Questions tagged [polygamma]

For questions about, or related to the polygamma function.

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486 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 $?. ...
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53 views

Two cool sums: Compute $\sum_{n=1}^\infty (-1)^{n}\frac{H_{2n+1}}{(2n+1)^3}$ and $\sum_{n=1}^\infty (-1)^{n}\frac{H_{2n+1}^{(2)}}{(2n+1)^2}$

How to prove $$S_1=\sum_{n=1}^\infty (-1)^{n-1}\frac{H_{2n+1}}{(2n+1)^3}=1+\frac{35}{128}\pi\zeta(3)+\frac{1}{48}\zeta(4)-\frac1{384}\psi^{(3)}\left(\frac14\right)$$ $$S_2=\sum_{n=1}^\infty (-...
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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} ...
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1answer
49 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}...
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82 views

An identity on Polygamma

I would like to know how to prove that: $$\psi^{(n)}(z)=(-1)^{n+1}n! \sum_{k=0}^{\infty}\frac{1}{\left ( k+z \right )^{n+1}}$$ I know that $\displaystyle \sum_{n=0}^{\infty}\frac{1}{n+z}=-\psi (z)$ ...
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1answer
153 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}...
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39 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 $x>1$.
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52 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} \...
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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}=\...
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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 ...
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68 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(\...
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68 views

Are these identities Newton series?

Newton series is the following expansion of a function: $$f(x)=\sum_{k=0}^\infty \binom{x}k \Delta^k [f]\left (0\right)=\sum_{n=0}^{\infty} {x\choose n} \sum_{k=0}^n{n\choose k}(-1)^{k-n}f(k)$$ Now ...
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171 views

Solving an integral (or series) equations system

Peace be upon you, In the question A late-diverging "approximating solution" for a system of functional equations, I have asked for an approximating solution for a system of functional ...
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1answer
81 views

Function related to Harmonic numbers, the Pascal triangle, Logarithmic integral and the Polylogarithm.

What function satisfies the following: Let the matrix: $$\displaystyle T = \left(\begin{matrix} 1&0&0&0&0&0&0&\cdots \\ 1&1&0&0&0&0&0 \\ 1&1&...
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88 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+...
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94 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 ...
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1answer
55 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 ...
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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 ...
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135 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 ...
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1answer
242 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)=...
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113 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+...
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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(...
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1answer
214 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)\...
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1answer
52 views

Solve for X in the given Equation(Gamma Curve)

I'm having a set of points in the form: ${(\frac A{255})}^{\frac 1x}=\frac B{255}$ I need to Find $x$ in the Equation.Where $A$ & $B$ are set of constants ranging from $0$ to $255$. Please ...
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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?
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37 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 $\epsilon$ ...
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25 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 ...
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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)...
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27 views

How can I get the value of k?

I have to get the value of k in this equation: $\frac{(\lambda T)^k [ln(\lambda T)-\psi(k+1)]}{\Gamma(k+1)}=0$, where $\psi$ is the digamma function. Since $\Gamma(k+1)$ is in the denominator and the ...
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527 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 ...