# Questions tagged [polygamma]

For questions about, or related to the polygamma function.

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### 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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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}... 1answer 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$\displaystyle \sum_{n=0}^{\infty}\frac{1}{n+z}=-\psi (z)... 1answer 153 views ### Double series of Harmonic Numbers In a solution presented here a series involving the product of Harmonic numbers is involved. The intent of the problem is to determine a form of the series \begin{align} \sum_{n=1}^{\infty} \frac{H_{n}... 0answers 39 views ### Digamma function question I just learned online about Polygamma functions, and I want to know whatx$equals (and how to get it) when$\psi(x)=1$and$x>1$. 0answers 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: \int_0^\infty \frac{ \ln(t)}{t^n + 1}\:dt = -\frac{\pi^2}{n^2} \... 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 $$\hat{O}=\... 0answers 104 views ### Convexity of reciprocal polygamma Is the reciprocal of polygamma functions of odd order convex on \mathbb{R^+}, while that of even order above 0 concave? Plotting the functions suggest so, but I've been trying for days to come up ... 0answers 68 views ### Does this reduce down to the PolyGamma function? Does this reduce down to the PolyGamma function? H_n=$$\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(\... 0answers 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 ... 0answers 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 ... 1answer 81 views ### Function related to Harmonic numbers, the Pascal triangle, Logarithmic integral and the Polylogarithm. What function satisfies the following: Let the matrix: $$\displaystyle T = \left(\begin{matrix} 1&0&0&0&0&0&0&\cdots \\ 1&1&0&0&0&0&0 \\ 1&1&... 1answer 88 views ### On \lim_{x\to 0}\frac{-(1+\sqrt{-x})\psi^{(0)}(1-\sqrt{-x})+(1+\sqrt{-x})\psi^{(0)}(\sqrt{-x}+1)+2\psi^{(0)}(1+x)}{2(\sqrt{-x}-1)(\sqrt{-x}+1)x} I've copy the identity in my Question from the solution of Wolfram Alpha online calculator. The expression is tedious to write thus I hope that there are no typos. When you type the code sum 1/((k+... 1answer 94 views ### Recurrence relation for the polygamma function of negative order? I know the recurrence relation for the Polygamma function is$$\psi^{(m)}(x+1)=\psi^{(m)}(x)+\frac{(-1)^mm!}{x^{m+1}}$$Does such a recurrence formula exist for negative integer m? I am using the ... 1answer 55 views ### Logarithmic Sum Is there a closed form for the following sum?$$\sum_{n=0}^{\infty}\sum_{m=1}^{\infty}(-1)^{m+n}\frac{\ln(m+n)}{(m+n)}$$According to https://www.mathmash.org/contestprob.php?prob=227 it has a closed ... 1answer 378 views ### Bounds on the real and imaginary parts of the digamma function \psi Let \psi be the digamma function given by$$\psi (z)=\left.\frac {d}{dt}\log\Gamma (t)\right|_{t=z}. $$I wonder does anyone know of any lower and/or upper bounds on the real and imaginary parts ... 1answer 135 views ### \sum_{n=1}^{4000000} \frac{1}{n^3} very quick. Some days ago I have tried to find the sum of the first milion terms of the infinite sum \zeta(3) = \sum_{n=1}^\infty\frac{1}{n^3} (Apéry's constant) on Wolfram Programming Lab (Open Cloud), an ... 1answer 242 views ### Series Representation of Gamma Function The \Gamma(x) is function That has derivatives in the polygamma form. Can those derivatives be used to make a Taylor series? I've tried but I got stuck as soon as I find out That \Psi^1(1)=\zeta(2)=... 1answer 113 views ### Evaluate an infinite series involving the polygamma function OR first derivative of the hurwitz zeta function Can we find a closed form for$$\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+... 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 $$\psi^{(-2)}\left(x\right)-\psi^{(-2)}\left(... 1answer 214 views ### Are the real and imaginary parts of Riemann zeta equal to each other by multiplication of a Riemann Siegel theta function expression? It appears that the real part of Riemann zeta is related to the imaginary part by this formula:$$\Re\left(\zeta \left(\frac{1}{2}+i t\right)\right)=\frac{\Im\left(\zeta \left(\frac{1}{2}+i t\right)\... 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 ... 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? 1answer 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$\epsilon$... 0answers 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 ... 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)...
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 ...