Questions tagged [polygamma]

For questions about, or related to the polygamma function.

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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=\...
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:...
0
votes
0answers
17 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.
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 ...
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 ...
0
votes
1answer
29 views

Derivative of polygamma function

I am working on my Matlab homework and I have to make a derivative of function $f(x)=\psi (x)\cdot \sin (x)$ , where $\psi(x)$ is polygamma function. What the derivative of $\psi(x)$ will be?
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)-\...
1
vote
0answers
36 views

Digamma function question

I just learned online about Polygamma functions, and I want to know what $x$ equals (and how to get it) when $\psi(x)=1$ and $x>1$.
0
votes
1answer
31 views

Dimensional regularization and expansion of gamma function

In my calculations, I used dimensional regularization, i.e. replace $d\rightarrow d-\epsilon$ and calculated the divergent integral. Then, I would like to expand the answer into seriers by $\epsilon$ ...
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 ...
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
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}...
1
vote
0answers
50 views

How to show: $\psi^{(0)}\left(\frac{1}{n}\right) - \psi^{(0)}\left(1 - \frac{1}{n}\right) = -\pi\cot\left(\frac{\pi}{n}\right)$ [duplicate]

Based on a result I found recently and in conjunction with methods I've observed on MSE I was able to show that: \begin{equation} \int_0^\infty \frac{ \ln(t)}{t^n + 1}\:dt = -\frac{\pi^2}{n^2} \...
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}^\...
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} ...
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 ...
1
vote
1answer
47 views

Is the polygamma function of order $2$ non-negative (or negative) for all $x >0$?

The polygamma function of order $2$ is defined as $$\psi^{(2)}(z)= \frac{d^2}{dz^2} \psi(z) = \frac{d^{3}}{dz^{3}} \ln\Gamma(z)$$ where $\Gamma(z)$ is the usual gamma function: $\int_0^\infty x^{z-1}...
0
votes
1answer
134 views

$\sum_{n=1}^{4000000} \frac{1}{n^3}$ very quick.

Some days ago I have tried to find the sum of the first milion terms of the infinite sum $\zeta(3) = \sum_{n=1}^\infty\frac{1}{n^3}$ (Apéry's constant) on Wolfram Programming Lab (Open Cloud), an ...
0
votes
1answer
92 views

Recurrence relation for the polygamma function of negative order?

I know the recurrence relation for the Polygamma function is $$\psi^{(m)}(x+1)=\psi^{(m)}(x)+\frac{(-1)^mm!}{x^{m+1}}$$ Does such a recurrence formula exist for negative integer $m$? I am using the ...
0
votes
1answer
68 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 ...
0
votes
1answer
54 views

Logarithmic Sum

Is there a closed form for the following sum? $$\sum_{n=0}^{\infty}\sum_{m=1}^{\infty}(-1)^{m+n}\frac{\ln(m+n)}{(m+n)}$$ According to https://www.mathmash.org/contestprob.php?prob=227 it has a closed ...
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{...
0
votes
1answer
229 views

Series Representation of Gamma Function

The $\Gamma(x)$ is function That has derivatives in the polygamma form. Can those derivatives be used to make a Taylor series? I've tried but I got stuck as soon as I find out That $\Psi^1(1)=\zeta(2)=...
1
vote
1answer
151 views

Double series of Harmonic Numbers

In a solution presented here a series involving the product of Harmonic numbers is involved. The intent of the problem is to determine a form of the series \begin{align} \sum_{n=1}^{\infty} \frac{H_{n}...
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 ...
0
votes
1answer
112 views

Evaluate an infinite series involving the polygamma function OR first derivative of the hurwitz zeta function

Can we find a closed form for $$\sum _{k=1}^\infty\frac{\left(-1\right)^k}{2k-1}\left(2k^2-k+8k^2P_1(k)-16kP_2(k)+16P_3(k)\right)$$ where $$P_n(k)=\psi^{(-n)}\left(k+\frac12\right)-\psi^{(-n)}\left(k+...
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(...
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
209 views

Are the real and imaginary parts of Riemann zeta equal to each other by multiplication of a Riemann Siegel theta function expression?

It appears that the real part of Riemann zeta is related to the imaginary part by this formula: $$\Re\left(\zeta \left(\frac{1}{2}+i t\right)\right)=\frac{\Im\left(\zeta \left(\frac{1}{2}+i t\right)\...
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 ...
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-...
1
vote
0answers
67 views

Infinite Gamma Derivative Identity

We have $$ \Gamma(z)=\int_0^\infty x^{z-1}e^{-x}\;dx \tag{1} $$ We also have $$ \frac{d^n}{dz^n}\, x^{z-1}e^{-x}=\log(x)^n e^{-x}x^{z-1}, \;\; z>1 \tag{2} $$ If we create an operator $$ \hat{O}=\...
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. ...
0
votes
0answers
24 views

How to sum this vaguely zeta-looking function to QGammafunctions?

For $q>1$ and $s\ge{}1$, I'm trying to express $$\sum_{n=1}^\infty \dfrac{1}{(1-q^n)^s}$$ in terms of the Qpolygamma function. Just from its definition, it's clear to see that for $s$ the ...
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}\...
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 ...
-1
votes
1answer
519 views

Solve the system of equations in the maximum likelihood estimation of Gamma distribution parameters

I'm trying to calculate the two parameters of the Gamma distribution by solving the system of two equations obtained by differentiating the ...
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 ...
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 ...
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-...
0
votes
0answers
61 views

Does the following digamma/trigamma inequality hold? And can it be formally shown?

Suppose $x > \frac{1}{2}$. Let $\psi^{(0)}$ and $\psi^{(1)}$ denote the digamma and trigamma functions, respectively. Does the following inequality hold for any such $x$? $\psi^{(0)}(x) - \psi^{(0)...
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
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 $$\zeta '(-1, x)- \zeta '(-1, 1-x) $$
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}...
0
votes
1answer
87 views

On $\lim_{x\to 0}\frac{-(1+\sqrt{-x})\psi^{(0)}(1-\sqrt{-x})+(1+\sqrt{-x})\psi^{(0)}(\sqrt{-x}+1)+2\psi^{(0)}(1+x)}{2(\sqrt{-x}-1)(\sqrt{-x}+1)x}$

I've copy the identity in my Question from the solution of Wolfram Alpha online calculator. The expression is tedious to write thus I hope that there are no typos. When you type the code sum 1/((k+...
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}...
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. ...
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 ...
1
vote
0answers
66 views

Does this reduce down to the PolyGamma function?

Does this reduce down to the PolyGamma function? $H_n=$ $$\lim_{s\to 0} \, \left(-\frac{\left(\frac{1}{s}+1\right)^n (s+1)^{-n} \left(\sum _{k=0}^{\infty } \frac{\left(-\frac{1}{s}\right)^k \left(\...
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\...