Questions tagged [polygamma]
For questions about, or related to the polygamma function.
152
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Symbolic differentiation Gamma function (rings - algebra)
I was after an expression for the the nth derivative of the Gamma function and I managed to find this
Symbolic differentiation Gamma
which reads as
\begin{equation}
\Gamma^{(n)}(z) = \Gamma(z) R(n,z)
\...
- 13
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2
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Is it possible to find the $n$th derivative of Gamma function?
By repeatedly differentiating $\Gamma(x)$, I noticed that
$$\frac{d^{n}}{{dx}^{n}}\Gamma(x)=\sum_{k=0}^{n-1}\binom{n-1}{k}\psi^{(n-k-1)}(x)\,\frac{d^{k}}{{dx}^{k}}\Gamma(x),$$
where $\psi^{(a)}(x)$ is ...
- 25.8k
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Inequality involving trigamma function $\psi'$
By chance in my ongoing research, I discovered the following inequality,
$$a^2\psi'(ax-1) + b^2\psi'(bx-1)\geq 2\psi'(x-1),$$
where $\psi'$ is the trigamma function $\psi'(x) = \frac{\mathrm{d}^2}{\...
- 181
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How to prove this inequality involving trigamma functions?
While solving a problem I succeeded to reduce it to the following inequality:
$$
\forall \{a,b,z\in\mathbb R_+,\ a\ne b\}:\quad 0<\frac1{a-b}\int_0^\infty\frac{t(a^2e^{-azt}-b^2e^{-bzt})}{1-e^{-t}}...
- 26.7k
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Continued fraction of Laplace transform
I first learned of the below identity from MathWorld and the works of Ramanujan, but it's completely crazy with polygammas and Laplace transforms of hyperbolic trig. It seems weird that the Laplace ...
9
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Prove the zeta log gamma integral $\int_0^1{\zeta(1-n,1-t)\ln\Gamma{(t)}}dt = \frac{H_n\zeta(-n)+\zeta’(-n)}{n}$
How to prove that $\displaystyle\int_0^1{\zeta(1-n,1-t)\ln\Gamma{(t)}}dt = \frac{H_n\zeta(-n)+\zeta’(-n)}{n}$?
For integer values Wolfram Alpha gave me solutions to the integral in the form on the ...
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Does a closed-form expression exist for $ \int_0^\infty \ln(x) \operatorname{sech}(x)^n dx $?
I am trying to find a closed-form expression for the following integral
$$
\int_0^\infty \ln(x) \operatorname{sech}(x)^n dx
$$
There are specific values that I would like to generate
(Table of ...
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3
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Calculation of a derivative of a function related to the Euler Gamma function [closed]
Let $F$ be defined by
$
F(x)=\frac{\Gamma(\frac{1 + x}2)}{\sqrt π\ \Gamma(1 + \frac x2)}.
$
I believe that $F'(0)=-\ln 2$, but I do not have a proof. Is there an easy way to get that result?
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Series involving derivative of Riemann Zeta function: $\displaystyle \sum_{k=2}^{\infty}\zeta'(k)x^{k-1}$
1. Question
Could anyone recommend a useful method for approaching the following series?
$$\sum_{k=2}^{\infty}\zeta'(k)x^{k-1}$$
Where $\zeta(z)$ is the Riemann Zeta function.
I've seen that there are ...
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60
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Resummation of the following series
Recently, I encountered the following series:
\begin{equation}
\mathcal{I} = \sum_{q=1}^{\infty}\frac{\Gamma \left(q+\frac{1}{2}\right)^2 \left(2 q \psi
^{(0)}(q)-2 q \psi
^{(0)}\left(q+\frac{1}{...
3
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closed form for $\psi^{(-2)}\left(\frac{1}{4} \right)$
the polygamma function of negative order exist in a lot of complicated integrals like this integral:
$$\int_0^{\frac{1}{4}} x \psi(x) dx=\left(x \psi^{(-1)}(x) \right)^{\frac{1}{4}}_0-\int_0^{\frac{1}{...
- 1,637
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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 ...
- 11.5k
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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 ...
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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>...
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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 ...