there is a closed formula for the following generalized hypergeometric function at $z=1$: $${}_4F_3\left(a,a,a,a;a+1,a+1,2a;1\right)$$ Specifically, I would like to have a formula in ...
2
votes
1answer
57 views

Path containing zeros of all derivatives

Following @mercio's comments, I've rewritten my question in terms of zeros instead of saddles. Also, after more careful consideration, I've decided that perhaps the path I seek might not depend on the ...
2
votes
0answers
93 views

Relation between the derivative of hurwitz zeta and trigamma

How to prove that $$\zeta '(-1, 1/3)- \zeta '(-1, 2/3) = \frac{\psi ' (1/3)}{6 \sqrt{3} \pi}-\frac{\pi}{9 \sqrt{3}}$$ Does there exist a general formula for $$\zeta '(-1, x)- \zeta '(-1, 1-x) $$
2
votes
1answer
91 views

Is there a nice solution to the equation $\Psi(x)=\ln(\pi)$ with a positive real $x$?

I tried to find a nice solution to the following equation: $$ \Psi(x)=\ln(\pi) $$ with $x\in\Bbb R_{\ge0}$ and where $\Psi(x)=\frac{\Gamma'(x)}{\Gamma(x)}$. Is there a nice expression for x satisfying ...
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 $R(n,x)=\psi(x)\,R(-1+n,x)+R^{(0,1)}(-1+n,x)$ ...
1
vote
1answer
385 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}...
1
vote
1answer
482 views

Numeric approximation for fitting a Gamma Distribution with a single parameter

Given a series of $N$ observations $\left(x_1, \ldots, x_N\right)$ that follow a Gamma distribution with a single parameter, $ \text{Gamma}(k, k)$, what is the maximum likelihood estimate of $ k $?. ...
1
vote
1answer
39 views

Polygamma expression for $\frac{\Gamma^{(k)}(z)}{\Gamma(z)}$?

I'm trying to simplify $$\frac{\Gamma^{(k)}(z)}{\Gamma(z)}$$ for $k=1,2,\cdots$, using polygamma notation Try I've calculated a few, using $$\Gamma^{(k)}(z) = \int_0^\infty (\log x)^k x^{z-1} ...