Questions tagged [gamma-function]

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

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39 views

How can I show that this product is equal to a product of Gamma functions? [migrated]

$\prod_{n=0}^{x-1}\left( 1+\frac{a}{an+b}\right) = \frac{\Gamma\left(\frac{a}{b}\right)\Gamma\left(x+\frac{a+b}{b}\right)}{\Gamma\left(\frac{a+b}{b}\right)\Gamma\left(x+\frac{a}{b}\right)}$ I found ...
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1answer
146 views

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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531 views

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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2answers
584 views

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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0answers
33 views

Bounding the working precision required in Spouge's Approximation

Spouge's approximation for the gamma function is $\Gamma(z+1) = (z+a)^{z+\frac{1}{2}}e^{-z-a} \left(c_0 + \sum_{k=1}^{a-1} \frac{c_k}{z+k} + \epsilon_a(z) \right)$ where the coefficients are given ...
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2answers
92 views

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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1answer
113 views

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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2answers
281 views

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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189 views

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(...
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1answer
150 views

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\...
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1answer
86 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 ...
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1answer
76 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 ...
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1answer
127 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 $d=3$) ...
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1answer
171 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 ...
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1answer
162 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 ...
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132 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{...
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93 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,\...
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1answer
107 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})^{\...
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1answer
312 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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1answer
112 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.
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2answers
164 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)}{\...
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1answer
100 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-...
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1answer
124 views

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 ...
4
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1answer
191 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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150 views

Gosper's Beta function identities

According to the last paragraph in Mathworld's Beta function article http://mathworld.wolfram.com/BetaFunction.html, Gosper found some multiplication formulas for the Beta function, but it does not ...
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2answers
436 views

A tantalizing Gamma quotient to challenge the Rohrlich-Lang Conjecture

The Rohrlich-Lang Conjecture for polynomial relations in Gamma values predicts that all polynomial relations between Gamma values over $\mathbb Q$ come from the functional equations satisfied by the ...
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1answer
228 views

Complicated bound after using Stirling's approximation

I have this inequality $$\frac{1}{a}\exp\bigl\{-\frac{4}{h^2}\bigr\} \geq \frac{1}{f}$$ where $$ a \leq \Bigl(\pi^{d/2}\Gamma(\frac{1}{2}d+1)^{-1} + 1\Bigr) \left(\frac{h^{d+1}}{2} \Gamma \left(\frac{...
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1answer
323 views

Upper bound of the fraction of gamma functions

Is there a simple upper bound of the following fraction of gamma functions for any $a,b\geq1/2$: $$\left(\frac{\Gamma(a+b)}{a\Gamma(a)\Gamma(b)}\right)^{1/a}$$ An upper bound in the following form is ...
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0answers
111 views

Brylinski Beta Function Calculation

I've recently read the paper written by Brylinski on the Beta function of a knot, where he gave the example of a trivial "circular" knot. Having a physics background, and not being formally introduced ...
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3answers
187 views

Estimate for the binomial coefficients and bounds from below for the Beta function

Let $n\ge p\in \mathbb N$ and let $\binom{n}{p}$ be the binomial coefficient. I believe that $$ \binom{n}{p}\le 2^n\sqrt\frac{2}{π n}. $$ Question: is that true? Of course I would like it as a non-...
2
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1answer
142 views

Monotonicity of the regularized incomplete gamma function

In the theory of chi-squared distribution in statistics, for the random variable $X$ following $\chi^2 (k)$, the probability that $X$ is lower than its expactation is given by $$ P(X\le k) = \frac{\...
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0answers
105 views

Relation between Gross-Koblitz and Chowla-Selberg formulas

The Chowla-Selberg formula relates the eta function with values of the gamma function at rational numbers. The eta function appears, at least in the proofs I have seen, related to values of $L$-...
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1answer
500 views

Series involving factorials

Playing around with this series for natural values of $a,b$, it appears that more generally for $c\in\mathbb N$, $$\sum_{k=0}^\infty \frac{ (a+k)! \ (b+k)!}{k!\ (a+b+c+ k+1)! }=\frac{a!\ b!\ (c-1)!}{...
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2answers
435 views

Solution of the functional equation $f(x+1)=g(x)f(x)$

In the paper (http://www.sciencedirect.com/science/article/pii/S0022247X97953439), Webster obtained a unique solution of the functional equation $f(x+1)=g(x)f(x)$ (where $f,g:\mathbb{R}^+\rightarrow \...
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0answers
116 views

The relationship between $\Gamma (a)$ and $\Gamma (a+\frac{1}{2})$ [closed]

Does anybody know the relationship between the $\Gamma (a)$ and $\Gamma (a+\frac{1}{2})$? Is there an equation or an approximate equality between these two?
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102 views

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 ...
3
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1answer
235 views

A generalized logarithmic function

Consider the function $$f_\epsilon(x):=\int_0^1 \frac{(1-z)^x-1}{\log(1-\epsilon z)}\, dz,$$ defined for a parameter $\epsilon \in (0,1]$ and $x \geq 0$. When $\epsilon=1$, this is $\log(1+x)$. One ...
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0answers
460 views

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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2answers
589 views

Intuition behind the Riemann $\zeta$ functional equation

Let $\Lambda(s)=\pi^{-s/2} \Gamma\left(\frac{s}{2}\right)\zeta(s).$ Then $\Lambda\left(\frac{1}{2} + s\right) = \Lambda\left(\frac{1}{2} - s\right)$ constitutes the functional equation for the Riemann ...
5
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3answers
531 views

Asymptotic behavior of integral with gamma functions

Consider the following function defined for complex numbers $z\in\mathbb{C}$ with $\Re(z)\geq \frac{1}{2}$: $$F(z)=\frac{1}{5^{\Re(z)}}\int_0^\infty \left| \frac{\Gamma(z+ix)\Gamma(z-ix)}{\Gamma(z)^2}...
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0answers
151 views

Reflection Formulas for the $\Gamma$ Function

We have $$\begin{align} &\Gamma\Big(1~+~x\Big)~\cdot~\Gamma\Big(1-x\Big)~=~\frac{\pi x}{\sin\pi x} \\\\ &\Gamma\Big(\tfrac12+x\Big)~\cdot~\Gamma\Big(\tfrac12-x\Big)~=~\frac\pi{\cos\pi x} \\\...
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2answers
221 views

An integral identity relate to the Gamma function or the Beta function

I encountered the following identity in a paper on number theory, $$\int_{-\infty}^{\infty}\frac{dW}{(W+i)^{\frac{3}{2}}(W^2+1)^s}=\frac{e^\frac{-3\pi i}{4}\sqrt{2}\pi \Gamma(2s+\frac{1}{2})}{2^{2s}\...
9
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1answer
379 views

“unexpected” residue formula for $\Gamma^3(s)/(\Gamma(3s)(e^{2\pi is}-1)) $

There is a related problem in my current work: to find the residue of the following function at any negative integer $s=-n$: $$f(s)=\frac{\Gamma^3(s)}{\Gamma(3s)(e^{2\pi is}-1)}$$ It seems to be a ...
2
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0answers
458 views

About the Stirling's approximation of the incomplete gamma function $\left|\gamma\left(a+ib,z\right)\right|$

Let $a+ib$ be a complex number. It is well-known that $$\left|\Gamma\left(a+ib\right)\right|\sim\sqrt{2\pi}e^{-\pi\left|b\right|/2}\left|b\right|^{a-1/2}$$ for any fixed $a$ and $\left|b\right|\...
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1answer
103 views

Independence of Gamma and Beta random variables with common term

Given $\textbf{P}$ independent and identically distributed random variables, $X_1, X_2, ..., X_P \sim \Gamma(M,2c)$ how can we prove that: $$U = X_1 + X_2 + ... + X_P$$ and $$V = \frac{X_1}{X_1 + ...
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0answers
295 views

Generalization of Trigonometric, Hyperbolic, $\Gamma$ and $\zeta$ Functions

It is becoming increasingly clear that the expression $~\displaystyle\prod_{n\in\mathbf M}\left[1-\frac{z^s}{(n-a)^s}\right],~$ with $~|\mathbf M|=\aleph_0,~$ $a\not\in\mathbf M,~$ and $~\Re(s)>1,...
5
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0answers
351 views

integrating with respect to parameters in beta function

I would like to evaluate an integral: $$\int_t^1\frac{1}{B(1+s\phi,1+\phi(1-s))}p^{s\phi}(1-p)^{\phi(1-s)}ds,$$ where $B(a,b)$ is a beta function and $p\in(0,1)$ and $\phi>0$ are some parameters. ...
1
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1answer
271 views

Is this a valid way of generating the non-trivial zeros of $\zeta(s)$?

This question is migrated from this one on MSE and rephrased more simply. In this question the following closed form was derived, with $0 < \sigma<1$ and $\sigma,x \in \mathbb{R}$: $$ \frac{1}...
-1
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1answer
144 views

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 ...
2
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0answers
89 views

Gamma function in terms of a linear function

I have noticed that the plot of the function $[\Gamma(x+1)]^{1/x}$ for $x > 0$ looks like a straight line at all scales. This implies that $\Gamma(x+1) \approx ((1-e^{-\gamma})x+e^{-\gamma})^x$ for ...