Questions tagged [polygamma]

For questions about, or related to the polygamma function.

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Prove the zeta log gamma integral $\int_0^1{\zeta(1-n,1-t)\ln\Gamma{(t)}}dt = \frac{H_n\zeta(-n)+\zeta’(-n)}{n}$

How to prove that $\displaystyle\int_0^1{\zeta(1-n,1-t)\ln\Gamma{(t)}}dt = \frac{H_n\zeta(-n)+\zeta’(-n)}{n}$? For integer values Wolfram Alpha gave me solutions to the integral in the form on the ...
tyobrien's user avatar
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3 votes
1 answer
155 views

Does a closed-form expression exist for $ \int_0^\infty \ln(x) \operatorname{sech}(x)^n dx $?

I am trying to find a closed-form expression for the following integral $$ \int_0^\infty \ln(x) \operatorname{sech}(x)^n dx $$ There are specific values that I would like to generate (Table of ...
Gabriel Demirdag's user avatar
0 votes
3 answers
65 views

Calculation of a derivative of a function related to the Euler Gamma function [closed]

Let $F$ be defined by $ F(x)=\frac{\Gamma(\frac{1 + x}2)}{\sqrt π\ \Gamma(1 + \frac x2)}. $ I believe that $F'(0)=-\ln 2$, but I do not have a proof. Is there an easy way to get that result?
Bazin's user avatar
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4 votes
1 answer
431 views

Series involving derivative of Riemann Zeta function: $\displaystyle \sum_{k=2}^{\infty}\zeta'(k)x^{k-1}$

1. Question Could anyone recommend a useful method for approaching the following series? $$\sum_{k=2}^{\infty}\zeta'(k)x^{k-1}$$ Where $\zeta(z)$ is the Riemann Zeta function. I've seen that there are ...
Math Attack's user avatar
0 votes
0 answers
57 views

Resummation of the following series

Recently, I encountered the following series: \begin{equation} \mathcal{I} = \sum_{q=1}^{\infty}\frac{\Gamma \left(q+\frac{1}{2}\right)^2 \left(2 q \psi ^{(0)}(q)-2 q \psi ^{(0)}\left(q+\frac{1}{...
Alessandro Pini's user avatar
3 votes
1 answer
120 views

closed form for $\psi^{(-2)}\left(\frac{1}{4} \right)$

the polygamma function of negative order exist in a lot of complicated integrals like this integral: $$\int_0^{\frac{1}{4}} x \psi(x) dx=\left(x \psi^{(-1)}(x) \right)^{\frac{1}{4}}_0-\int_0^{\frac{1}{...
Faoler's user avatar
  • 1,189
0 votes
0 answers
29 views

Distributions of a product and of sums of products of iid, standard normal random variables.

It is known how to compute the distribution of a product of independent random variables. The results boils down to evaluating the inverse Mellin transform of a product of Mellin transforms of the ...
Przemo's user avatar
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4 votes
0 answers
72 views

Wolfram wishlist: series of $\Gamma(z)$ in $z=-m$. Cycle index of symmetric groups.

I found out that Wolfram has a wish list of formulas it is researching. The first point is "Series for the gamma function" We are searching for general formulas for the series expansion of ...
Math Attack's user avatar
1 vote
0 answers
16 views

Generalization of complete Bell polynomial $B_n(x_1,...,x_n)\mapsto B_{\nu}(f(\nu))$

I want to propose a question that may not have a solution Here I had asked a question: if it was possible to define the integral representation of the polygamma function $\psi^{(\nu)}(x)$ for $\nu>...
Math Attack's user avatar
1 vote
0 answers
42 views

Integral representation of $\psi^{(\nu)}(z)$ for $\nu>0$ but $\nu\not\in\mathbb{N}$

I need the integral representation of the polygamma function for $n>0$ but $n\not\in\mathbb{N}$. I searched both among Wolfram functions and Digital Library of Mathematical Functions On Wolfram I ...
Math Attack's user avatar
0 votes
0 answers
40 views

What is the intersection point of the graphs of digamma and trigamma functions?

We have the series expansions of digamma and trigamma functions for $x>0$, $$\psi^{(0)}(x)=-\gamma-\frac1x+\sum_{k=1}^{\infty}(\frac1k-\frac1{x+k})$$ and $$\psi^{(1)}(x)=\sum_{k=0}^{\infty}\frac{1}{...
Bob Dobbs's user avatar
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1 vote
1 answer
41 views

Asymptotic behavior of $\psi^{(n)}(1/2+ix)$ as $n\to\infty$?

I'm trying to determine the behavior of an asymptotic series whose $n$th term (where $n$ is even, and all odd terms vanish) is proportional to $$\psi^{(n)}\left(\frac12+ix\right),$$ where $\psi^{(n)}$ ...
WillG's user avatar
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2 votes
1 answer
156 views

Integral representation of Bell Polynomial?

From Wolfram Alpha: https://functions.wolfram.com/IntegerFunctions/BellB/07/01/0001/, we have an integral representation for Bell numbers as: $B_n = \frac{2n!}{\pi e} \displaystyle{\int_{0}^{\pi}} e^{...
BBadman's user avatar
  • 167
1 vote
1 answer
325 views

Closed form expression for an integral

Let $\psi_q(z)$ be the q-DiGamma function defined for a complex variable $z$ with $\Re(z)>0$ as $$\psi_q(z)=\frac{1}{\Gamma_q(z)}\frac{\partial}{\partial z} (\Gamma_q(z))$$ where $\Gamma_q(z)$ is ...
Max's user avatar
  • 346
4 votes
1 answer
143 views

Closed form expression for $\psi_{e^{\pi}}^{(3)}(1-i)$

Let $\psi_q(z)$ be the q-DiGamma function defined for a real variable $\Re(z)>0$ as $$\psi_q(z)=\frac{1}{\Gamma_q(z)}\frac{\partial}{\partial z} (\Gamma_q(z))$$ where $\Gamma_q(z)$ is the q-Gamma ...
Max's user avatar
  • 346
0 votes
1 answer
131 views

Closed form expression for $\psi_{e^{\pi}}^{(3)}(1)$

Let $\psi_q(x)$ be the q-DiGamma function defined for a real variable $x>0$ as $$\psi_q(x)=\frac{1}{\Gamma_q(x)}\frac{\partial}{\partial x} (\Gamma_q(x))$$ where $\Gamma_q(x)$ is the q-Gamma ...
Max's user avatar
  • 346
2 votes
1 answer
57 views

Can somebody show me how to derive the identity relating the polygamma function and the lerch transcendent

How do u derive this identity below? $$\Phi(-1,m+1,z)=\frac{1}{(-2)^{m+1}m!}\left(\psi^{(m)}\left(\frac{z}{2}\right)-\psi^{(m)}\left(\frac{z+1}{2}\right)\right)$$ If Lerch transcendent is defined as ...
Richie's user avatar
  • 49
1 vote
1 answer
80 views

Computing solutions to polygamma function of order 1?

I want to preface this with the fact that I am WAY out of my depth with my mathematical familiarity with these topics. While trying to figure out how to compute the solution to the polygamma function ...
YaGoi Root's user avatar
0 votes
1 answer
121 views

Closed form for $\rm{Li }_2\left( -{\frac {i\sqrt {3}}{3}} \right)$

In my personal research with Maple i find this closed form : $$\operatorname{Li }_2\left( -{\frac {i\sqrt {3}}{3}} \right)={\frac {{\pi}^{2}}{24}}+{\frac {\ln \left( 2 \right) \ln \left( 3 \right) }...
Dens's user avatar
  • 303
3 votes
1 answer
114 views

Sum of Fermi-Dirac integrals with opposite chemical potentials: closed form (Le Bellac eq. 1.13)

I am trying to reproduce the result of eq. (1.13) in Le Bellac's Thermal Field Theory book to compute the grand canonical potential of a gas of massless fermions: $$ Ω = - \frac{V T^4}{6 π^2} \int_0^\...
ShineOn's user avatar
  • 31
1 vote
0 answers
107 views

Evaluate the limit with the Appell F1 function

I encountered the limit with the form $$ \lim_{r\rightarrow 1^+}\frac{1-\frac{1}{r^{d-3}}}{e^{2\kappa r_s(r)}},\quad r_s(r)\equiv r~ F_1\left(-\frac{1}{d-3};-\frac12,1;\frac{d-4}{d-3},-\frac{Q}{r^{d-3}...
Louis Chou's user avatar
7 votes
2 answers
213 views

Great difficulty in finding the residues of $\frac{\mathrm{Log}\Gamma\left(\frac{z+ai}{2\pi i}\right)}{\cosh(z)+1}$

$\newcommand{\log}{\operatorname{Log}}\newcommand{\res}{\operatorname{Res}}\newcommand{\d}{\mathrm{d}}$Let $\Lambda(z)=\log\Gamma(z)$, $a\gt0$, let $\psi$ denote digamma. It is written here, Page 49, ...
FShrike's user avatar
  • 38.9k
1 vote
0 answers
70 views

Simplification of a difficult identity involving the digamma function

EDIT: The question here is not for the reader to laboriously scan all the working, but rather to suggest ways to continue the train of thought; for example, there is perhaps a closed form of the ...
FShrike's user avatar
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6 votes
3 answers
365 views

In an attempt to find $I = \int_0^\infty \frac{t}{e^t-1}dt$

I was trying to solve $I = \int_0^\infty \frac{t}{e^t-1}dt$ My approach I took the more general form of integral $f(s) = \int_0^{\infty}\frac{e^{-st}}{e^t-1}dt$ the same way as How to evaluate the ...
user avatar
4 votes
1 answer
216 views

New trigamma identity for $\Psi_1(\frac3{20})+6\,\Psi_1(\frac15)+10\,\Psi_1(\frac25)-\Psi_1(\frac1{20})$

I play with Maple, and I find this relation for the trigamma function: $$\begin{align} \Psi_1\left({\frac{3}{20}}\right)+6\,\Psi_1\left(\frac15\right)+ 10\,\Psi_1\left(\frac25 \right)-\Psi_1\left(\...
Dens's user avatar
  • 303
2 votes
0 answers
61 views

Polygamma sum problem ...

Hello guys i have a problem evaluating the following sum $$\sum_{n=1}^{+\infty}\frac{n(n+1)}{2}\frac{4x(3\pi ^{2}(n+1)^{2}+x^{2})}{(x^{2}-\pi ^{2}(n+1)^{2})^{3}}$$ It is obviously of the polygamma ...
Vuk Stojiljkovic's user avatar
0 votes
1 answer
58 views

Polygamma sum problem

I have a problem evaluating the following sum, $$\sum_{n=1}^{+\infty}\frac{4nx(3\pi ^{2}(n+1)^{2}+x^{2})}{(x^{2}-\pi ^{2}(n+1)^{2})^{3}}$$ The sum obviously is of the form of a polygamma function. ...
Vuk Stojiljkovic's user avatar
5 votes
0 answers
116 views

Ask for a proof of logarithmically complete monotonicity of a power-exponential function involving the difference of the psi and logarithmic functions

It is common knowledge that the classical Euler gamma function $\Gamma(z)$ can defined by \begin{equation*} \Gamma(z)=\int^\infty_0t^{z-1} e^{-t}\textrm{d}t, \quad \Re(z)>0 \end{equation*} and the ...
qifeng618's user avatar
  • 1,611
8 votes
1 answer
492 views

What is the general formula of the sum $\sum_{k=0}^{n}(-1)^{k} \binom{n}{k}\binom{k/2}{m}$ for $m,n\in\mathbb{N}$?

The classical Euler's gamma function $\Gamma(z)$ can be defined by \begin{equation} \Gamma(z)=\lim_{n\to\infty}\frac{n!n^z}{\prod_{k=0}^n(z+k)}, \quad z\in\mathbb{C}\setminus\{0,-1,-2,\dotsc\}. \end{...
qifeng618's user avatar
  • 1,611
1 vote
0 answers
59 views

closed form in terms of a polygamma function? $ f(2,2)=\sum \sum \exp\big(- n^2k^2 \big)? $

Is there a closed form for:$$ f(2,2)=\sum_{n=1}^\infty \sum_{k=1}^\infty \exp\big(- n^2k^2 \big)? $$ Note that I'm defining a function: $$f(x,y)=\sum_{n=1}^\infty \sum_{k=1}^\infty\exp\big(-n^xk^y\...
John Zimmerman's user avatar
2 votes
1 answer
123 views

A curious summation.

Months ago I was interested in calculating new infinite summations and was able to make a nice summation and also find its closed form as follows : $$\sum \limits_{n=0}^{\infty } \frac{(-1)^{0+1+2+..+....
Amrit Awasthi's user avatar
8 votes
6 answers
375 views

Evaluation of $\sum_{n=1}^{\infty}\frac{1}{n(2n+1)}=2-2\ln(2)$

I came across the following statements $$\sum_{n=1}^{\infty} \frac{1}{n(2 n+1)}=2-2\ln 2 \qquad \tag{1}$$ $$\sum_{n=1}^{\infty} \frac{1}{n(3 n+1)}=3-\frac{3 \ln 3}{2}-\frac{\pi}{2 \sqrt{3}} \qquad \...
Ricardo770's user avatar
  • 2,751
0 votes
1 answer
49 views

Is there a general summation formula for the polygamma function at $z=1/2$? i.e $\psi^{(s)}(\frac{1}{2})$ for all s.

For $s>0$ one has $ \psi^{(s)}(\frac{1}{2}) = s! \cdot \zeta(s+1, \frac{1}{2}) \cdot (-1)^{s+1} $. E.g. $ \psi^{(1)}(\frac{1}{2}) = 3 \cdot \zeta(2) $ $ \psi^{(2)}(\frac{1}{2}) = -14 \cdot \zeta(...
DecarbonatedOdes's user avatar
1 vote
1 answer
49 views

a limit containing a digamma function

I wonder if the following limit is correct and how to prove it $$ \lim _{x \rightarrow+\infty} \frac{4}{p^{2}} x^{4}\left(\frac{1}{x^{2}}+\frac{2}{x}(\log 2+\Psi(x))+\Psi^{\prime}(x)\right)>0 $$ ...
Bin Wang's user avatar
1 vote
2 answers
119 views

Simplify this expression $e^{\psi\left(\frac12+\frac{i}{2\sqrt{3}}\right)+\psi\left(\frac12-\frac{i}{2\sqrt{3}}\right)}$?

Is it possible to simplify this constant expression $e^{\psi\left(\frac12+\frac{i}{2\sqrt{3}}\right)+\psi\left(\frac12-\frac{i}{2\sqrt{3}}\right)}$? Here $\psi(x)$ is digamma function. Particularly, ...
Anixx's user avatar
  • 9,040
4 votes
2 answers
187 views

A convergent series for the Trigamma function $\psi_1(n) =\sum_{k=n}^{\infty} \frac1{k^2} $

I just came up with the following convergent series for the Trigamma function defined by $\psi_1(n) =\sum_{k=n}^{\infty} \frac1{k^2} $. \begin{align*} \psi_1(n) &=\lim_{m \to \infty} \sum_{j=1}^m \...
marty cohen's user avatar
2 votes
1 answer
207 views

Another bizarre sum involving a binomial coefficient and inverse powers of integers.

In the attempt to answer Binomial identity involving Harmonic numbers we stumbled on the following problem. Let $i\ge 0 $ and $k \ge i+2 $ and $p \ge 1$ be integers. Consider a following sum: \begin{...
Przemo's user avatar
  • 11.2k
0 votes
0 answers
80 views

Partial sum of polygamma

Do you know what kind of process use mathwolfram in determining \begin{align} &\sum_{m = 1}^{n}\left[\frac{1}{-s + m} + \frac{1}{s + m}\right] \\[3mm] = &\ \Psi_{0}\left(-s + n + 1\right) + \...
G.M.'s user avatar
  • 9
0 votes
1 answer
87 views

Polygamma function approximation to infinity

Be $\psi_0(s)$ the polygamma function of order zero in $s \in C$. Do you think is correct to write $ \psi_{0}(s) + \psi_{0}(-s) = \sum_{n=1}^{\infty} \frac{1}{-s+n} + \sum_{n=1}^{\infty} \frac{1}{s+n} ...
G.M.'s user avatar
  • 9
0 votes
1 answer
237 views

Is there a decomposition for the digamma function as a sum of digamma functions?

Let $\psi(x)$ denote the digamma function $$ \psi(x)=\Gamma(x)\frac{\partial}{\partial x} \Gamma(x). $$ Consider $x=x_1 +x_2+\dots +x_m$, where $x_j>0$, for $j=1, \ldots,m$. Is there any formula to ...
Jack London's user avatar
  • 1,736
2 votes
1 answer
58 views

What is the root of $\sum _{i=1}^{m-1} i^k=m^k$ given integer k?

Let $m'$ be the root of $\sum _{i=1}^{m-1} i^k=m^k$ given integer $k$ (solving for $m$). What is $m'$?
Andrey Mitin's user avatar
2 votes
1 answer
97 views

Derivatives of Polygamma Functions

I would like to know if there's a quicker way to verify: $$\partial_z^{n-1}\psi(tz)t^n = \partial_z^n\ln[\Gamma(tz)], \,\,\\ n \in \mathbb{N}^+, t \in \mathbb{C}_+\tag{1}\label{1}$$ That's true for $t=...
Mr. N's user avatar
  • 516
1 vote
1 answer
184 views

Upper bound for the nth derivative of $\Gamma(x)^n$

I was trying to find an upper bound for $$ \frac{d^n}{ds^n} \Gamma(s)^{n}|_{s=1}$$ yet, I only get the bound for the nth derivative of gamma, as follow: First, the integral of the nth derivative of ...
Marcos Benício's user avatar
3 votes
2 answers
212 views

Prove $\lim_{n\mapsto 0}[(\psi(n)+\gamma)\psi^{(1)}(n)-\frac12\psi^{(2)}(n)]=2\zeta(3)$

How to prove that $$\lim_{n\mapsto 0}[(\psi(n)+\gamma)\psi^{(1)}(n)-\frac12\psi^{(2)}(n)]=2\zeta(3)\ ?$$ I encountered this limit while I was trying to solve $\int_0^1\frac{\ln x\ln(1-x)}{x(1-x)}dx$ ...
Ali Shadhar's user avatar
  • 25.2k
0 votes
2 answers
177 views

How can we derive the asymptotic expansion for the second derivative of the gamma-function?

We can expression the first derivative of the gamma function as: $$\Gamma'(s) \sim -\frac{1}{s^2}+\frac{6\gamma^2+\pi^2}{12}+O(s)$$ but what about the second derivative? I do not know how to approach ...
zalm's user avatar
  • 125
0 votes
0 answers
70 views

Log gamma integral with a slightly more complicated polynomial factor

I need some help figuring out how to evaluate this integral. $$\frac{1}{n!}\int_0^1 (x-t)^n \ln\Gamma(t) \, dt$$ A similar integral is known due to the work of Victor Adamchik. But I haven’t been ...
tyobrien's user avatar
  • 3,469
1 vote
3 answers
159 views

How to Prove : $ \gamma +\ln\left(\frac{\pi}{4}\right) = \sum_{n=2}^{\infty} \frac{(-1)^{n} \zeta{(n)}}{2^{n-1}n} $

How to Prove : $$ \gamma +\ln\left(\frac{\pi}{4}\right) = \sum_{n=2}^{\infty} \frac{(-1)^{n} \zeta{(n)}}{2^{n-1}n} $$ I have tried looking at Series definitions of the Polygamma function from which ...
No-one Important's user avatar
0 votes
1 answer
86 views

Find the function $f(x)=\sum_{n=1}^{\infty}\frac{H_{n-1}(-x)^n}{n!}$

I want to find the function defines by : $$f(x)=\sum_{n=1}^{\infty}\frac{H_{n-1}(-x)^n}{n!}$$ Where $H_n=1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}$ is the Harmonic series. My work We have the ...
DesmosTutu's user avatar
  • 3,054
1 vote
3 answers
91 views

How can we evaluate $\sum_{k\geq 0} \frac{1}{(2k+1)^3}$?

I have been looking to evaluate $$\mathcal{A} = \sum_{k=0}^\infty \frac{1}{(2k+1)^3}.$$ We can represent our sum in terms of the Hurwitz zeta function; namely, $$\mathcal{A} = \zeta\left(\frac{1}{2},...
user avatar
1 vote
2 answers
66 views

Decomposition of $\psi^{(n)}(1)$ in terms of $\psi^{(n)}(k)$

Accidentally run into this identity: \begin{align} \psi^{(n)}(1) &= 2^{n+1}\, \sum_{k = 2}^\infty (-1)^k\,\psi^{(n)}(k) \tag{1}\label{1} , \end{align} its variation \begin{align} 2^{-n-1} &...
g.kov's user avatar
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