All Questions
Tagged with closed-form gamma-function
25
questions with no upvoted or accepted answers
7
votes
0
answers
194
views
The closed-form of $1-5\left(\frac{1}{2}\right)^k+9\left(\frac{1\cdot3}{2\cdot4}\right)^k-13\left(\frac{1\cdot3\cdot5}{2\cdot4\cdot6}\right)^k+\dots$?
(A related MSE question by P. Singh.) First define,
$$F_k = 1-5\left(\frac{1}{2}\right)^k+9\left(\frac{1\cdot 3}{2\cdot 4}\right)^k-13\left(\frac{1\cdot 3\cdot 5}{2\cdot 4\cdot 6}\right)^k+17\left(\...
- 54.9k
4
votes
0
answers
96
views
Evaluate $\sum_{n=0}^{\infty} \frac{(a)_n (b)_n}{(2b)_n} \frac{\psi^{(0)}\left (n+ \frac{a}2+1 \right ) }{\Gamma\left (n+ \frac{a}2+1 \right ) }$
Define
$$S(a,b)=\sum_{n=0}^{\infty} \frac{(a)_n (b)_n}{(2b)_n} \frac{\psi^{(0)}\left (n+ \frac{a}2+1 \right ) }{\Gamma\left (n+ \frac{a}2+1 \right ) },$$
where $\psi^{(0)}(x)=\frac{\Gamma^\prime(x)}{\...
- 5,370
4
votes
0
answers
281
views
Non-recursive closed-form of the coefficients of Taylor series of the reciprocal gamma function
The reciprocal gamma function has the following Taylor series.
$$\frac{1}{\Gamma(z)}=\sum_{k=1}^{\infty}a_kz^k,$$
where the $a_k$ coefficient are given by the followint recursion.
$a_1=1$, $a_2=\...
- 12.4k
3
votes
0
answers
53
views
Find $\prod_{k=1}^n \frac{\Gamma (a_k/m)}{\Gamma (b_k/m)}$ algorithmically
It sometimes happens that
$$\prod_{k=1}^n \frac{\Gamma (a_k/m)}{\Gamma (b_k/m)}$$
is algebraic for positive integers $m,n,a_k,b_k$. For example,
$$\frac{\Gamma\left(\frac{1}{24}\right)\Gamma\left(\...
- 667
3
votes
0
answers
173
views
Question on a closed-form expression related to the harmonic number $H_n$
In this question the notation $\tilde{f}(x)$ refers to an analytic representation of the summatory function
$$f(x)=\sum\limits_{n=1}^x a(n)\tag{1}$$
that converges to
$$\underset{\epsilon\to 0}{\text{...
- 7,631
3
votes
0
answers
84
views
Closed form for infinite series $\sum_{n=1}^\infty \prod_{j=1}^n \left[1-(\tfrac{j}{n}u+v)^2\right]^{-1} x^n$
I've been struggling to find a closed form for the following series that depends on $u$ and $v$ as parameters:
$$
f(u,v,x) = \sum_{n=1}^\infty c_n x^n
$$
with
$$
c_n = \frac{1}{\prod_{j=1}^n \left[...
- 207
3
votes
0
answers
134
views
Is there a known transformation between $_2F_1\big(\tfrac12,\tfrac12;1;z\big)$ and $_2F_1\big(\tfrac12,\tfrac12;1;z^2\big)$?
In this post, the OP seeks a closed-form for,
$$A=\,_2F_1\big(\tfrac12,\tfrac12;1;\tfrac19\big)=1.02966\dots$$
Using the transformation,
$$\,_2F_1\big(\tfrac12,\tfrac12;1;z\big) = \tfrac2{1+\sqrt{1-z}}...
- 54.9k
3
votes
0
answers
160
views
Two complementary continued fractions that are algebraic numbers
Define the two similar continued fractions,
$$x=\cfrac{1}{km\color{blue}+\cfrac{(m-1)(m+1)} {3km\color{blue}+\cfrac{(2m-1)(2m+1)}{5km\color{blue}+\cfrac{(3m-1)(3m+1)}{7km\color{blue}+\ddots}}}}\tag1$$...
- 54.9k
3
votes
0
answers
124
views
Closed form of specific series
I'm working on a problem that involves the integrals of various Bessel functions that Mathematica can't symbolically handle. I've managed to grind out the transformations and integrals by hand, and ...
- 330
2
votes
0
answers
52
views
How to define double factorial for non positive integers?
I studied double factorial which known for natural number
$$ n!!=n(n-2)!! , 1!!=0!!=1$$
So we have for $n\in N$
$$ (2n)!!=2^n n! , (2n+1)!!=\frac{(2n+1)!}{2^n n!}$$
but I found on Math-World formula ...
- 1,637
2
votes
0
answers
238
views
Showing $\int_{0}^{1}\frac{E(\tfrac{x}{\sqrt{x^2+8}})}{\sqrt{8-7x^2-x^4}}dx=\frac{1}{3}K(\frac{1}{\sqrt{2}})E(\frac{1}{\sqrt{2}})$
Context
$\begin{align}
K(k)=\int_{0}^{\pi/2}\frac{dt}{\sqrt{1-k^2\sin^2t}}\tag{1}
\end{align}$
and
$\begin{align}
E(k)=\int_{0}^{\pi/2}\sqrt{1-k^2\sin^2t}dt\tag{2}
\end{align}$
the complete elliptic ...
- 323
2
votes
0
answers
238
views
Closed form expression for a sum
I want to calculate a sum of the form $$\sum_{k=0}^m \frac{\Gamma[m+1+\alpha-k]^2}{\Gamma[m+1-k]^2}\frac{\Gamma[x+k]}{\Gamma[x]k!}$$ where $m>0$ and belongs to integers and $\alpha$ takes half ...
- 21
1
vote
0
answers
52
views
Does the rest of this family of continued fractions have closed forms?
The pattern for the continued fractions below is quite straightforward. $F_1$ has numerators with all the integers but,
$F_2\; \text{is missing}\; 2m+1 = 3,5,7,\dots\\
F_3\; \text{is missing}\; 3m+1 = ...
- 54.9k
1
vote
0
answers
43
views
Computing $\frac{d^{n-1}}{dz^{n-1}}\frac{z}{\ln\left(z!\right)}^{n}$
I am trying to compute
$$\frac{d^{n-1}}{dz^{n-1}}\left(\frac{z}{\ln\left(z!\right)}\right)^{n}$$
The problem arises when dealing with inversion formulae. My question is, can this expression be ...
- 11
1
vote
0
answers
95
views
Is there a closed form for the binomial expression $\binom{-1/m}{k} $?
I'm interested in binomial coefficients of the form $$\binom{-1/m}{k} ,$$ where $m$ is a positive integer.
For $m=2$, it holds that \begin{align} \binom{-1/2}{k} &= (-4)^{-k} \binom{2k}{k} \qquad ...
- 7,148