Questions tagged [gamma-function]
used only for functions based on gamma, not functions with some obscure relation to gamma
146
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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 ...
- 59
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vote
2
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Power series of ratio of Gamma functions
Let $a>1$ and define $G_a(x)=\sum\limits_{n=0}^{+\infty} \frac{\Gamma(\frac{2n+1}{a})}{\Gamma(2n+1)\Gamma(\frac{1}{a})}x^n$ where $\Gamma$ is the Gamma function. This series is convergent on $\...
- 29
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answers
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Motivation behind the Bohr-Mollerup Theorem relating the Gamma function
In Wikipedia, it states about the Bohr-Mollerup Theorem:
The theorem was first published in a textbook on complex analysis, as Bohr and Mollerup thought it had already been proved.
If anyone knows, ...
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Stirling's formula and a Gamma function relation
I am trying to understand a paper by by A. Booker on poles of Artin $L$-functions where in one of the lemmas he uses the following identity, derived using Stirling's formula:
$$
\frac{\Gamma(s/2)^2}{2^...
- 11
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239
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Extended binomial coefficients and the gamma function
For which $(a,b,n) \in \mathbb{Z}^3$ satisfying $a+b=n$ does $\frac{\Gamma(z+1)}{\Gamma(x+1)\Gamma(y+1)}$ approach a limit as $(x,y,z) \rightarrow (a,b,n)$ in $\mathbb{C}^3$, and what is that limit? (...
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A question on (trigonometric) prime counting function and twin prime counting function
(I myself don't think the following question is suitable for this forum but as it contains something related to twin primes, I asked here.Please help accordingly)
Consider the following sum:
$$S(t)=\...
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Addition formulas for q-analogs of trigonometric functions
Sine and Cosine functions possess notable formulas for addition of angles
$$
\sin(a+b) = \sin(a)\cos(b) + \cos(a)\sin(b) \qquad \text{or} \qquad \cos(a+b) = \cos(a)\cos(b) - \sin(a)\sin(b).
$$
One can ...
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answer
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Gamma function and the somewhat extended version of Bohr-Mollerup theorem
The Gamma function $\Gamma$ is defined by
\begin{equation*}
\Gamma(x)=\int_{0}^\infty t^{x-1}e^{-t} \,\mathrm{d}t,
\end{equation*}
for $x>0$. It satisfies the well-known functional equation
$$\...
- 143
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0
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Nature of function as $x\rightarrow\infty$
I'm studying the limits and applicability of Abel Plana summation for different test functions (class of functions). In doing so this just pops out and couldn't handle the said integral so asked here (...
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Simplification of $\sum_{m=0}^\infty \text B_z(m+a,b-m)x^m,\sum_{m=0}^\infty \frac{\text B_z(m+a,b-m)x^m}{m!}$ in terms of Kampé de Fériet function
Here is the goal sum where the Pochhammer Symbol with the Incomplete Beta function series
$$\sum_{m=0}^\infty \frac{\text B_z(m+a,b-m)x^m}{m!}=\sum_{m=0}^\infty z^{m+a}\sum_{n=0}^\infty\frac{(1-(b-m))...
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Sum of ratio of Gamma functions
I was wondering what the following expression
$$\frac{\Gamma(F+1)}{\Gamma(F+\alpha)}\frac{\Gamma(\alpha)}{\Gamma(\alpha/K)^K}\sum_{\sum_k F_k^C = F^C} \prod_{k = 1}^K\frac{\Gamma(F_k^+ + F_k^C+\alpha/...
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The Basel problem revisited?
In the Basel problem, the $sinc$ function is considered at the Wikipedia page.
Let me try to make an alternative function definition:
$$f(x) = \prod_{n=1}^\infty \left ( 1+ \frac{x^3}{n^3} \right ) = \...
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$\sum_{k=0}^{\infty} {\frac{(-1)^{k} \Gamma(\frac{2k+n+m}{2})\,\Gamma(\frac{2k+m-1}{2})}{k!\Gamma(k+\frac{m}{2})}} z^{2k}$ is an elementary function
I try to calculate the following series
\begin{align*}
S_{n,m}(z)=\sum_{k=0}^{\infty} {\frac{(-1)^{k} \Gamma(\frac{2k+n+m}{2})\,\Gamma(\frac{2k+m-1}{2})}{k!\Gamma(k+\frac{m}{2})}} \, z^{2k},
\end{...
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Upper bound for the complex Beta function
The question is almost the same as here.
What is the upper bound for a complex Beta function $\displaystyle B(s,z)=\frac{\Gamma(s) \Gamma(z)}{\Gamma(s+z)}$ with $0<Re(s)<1$ and $0<Re(z)<1$;...
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1
answer
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Sum of reciprocal of Pochhamer symbols through multiples of a natural L
In a question in StackExchange (https://math.stackexchange.com/questions/4236635/sum-of-quotients-of-gamma-functions), I asked if there is a closed expression for the following sum related with a ...
1
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1
answer
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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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Upper bound an integral
I am trying to upper bound the following integral $$\int_{0}^{1} \left| (1-(x-a^2))^n - (1-(x-b)^2)^n \right|dx,$$ where $x \in [0,1]$, $a$ and $b$ are fixed constants, and $n$ is a possitive integer.
...
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Beta function, harmonic numbers, and integral values
A problem of Mathematical Physics that I am working on involves the computation of a certain integral. Part of the result reads:
$$
I_k:= [\beta_x( -1 - k, 0) + H_{-2 - k}]x^k
$$
where $\beta_x( -1 - ...
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votes
2
answers
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Hankel determinant of incomplete gamma functions
I have some expressions that involve Hankel determinants of incomplete gamma functions. They are of the ($r \times r$ form)
I'd like to evaluate these determinants. Elementary operations help, but ...
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answers
158
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definite integral with incomplete gamma function and exponential
While working with electron density computations in quantum chemistry, I encountered the following improper integral:
$$
I(k, n) = \int\limits_0^\infty \Gamma\left(\frac{3}{n},\ k r^n\right) r \exp(-k ...
- 31
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votes
3
answers
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Method to evaluate an infinite sum of ratio of Gamma functions (how does Mathematica do it?)
This question arose from Amdeberhan's question, the evaluation of a double integral, which can be reduced to the evaluation of this series:
$$\sum _{n=0}^{\infty } \frac{\Gamma \left(n+\frac{1}{2}\...
- 147k
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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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how to numerically evaluate $\int_{0}^{\infty} \frac{1}{x!} dx$ [closed]
So I was graphing the equation $ y=\frac{1}{x!} $ for $ x \geq 0$ and tried the integral:
$$\int_{0}^{\infty} \frac{1}{x!} dx$$
$$\int_{0}^{\infty} \frac{1}{\Gamma(x+1)} dx$$
$$\int_{0}^{\infty} \frac{...
- 57
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Analytic continuation of convergent integral
I was trying to solve the following integral:
$$I = \oint _{|z|=1}\frac{dz}{2 \pi i z}\int_{0}^{\infty} dr \dfrac{e^{-\tfrac{r^2}{z^2}}r^{2n+1}}{z^2(z-1)} $$
The singular structure in the $z$ ...
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An interesting infinite product involving the factorial function with connection to the K and gamma function
I have posted this question in StackExchange, but it didn't get any answers there. This question is important for my research. I got stuck on an infinite product which even WolframAlpha can't answer. ...
user167505
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Is $\Gamma(s, x=s-1)/\Gamma(s)$ decreasing for real $s>1$? Is $\Gamma(s, x=s)/\Gamma(s)$ increasing?
This has received no full solution at StackExchange.
As per https://dlmf.nist.gov/8.10#E13 we have
$$\frac{\Gamma\left(n,n\right)}{\Gamma\left(n\right)}<\frac{1}{2}<\frac{\Gamma%
\left(n,n-1\...
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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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answers
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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 ...
- 17.2k
2
votes
2
answers
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The exact constant in the simple bound of the fraction of Gamma functions
In the question : Upper bound of the fraction of Gamma functions the asymptotic upper bound for the fraction of Gamma functions have been established:
$$\left(\frac{\Gamma(a+b)}{a\Gamma(a)\Gamma(b)}\...
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1
answer
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Inequalities involving Gamma function
I am having some difficulty in proving the following inequality:
\begin{equation*}
\frac{1-e^{-\gamma b}}{b^\eta}-\frac{1-e^{-\gamma s}}{s^\eta}\geq \gamma(1-\eta)\int^b_sy^{-\eta}e^{-\gamma y}dy
\end{...
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Closed form of $\prod_{i=0}^{N}\big(i!\big)^{{N}\choose{i}}$
I have made a question here about closed form of the following:
$$\prod_{k=0}^{N}\big(k!\big)^{{N}\choose{k}}$$
I know that there is a known closed form for,
$$\prod_{i=0}^{N}\big(i!\big)=G(N+2)$$
...
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votes
2
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Interpolating asymptotic expression for logarithm of middle binomial sums
Define $S(k,2n)=\sum_{i=-k}^k\binom{2n}{n+i}$ at every $k\in\{0,\dots,n\}$.
We know $$\ln(S(\gamma\mbox{ } n^\gamma,2n))\asymp(2n\ln2-\frac12\ln\pi-\frac12\ln n)$$ at $\gamma\rightarrow0$ and $$\ln(S(...
- 1,708
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Logarithm of an integral involving generalized real binomial coefficients
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\...
- 1,708
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1
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152
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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 ...
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votes
1
answer
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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 ...
- 105
1
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1
answer
161
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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 $d=3$) ...
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votes
1
answer
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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 ...
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votes
1
answer
287
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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 ...
user147095
2
votes
0
answers
137
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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{...
- 1,513
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0
answers
101
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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:
$$\Upsilon(s,\...
- 11
1
vote
1
answer
142
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A generalization of gamma function
For $\alpha\in\mathbb{C}$, I defined the "complex-weighted" Hurwitz zeta function
\begin{eqnarray*}\displaystyle \zeta^{\alpha}(s,w)=\frac{1}{\Gamma(s)}\int_0^{\infty} \frac{e^{-wt}}{(1-e^{-t})^{\...
- 11
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votes
1
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554
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On the integral $\int_0^1\log(x!)dx$ revisited
I was interested in an integral that I known from [1], it is
$$\int_0^1 \log(x!)dx.$$
I tried to get such closed-form using myself ideas and symbolic calculations, also with the help of Wolfram ...
- 37
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votes
1
answer
122
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Is there a name for this family of integral?
This one: $\int_{0}^{\bar{x}}e^{-x^{a}}x^{b}(1-x)^{c}dx,a,b,c\ge0$. When $a=1,c=0,\bar{x}=\infty$ it is the gamma function.
- 61
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votes
2
answers
178
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Stirling's approximation for normalized $\Gamma$
Let
$$
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)}{\...
- 189
1
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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-...
- 5,580
1
vote
1
answer
139
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An equation with Gamma Euler function in critical strip
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 ...
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