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# Questions tagged [gamma-function]

used only for functions based on gamma, not functions with some obscure relation to gamma

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### Inverse Mellin transform of 3 gamma functions product

I want to calculate the inverse Mellin transform of products of 3 gamma functions. $$F\left ( s \right )=\frac{1}{2i\pi}\int \Gamma(s)\Gamma (2s+a)\Gamma( 2s+b)x^{-s}ds$$ Above contour integral has ...
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### Hankel determinant of incomplete gamma functions

I have some expressions that involve Hankel determinants of incomplete gamma functions. They are of the ($r \times r$ form) I'd like to evaluate these determinants. Elementary operations help, but ...
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### Computing the $p$-adic gamma function $\Gamma_p$

Let $p>2$ be a prime. For $n \in \mathbb{Z}^+$ we can define \begin{equation} F(n) = (-1)^n \prod_{1<i<n, p\nmid i} i. \end{equation} Since $\mathbb{Z}$ is dense in $\mathbb{Z}_p$, we can ...
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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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### Inverse mellin transform

Let $K_1(t)$ be the K-Bessel function, then we have $\int_{0}^{\infty}K_v(y)y^s\frac{dy}{y}=2^{s-2}\Gamma(\frac{s+v}{2})\Gamma(\frac{s-v}{2})$ See page 106 of Bump's book Automorphic forms and ...
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### What is the value of this product (with gamma and zeta function)?

Since $\zeta(s)=0$ and $\Gamma(s)$ has poles at even negative numbers $-2, -4, -6...$ Can we find a value for $\frac{1}{\zeta(s)\Gamma(s)}$ using L'Hôpital's rule?
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### Any simplification of this inequality if it is true? :For $t\geq 1.22$: $|\zeta(0.5+it)|\leq 0.5 \frac{|\Gamma(0.5+it)|}{|\Gamma(-0.5+it)|}$

When I tried to give bounds for $\zeta(0.5+it)$ using some transformations over Gamma function using the function $f(x)=\exp(-n x)$ over the range $(0,+\infty)$ , For $Re(s)=\frac12$ and $t >0$ ...
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### Reference request: proofs of integrals presented in Erdélyi's *Table of Integral Transforms*

I have asked related questions on MSE and received no answer, so: I am looking for proofs of the integrals presented in Erdélyi's Table of Integral Transforms vol. i-ii, specifically proofs of the ...
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### Formula for volume of $n$-ball for negative $n$

Does the expression $$\frac{\pi^{\frac{n}{2}}}{\Gamma(\frac{n}{2}+1)}R^n,$$ which gives the volume of an $n$-dimensional ball of radius $R$ when $n$ is a nonnegative integer, have any known ...