Questions tagged [poly-gamma-function]
The polygamma function may be represented as $$\begin{align} \psi^{(m)}(z)&= (-1)^{m+1}\int_0^\infty\frac{t^m e^{-zt}} {1-e^{-t}}\ dt\\ &=-\int_0^1\frac{t^{z-1}}{1-t}\ln^mt\ dt \end{align}$$ which holds for $Re z >0$ and $m > 0$. For $m = 0$ see the digamma function definition.
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Class of differentiated Gamma functions: are there any algebras where they are elementary?
There is a class of differentiated gamma functions (DGFs) that basically can be expressed via derivatives of Gamma function.
They include the Gamma function, Polygamma function, and Hurwitz Zeta ...
4
votes
1answer
359 views
On the integral $\int_0^1\log(x!)dx$ revisited
I was interested in an integral that I known from [1], it is
$$\int_0^1 \log(x!)dx.$$
I tried to get such closed-form using myself ideas and symbolic calculations, also with the help of Wolfram ...
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3answers
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A challenging inequality that involves the digamma function and polygamma functions
Let $f(x)=x \psi(x+1)$, where $\psi$ is the digamma function. Define
$$g(x)=(f(ax)+f((1-a)x)-f(x))-(f(ax+by)+f((1-a)x+(1-b)y)-f(x+y)),$$
where $0\le a,b\le 1$ and $x,y\ge 0$.
How to show that $g(x)$ ...
4
votes
1answer
200 views
Proving two inequalities involving the gamma and digamma functions
I'm having trouble proving the following inequality:
$$\forall p>1 \quad \forall m\geq 0 \quad \dfrac{m^2\Gamma(\dfrac{2m}{p})\Gamma(\dfrac{2m}{q})}{\Gamma(\dfrac{2m+2}{p})\Gamma(\dfrac{2m+2}{q})}\...
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1answer
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Two integral representations for $\zeta(3)$ from Zurab's integral and standard formulas for the gamma function
This morning I wrote with the help of a CAS, and integral representation for the Apéry's constant $\zeta(3)$ and some standard formulas two formulas involving this constant. I would like to know if ...
2
votes
0answers
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Generalized hypergeometric function at $z=1$
I wonder if there is a closed formula for the following generalized hypergeometric function at $z=1$:
$${}_4F_3\left(a,a,a,a;a+1,a+1,2a;1\right)$$
Specifically, I would like to have a formula in ...
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The inverse of the digamma function
The gamma function is increases on the interval $(x_0, \infty),$ where $x_0$ denotes the unique zero of the digamma function on the positive half line.
The inverse function of gamma function defined ...
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Closed form for sum involving digamma? [closed]
Let $\Gamma(n)$ be Euler's Gamma function and $\psi_0$ = $\frac{\Gamma'(n)}{\Gamma(n)}$ be the Digamma function.
Is there a closed form for
$$\sum_{n=1}^{\infty} \frac{\psi_0(n)}{n^2}=?$$
I've done ...
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2answers
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Closed form for $\sum_{n=1}^{\infty} \frac{1}{1+n+n^2+\cdots+n^a}$
I am looking at a certain sequence, and consequently I am wondering if anyone knows if there happens to exist a closed form solution for this sum:
$$\sum_{n=1}^{\infty} \frac{1}{1+n+n^2+\cdots+n^a}$$
...
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Are all complex zeros of $\Psi^{(s)}(1) \pm \Psi^{(1-s)}(1)$ on the critical line?
The balanced polygamma function $\Psi^{(s)}(x)$ for $x=1$ can be expressed as:
$$\Psi^{(s)}(1)=\dfrac{\big(\Psi(-s)+\gamma\big)\,\zeta(s+1)+\zeta'(s+1)}{\Gamma(-s)}$$
Note $\Psi(s)$ is the digamma ...
