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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How to prove negativity of a $3\times3$ determinant whose elements involve trigamma, tetragamma, and pentagamma functions?

The classical Euler gamma function can be defined by the integral \begin{equation*} \Gamma(z)=\int_0^{\infty}t^{z-1}\operatorname{e}^{-t}\operatorname{d}t, \quad \Re(z)>0. \end{equation*} Its ...
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How to extend this sum involving generalized harmonic numbers?

It is well-known since Euler that the Generalized harmonic numbers, defined for $n\in\mathbb N$ by $$H_n^{(r)}=\sum_{k=1}^n\frac1{k^r},$$ can be naturally extended for non integer $n$ in terms of ...
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Polygamma function in mathematical physics

Are there situations in which the polygamma pops up naturally in a mathematical physics context? In particular: are there examples of potentials having some interest for which the dependence on the ...
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How to prove the convexity of a simple function involving a ratio of two polygamma functions?

Let \begin{equation*} \Gamma(z)=\int_0^{\infty}t^{z-1}\textrm{e}^{-t}\textrm{d}t, \quad \Re(z)>0 \end{equation*} and $$ \psi(z)=[\ln\Gamma(z)]'=\frac{\Gamma'(z)}{\Gamma(z)}. $$ In the literature, ...
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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 ...
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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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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 $g(x)$ ...
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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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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 ...
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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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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 ...
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