Questions tagged [polygamma]
For questions about, or related to the polygamma function.
132
questions
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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 ...
- 25
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0
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39
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How do you solve for x in this equation which includes the polygamma function.
How do you solve for x given:
$$
0 = 2x+18 - \int_{0}^{\infty}\left(t^x * e^\left(-t\right) * ln(t)\right)
$$
The background is I have this function:
$f\left(x\right)\ =x^{2}+\ 19x\ -\ x!$
I took the ...
0
votes
1
answer
64
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Closed form for $\rm{Li }_2\left( -{\frac {i\sqrt {3}}{3}} \right)$
In my personal research with Maple i find this closed form :
$$\operatorname{Li }_2\left( -{\frac {i\sqrt {3}}{3}} \right)={\frac {{\pi}^{2}}{24}}+{\frac {\ln \left( 2 \right) \ln \left( 3
\right) }...
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Sum of Fermi-Dirac integrals with opposite chemical potentials: closed form (Le Bellac eq. 1.13)
I am trying to reproduce the result of eq. (1.13) in Le Bellac's Thermal Field Theory book to compute the grand canonical potential of a gas of massless fermions:
$$
Ω = - \frac{V T^4}{6 π^2} \int_0^\...
- 31
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Evaluate the limit with the Appell F1 function
I encountered the limit with the form
$$
\lim_{r\rightarrow 1^+}\frac{1-\frac{1}{r^{d-3}}}{e^{2\kappa r_s(r)}},\quad r_s(r)\equiv r~ F_1\left(-\frac{1}{d-3};-\frac12,1;\frac{d-4}{d-3},-\frac{Q}{r^{d-3}...
- 11
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Great difficulty in finding the residues of $\frac{\mathrm{Log}\Gamma\left(\frac{z+ai}{2\pi i}\right)}{\cosh(z)+1}$
$\newcommand{\log}{\operatorname{Log}}\newcommand{\res}{\operatorname{Res}}\newcommand{\d}{\mathrm{d}}$Let $\Lambda(z)=\log\Gamma(z)$, $a\gt0$, let $\psi$ denote digamma.
It is written here, Page 49, ...
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Simplification of a difficult identity involving the digamma function
EDIT:
The question here is not for the reader to laboriously scan all the working, but rather to suggest ways to continue the train of thought; for example, there is perhaps a closed form of the ...
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3
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In an attempt to find $I = \int_0^\infty \frac{t}{e^t-1}dt$
I was trying to solve $I = \int_0^\infty \frac{t}{e^t-1}dt$
My approach
I took the more general form of integral $f(s) = \int_0^{\infty}\frac{e^{-st}}{e^t-1}dt$ the same way as How to evaluate the ...
user986763
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New trigamma identity for $\Psi_1(\frac3{20})+6\,\Psi_1(\frac15)+10\,\Psi_1(\frac25)-\Psi_1(\frac1{20})$
I play with Maple, and I find this relation for the trigamma function:
$$\begin{align}
\Psi_1\left({\frac{3}{20}}\right)+6\,\Psi_1\left(\frac15\right)+
10\,\Psi_1\left(\frac25 \right)-\Psi_1\left(\...
- 193
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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 ...
- 358
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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. ...
- 358
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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 ...
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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{...
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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: $$f(x,y)=\sum_{n=1}^\infty \sum_{k=1}^\infty\exp\big(-n^xk^y\...
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A curious summation.
Months ago I was interested in calculating new infinite summations and was able to make a nice summation and also find its closed form as follows :
$$\sum \limits_{n=0}^{\infty } \frac{(-1)^{0+1+2+..+....
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Evaluation of $\sum_{n=1}^{\infty}\frac{1}{n(2n+1)}=2-2\ln(2)$
I came across the following statements
$$\sum_{n=1}^{\infty} \frac{1}{n(2 n+1)}=2-2\ln 2 \qquad \tag{1}$$
$$\sum_{n=1}^{\infty} \frac{1}{n(3 n+1)}=3-\frac{3 \ln 3}{2}-\frac{\pi}{2 \sqrt{3}} \qquad \...
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Is there a general summation formula for the polygamma function at $z=1/2$? i.e $\psi^{(s)}(\frac{1}{2})$ for all s.
For $s>0$ one has $ \psi^{(s)}(\frac{1}{2}) = s! \cdot \zeta(s+1, \frac{1}{2}) \cdot (-1)^{s+1} $.
E.g.
$ \psi^{(1)}(\frac{1}{2}) = 3 \cdot \zeta(2) $
$ \psi^{(2)}(\frac{1}{2}) = -14 \cdot \zeta(...
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a limit containing a digamma function
I wonder if the following limit is correct and how to prove it
$$
\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
$$
...
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2
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Simplify this expression $e^{\psi\left(\frac12+\frac{i}{2\sqrt{3}}\right)+\psi\left(\frac12-\frac{i}{2\sqrt{3}}\right)}$?
Is it possible to simplify this constant expression $e^{\psi\left(\frac12+\frac{i}{2\sqrt{3}}\right)+\psi\left(\frac12-\frac{i}{2\sqrt{3}}\right)}$?
Here $\psi(x)$ is digamma function.
Particularly, ...
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A convergent series for the Trigamma function $\psi_1(n) =\sum_{k=n}^{\infty} \frac1{k^2} $
I just came up with
the following
convergent series
for the Trigamma function
defined by
$\psi_1(n)
=\sum_{k=n}^{\infty} \frac1{k^2}
$.
\begin{align*}
\psi_1(n)
&=\lim_{m \to \infty} \sum_{j=1}^m \...
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Another bizarre sum involving a binomial coefficient and inverse powers of integers.
In the attempt to answer Binomial identity involving Harmonic numbers we stumbled on the following problem.
Let $i\ge 0 $ and $k \ge i+2 $ and $p \ge 1$ be integers. Consider a following sum:
\begin{...
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Partial sum of polygamma
Do you know what kind of process use mathwolfram in determining
\begin{align}
&\sum_{m = 1}^{n}\left[\frac{1}{-s + m} + \frac{1}{s + m}\right]
\\[3mm] = &\ \Psi_{0}\left(-s + n + 1\right) + \...
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Polygamma function approximation to infinity
Be $\psi_0(s)$ the polygamma function of order zero in $s \in C$. Do you think is correct to write
$
\psi_{0}(s) + \psi_{0}(-s) = \sum_{n=1}^{\infty} \frac{1}{-s+n} + \sum_{n=1}^{\infty} \frac{1}{s+n} ...
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Is there a decomposition for the digamma function as a sum of digamma functions?
Let $\psi(x)$ denote the digamma function
$$
\psi(x)=\Gamma(x)\frac{\partial}{\partial x} \Gamma(x).
$$
Consider $x=x_1 +x_2+\dots +x_m$, where $x_j>0$, for $j=1, \ldots,m$. Is there any formula to ...
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What is the root of $\sum _{i=1}^{m-1} i^k=m^k$ given integer k?
Let $m'$ be the root of $\sum _{i=1}^{m-1} i^k=m^k$ given integer $k$ (solving for $m$).
What is $m'$?
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Closed formulae of polygamma function of negative order
Wolfram Alpha seems to output nice closed forms for the polygamma function of negative orders with whole and half arguments. I was interested in order $-2$ and I deduced the following
$$\psi^{(-2)}\...
- 3,193
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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=...
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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 ...
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2
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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 $\int_0^1\frac{\ln x\ln(1-x)}{x(1-x)}dx$ ...
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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 ...
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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 ...
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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 ...
- 1,154
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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 ...
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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,
$$\mathcal{A} = \zeta\left(\frac{1}{2},...
user753116
1
vote
2
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Decomposition of $\psi^{(n)}(1)$ in terms of $\psi^{(n)}(k)$
Accidentally run into this identity:
\begin{align}
\psi^{(n)}(1) &=
2^{n+1}\,
\sum_{k = 2}^\infty
(-1)^k\,\psi^{(n)}(k)
\tag{1}\label{1}
,
\end{align}
its variation
\begin{align}
2^{-n-1} &...
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2
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1
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Analytic continuation of $H_x^{(k)}=\sum_{n=1}^x \frac{1}{n^k}$?
Is there an analytic continuation of the generalised harmonic number $H_x^{(k)}=\sum_{n=1}^x \frac{1}{n^k}$ to the positive reals x, for $k>1$?
I can’t find anything useful through Google, just ...
1
vote
2
answers
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Prove $\lim_{x\to\infty}\sum_{n=1}^x x\log\left(1+\frac1{xn(an+1)}\right)= H_{\frac1a}$
How to prove that
$$\large\lim_{x\to\infty}\sum_{n=1}^x x\log\left(1+\frac1{xn(an+ 1)}\right)= H_{\frac1a}, \quad a\in \mathbb{R},\ |a|>1$$
This question is a formulated form of this problem.
...
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7
votes
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Evaluate $\int _0^{2 \pi }\int _0^{2 \pi }\log (3-\cos (x+y)-\cos (x)-\cos (y))dxdy$
How to prove
$$\small\int _0^{2 \pi }\int _0^{2 \pi }\log (3-\cos (x+y)-\cos (x)-\cos (y))dxdy=
-4 \pi ^2 \left(\frac{\pi }{\sqrt{3}}+\log (2)-\frac{\psi ^{(1)}\left(\frac{1}{6}\right)}{2 \sqrt{3} \pi ...
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Sum involving 2nd antiderivative of the digamma function.
Help evaluate the following sum:
$$\sum_{n=1}^{\infty}\left(\frac{1}{4n-1}-\frac{1}{4n-1}\ln\left(2n-1\right)+\frac{4}{\left(4n-1\right)\left(4n-3\right)}\left(\psi^{(-2)}\left(2n-\frac{1}{2}\right)-\...
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votes
3
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Is the closed form for $\sum_{k=1}^\infty\frac{\overline{H}_k}{k^m}$ known in the literature?
I managed to find
$$\sum_{k=1}^\infty\frac{\overline{H}_k}{k^m}=(1-2^{-m})\sum_{k=1}^\infty\frac{H_k}{k^m}-2^{-m}\sum_{k=1}^\infty\frac{H_k}{(k+1/2)^m}$$
$$=(1-2^{-m})\left[\left(1+\frac m2\...
- 22k
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vote
3
answers
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$\int_0^{\pi/2} \sec^a(t)\,dt= \frac{\sqrt{\pi}}{2\Gamma\left(1-\frac{a}{2}\right)}\Gamma\left(\dfrac{1-a}{2}\right)$
Inside the Wolfram Documentation page for the secant function, an identity is given which involves the gamma function, polygamma function, and Catalan's constant.
Notes on documentation page:
Some ...
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1
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Reference request: $q$-gamma, $q$-polygamma, $q$-Pochhammer.
I'm trying to solve a problem related to a few difficult series and am using Mathematica to hammer out the difficult bits. The problem is that QPochhammer,...
5
votes
3
answers
313
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Evaluation of $\int\limits ^{\infty }_{0}\frac{x}{\left( x^{2} +1\right)^2\left( e^{tx} +1\right)} dx$
I want to show that
$$\int\limits ^{\infty }_{0}\frac{x}{\left( x^{2} +1\right)^2\left( e^{tx} +1\right)} dx=\frac{\psi^{(1)}(\frac{t}{2\pi})-4\psi^{(1)}(\frac{t}{\pi})}{8\pi}t+\frac14
$$
where $\...
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votes
0
answers
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correct property of a Gamma function?
On Wikipedia, https://en.wikipedia.org/wiki/Gamma_function, I read that the following hold true for any positive integer $n$:
\begin{aligned}\Gamma \left({\tfrac {1}{2}}+n\right)&={(2n)! \over 4^{...
- 309
4
votes
1
answer
56
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The limit of the ratio of polygamma functions
I want to calculate this quantity:
$$\lim_{x \rightarrow \infty}\frac{\Psi_1 (x)}{\Psi_1 (x + y)}$$
where
$$\Psi_1 (x)=\frac{d^2}{dx^2}\log \Gamma (x)=\sum_{k=0}^{\infty}\frac{1}{(x+k)^2}. $$
I ...
- 43
2
votes
2
answers
122
views
Evaluation of a digamma series involving golden-ratio
Let $\varphi =\frac{1}{2} \left(\sqrt{5}+1\right), a=\tan \left(\frac{\sqrt{5} \pi }{2}\right)$, then how can one prove
$$\sum _{n=1}^{\infty } \frac{\psi ^{(0)}(n+\varphi)-\psi ^{(0)}\left(n-\frac{1}{...
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votes
1
answer
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Limit involving Polygamma function: $\lim_{x\to \infty} \sum_{k = 0}^\infty \frac{2x^2}{(x + k)^3}$
I don't remember where I got this identity: $$\lim_{x\to \infty} [(-1)^{k + 1} x^k \psi^{(k)}(x)] = (k - 1)!$$
where $\psi^{(k)}$ is the polygamma function.
I just need to find the limit above for $k ...
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4
votes
2
answers
455
views
The closed-form of $\sum_{n=0}^{\infty}\frac{(-1)^n H^{(2)}_{n}}{(2n+1)^2} $
How to Prove that
$$ \sum_{n=0}^{\infty}\frac{(-1)^nH^{(2)}_{n}}{(2n+1)^2} \;\;=\;\;\frac{7 \pi \; \zeta(3)}{4}-\frac{\zeta(2)G}{2}+\frac{45\zeta(4)}{8}-\frac{\Psi^{(3)}\big(\frac{1}{4}\big)}{...
- 476
6
votes
3
answers
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views
Integrating $\int_0^1\frac{\ln^2x\ln(1+x)}{1+x^2} dx$ using real methods
How to evaluate, without contour integration the following integral:
$$I=\int_0^1\frac{\ln^2x\ln(1+x)}{1+x^2}\ dx\ ?$$
@Cody mentioned in this solution that
$$I=\frac{\pi^{2}}{6}G+\frac{\pi^{3}}{...
- 22k
2
votes
1
answer
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Prove $\lim_{n\ \mapsto\ \infty}4^n \psi^{(2)}(2^n)-\psi^{(2)}(1)=2\zeta(3)-1$
Prove
$$\lim_{n\ \mapsto\ \infty}4^n \psi^{(2)}(2^n)-\psi^{(2)}(1)=2\zeta(3)-1\tag{1}$$
where $\psi^{(m)}(x)$ is the polygamma function and $\zeta$ is the Riemann zeta function.
This problem ...
- 22k