# 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 ...
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I could not find a closed form for this integral although I think it should have been studied. What is a good approximation to $I$ in $$I=\ln\Bigg(\int_{0}^y\binom{2m}{m(1+x)}dx\Bigg)$$ where $0\... 0 votes 1 answer 152 views ### The distribution of the power of the sum of inner products of two independent complex normal vectors If I have$\mathbf x_n=[x_0, x_1,... ,x_K]^T$and$\mathbf y_n=[y_0, y_2, ..., y_K]^T$, where$x,y\sim\mathcal C\mathcal N(\mathbf 0,\sigma^2\mathbf I)$. What is the distribution of the following ... 0 votes 1 answer 288 views ### The distribution of the sum of inner products of two independent complex normal vectors If I have$\mathbf x_n=[x_0, x_1,... ,x_K]^T$and$\mathbf y_n=[y_0, y_2, ..., y_K]^T$, where$x,y\sim\mathcal C\mathcal N(\mathbf 0,\sigma^2\mathbf I)$. What is the distribution of the following ... 1 vote 1 answer 161 views ### Asympotics of$\int_0^{\infty} \left[ Q(m,x)\right]^d dx$I'm interested in the asympotic behaviour (assuming an exact solution is intractable) of $$I(m,d)=\int_0^{\infty} \left[ Q(m,x)\right]^d dx$$ for fixed$d \in \mathbb{N}$(in particular, for$d=3$) ... 8 votes 1 answer 289 views ### Converse of a result of Koblitz and Ogus on algebraic products of gamma values Let$n$be a positive integer,$a_1,\ldots,a_{n-1}\in\mathbb{Z}$. Suppose that for every$u\in(\mathbb{Z}/n\mathbb{Z})^\times$, $$\tag{\star} \sum_{i=1}^{n-1} i a_{(ui\!\!\!\mod n)} = 0.$$ Then ... 3 votes 1 answer 287 views ### Real and imaginary parts of$\ln \Gamma(i b)$The imaginary part of the digamma function when its argument is pure imaginary is known as $$\Im\psi(\mathrm{i}b)=\frac{1}{2}b^{-1}+\frac{1}{2}\pi\coth{\pi b},$$ and its real part is much more ... 2 votes 0 answers 137 views ### Monotone coupling between "two-sided Gumbel" distributions I am interested in finding a monotone coupling between two random variables. Let$\alpha_1>\alpha_2$,$b<a$. Define the following two (non-normalized) densities on the whole real line: \begin{... 1 vote 0 answers 101 views ### The complex roots of$\left(\Gamma(\kappa s)+2^{1-s} \pi^{-s}\cos\left(\frac{\pi s}{2} \right)\Gamma(s)\,\Gamma(\kappa(1-s))\right)$Question: With$\kappa \in \mathbb{R}, \kappa \ge 1$, the function: $$\Upsilon(s,\kappa) = \Gamma(\kappa s)\zeta(s)+\Gamma\left(\kappa(1-s)\right) \zeta(1-s)$$ could be rewritten as: $$\Upsilon(s,\... 1 vote 1 answer 142 views ### A generalization of gamma function For \alpha\in\mathbb{C}, I defined the "complex-weighted" Hurwitz zeta function \begin{eqnarray*}\displaystyle \zeta^{\alpha}(s,w)=\frac{1}{\Gamma(s)}\int_0^{\infty} \frac{e^{-wt}}{(1-e^{-t})^{\... 5 votes 1 answer 554 views ### On the integral \int_0^1\log(x!)dx revisited I was interested in an integral that I known from , it is$$\int_0^1 \log(x!)dx.$$I tried to get such closed-form using myself ideas and symbolic calculations, also with the help of Wolfram ... -1 votes 1 answer 122 views ### Is there a name for this family of integral? This one: \int_{0}^{\bar{x}}e^{-x^{a}}x^{b}(1-x)^{c}dx,a,b,c\ge0. When a=1,c=0,\bar{x}=\infty it is the gamma function. 0 votes 2 answers 178 views ### Stirling's approximation for normalized \Gamma Let$$ H(s)=\frac{1}{2}s(1-s)\pi^{-s/2}\Gamma\left(\frac{s}{2}\right). $$Using Stirling's approximation for the Gamma function I would like to prove that$$ \frac{H(1/2+it)\overline{H}(1/2+it+iu)}{\... 1 vote 1 answer 148 views ### Good upper bound for$\Gamma(1-b,\log(a))-\Gamma(1-b,N\log(a))$, where$a,b \in (0, 1)$and$N \ge 1$Let$a,b \in (0, 1)$and$N \ge 1$, and consider the incomplete gamma function$x \mapsto \Gamma(1-a,x)$. Question Is there a simple bound (involving 'simple function's) for the expression$\Gamma(1-...
Let $$D=\{z \in \mathbb{C} : 0<\Re(z)<\frac{1}{2} \text{ or } \frac{1}{2}<\Re(z)<1 \}$$ that is the critical strip without critical line. I have to find if the following equation, with ...