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4
votes
1answer
151 views

Conjectured bound on Kummer's function (confluent hypergeometric function)

I believe the following bound to be correct: ${}_1F_1(-a;1+a;-z) a z^{-a} \gamma(a,z) \leq 1$ for real-valued a > 0 and z $\geq 0$. $\gamma(a,z)$ is the lower incomplete gamma function. Apart from ...
3
votes
1answer
133 views

Inequality for a gamma function

Let $s=\sigma+it$ and $\Gamma(s)$ be the Euler gamma function. Does the inequality holds? $$ \left|\frac{\Gamma(s)}{\Gamma(2-s)}\right|\leq |s|^{2(\sigma-1)},\, 1<\sigma<2,\, t>t_0>0. $$ ...
9
votes
0answers
449 views

Are the twin primes the only positive double zeros of this real function?

Agno's answer was extremely helpful. For $x \in \mathbb{R}, x \ge 1$ define $$ f(x) = \sin\frac{\pi(\Gamma(x)+1)}{\lfloor x \rfloor}$$ By Wilson's theorem the positive integer zeros of $f(x)$ are ...
3
votes
0answers
212 views

Alternating sums of powers of the lngamma ($\small f_p(x) = \sum \log(k!)^p x^k $ at $\small x=-1$)

I'm still fiddling with this recent question and come to a detail, whether I can find closed forms for the sums of the $\small lngamma() $ function. Precisely $$ \small f_p(x) = \sum_{k=0}^{\infty} ...
3
votes
0answers
143 views

Iterated Kumaraswamy distributions

The Kumaraswamy distribution has cdf $F(x;a,b) = 1-(1-x^a)^b$. Does anyone know any formulas or properties relating to iterations of this on itself, meaning $$ F_i(x;a,b) = 1-(1-F_{i-1}^a)^b$$ If ...
2
votes
0answers
133 views

Sum of binomial coefficients weighted by a lower incomplete regularized gamma function

The following summation turned up in the course of my research: $$S_n=\sum_{k=0}^n {n \choose k}\lambda^k P(k,t)$$ where $P(k,t)=\frac{1}{\Gamma(k)}\int_{0}^t e^{-x}x^{k-1}dx$ is the lower ...
2
votes
0answers
280 views

The Riemann Zeta Function summing over the Gamma Function

Has anyone studied a function of the form $$\eta(s) = \sum_{n=1}^{\infty} \frac{1}{\Gamma(n)^{s}} = \sum_{n=0}^{\infty}\frac{1}{k!^s}$$ This series is appearing in my research on the volumetric ...
2
votes
0answers
187 views

Coutour Integral of Gamma Functions

How do I solve the Integral $$ \frac{1}{2\pi j} \oint \frac{b^{ - s} \Gamma[2 + i - s] \Gamma[s] \Gamma[-1 - i + s]}{ (2 + i - s) \Gamma[3 + i - s]} \:\mathrm{d}s$$ This integral is an inverse ...
2
votes
0answers
332 views

Can any antidifference (indefinite sum) of a function be expressed in elementary functions and generalized polygamma function if its integral can be expressed in elementary functions?

If the integral or multiplicative integral of a function can be expressed with elementary functions, does it mean its indefinite sum (antidifference) or indefinite product respectively can be ...
1
vote
0answers
177 views

Convergence of $\sum_{n=1}^\infty \frac{\psi^{(1)}(1-n/\pi)}{n^3}$

Related to an open problem about another series. Set $$A= \sum_{n=1}^\infty \frac{\psi^{(1)}(1-n/\pi)}{n^3}$$ where $\psi^{(n)}(k)$ is the polygamma function. Does $A$ converge? The related ...
1
vote
0answers
227 views

Poles of products of Gamma functions

I want to know if there can be a general statement about the poles (Laurent expansion) of such products of Gamma functions as a function of $p \in \mathbb{R}$ in the limit $\epsilon \rightarrow 0$, ...
1
vote
0answers
437 views

Derivative of the regularized upper incomplete gamma function

Hello everyone! I have a question about the derivative of the regularized upper incomplete gamma function. Considering the gamma function and the incomplete gamma function \begin{eqnarray} ...
1
vote
0answers
190 views

Does $e^{az}/\Gamma(z)dz$ have a nice indefinite integral? Definite integral?

Now obviously we can just expand $\frac{1}{\Gamma(z)}$ into a power series and then integrate with $e^{az}$. But the coefficients of $\frac{1}{\Gamma(z)}$ are way too ugly. We can represent ...
1
vote
0answers
165 views

Integrating gamma products and quotients over a vertical line

The Wolfram functions collection contains a small number of integrals of products and quotients of terms $\Gamma(a_i\pm t)$ over a vertical line, all of which can be evaluated in terms of only gamma ...
0
votes
0answers
56 views

Recurrence formula for digamma function with rational number

It is well known that $\psi\left( x+N\right) =\psi\left( x\right) +\sum_{k=0}^{N-1}\frac{1}{x+k}$. Is there a recurrence formula for $\psi\left( x+\frac{p}{q}\right) $ where $\frac{p}{q}$ is ...
0
votes
0answers
81 views

Simplifying a finite difference sum that represents a model posterior to avoid numerical issues.

Problem Is it possible to simplify/rewrite the following expression, preferably without explicit sums, such that it can be computed without numerical issues when the $n_*$ are in the range of ...