Tagged Questions
1
vote
0answers
18 views
Simplification of Kampé de Fériet function
I was dealing with a convolution type integral
$$
\int^z_0 t^m {}_0F_1(;1;-t) \: {}_2F_3\Big( 1,1;2,m,m+1 ; -a t\Big) \:\mathrm{d}t
$$
By applying one of the identities in Exton's book, the solution ...
2
votes
3answers
69 views
What type of Hypergeometric series is this?
I am trying to find a closed form for the series
$$ \sum^\infty_{n=0} \frac{1}{n!} \frac{1}{n+1}(-z)^n {}_2F_2\left(m,n+1;\frac{1}{2},n+2; b z\right)$$
$m$ is a nonzero positive integer, and $b$, ...
13
votes
1answer
163 views
$\int_0^1\arctan\,_4F_3\left(\frac{1}{5},\frac{2}{5},\frac{3}{5},\frac{4}{5};\frac{1}{2},\frac{3}{4},\frac{5}{4};\frac{x}{64}\right)\,\mathrm dx$
I need help with calculating this integral:
$$\int_0^1\arctan\,_4F_3\left(\frac{1}{5},\frac{2}{5},\frac{3}{5},\frac{4}{5};\frac{1}{2},\frac{3}{4},\frac{5}{4};\frac{x}{64}\right)\,\mathrm dx,$$
where ...
3
votes
1answer
43 views
How to prove this identity for ${}_3F_2$ (Generalized Hypergeometric Function)?
This may look like homework, but it is not. I've found this identity (using Mathematica):
$$
{}_3F_2 \left( \matrix{1,1,1 \\ 2, e} ; 1 \right) = (e-1) \psi^{\prime}(e-1),
$$
valid for $e$ with ...
0
votes
0answers
22 views
$_{1}{{F}_{1}}\left( 1,c,(a+b)x \right)-\, _{1}{{F}_{1}}\left( 1,c,ax \right)=(?)$
I am looking for:
$$_{1}{{F}_{1}}\left( 1,c,(a+b)x \right)-\, _{1}{{F}_{1}}\left( 1,c,ax \right)=(?)$$
where $x$ is between $0$ and $T$ and $a,b,c>0$.
I know the asymptotic behavior as $x$ goes ...
2
votes
1answer
83 views
Prove an identity involving Gamma function and Gaussian hypergeometric function
The question asks to prove the following identity:
...
0
votes
0answers
27 views
Infinite series of Hypergeometric function
Any ideas how to find a closed form for the sum given by:
$$
\sum^\infty_{n=0}
\frac{1}{n!}
\frac{a^n b^{n+m}}{(m+n)^2 \Gamma(m+n)}
{}_2F_2 \left(m+n,m+n;m+n+1,m+n+1;-b\right)
$$
Given that both $a$ ...
3
votes
2answers
107 views
Bounds on $ \sum\limits_{n=0}^{\infty }{\frac{a..\left( a+n-1 \right)}{\left( a+b \right)…\left( a+b+n-1 \right)}\frac{{{z}^{n}}}{n!}}$
I have a confluent hypergeometric function as $ _{1}{{F}_{1}}\left( a,a+b,z \right)$ where $z<0$ and $a,b>0$ and integer.
I am interested to find the bounds on the value it can take or an ...
1
vote
1answer
105 views
Integration over a combination of confluent hypergeometric, power, and exponential functions
I am trying to work out this integral. If there is no closed form, can you think of any approximations to it?
$$\int_0^T e^{a (T-x)} (T-x)^{1+m+n} x^k \, _1F_1\Big(1+n;2+m+n;a (x-T) \Big) \, dx$$
...
5
votes
1answer
125 views
Finding x in $\frac{\,_2F_1(\frac{1}{5},\frac{4}{5},\,1,\,1-x)}{\,_2F_1(\frac{1}{5},\frac{4}{5},\,1,\,x)} = \sqrt{n}$
I was trying to find a closed-form for $0<x<1$ in,
$$\frac{\,_2F_1(\frac{1}{m},\,1-\frac{1}{m},\,1,\,1-x)}{\,_2F_1(\frac{1}{m},\,1-\frac{1}{m},\,1,\,x)} = \sqrt{n}$$
where $\,_2F_1(a,b,c,z)$ ...
0
votes
1answer
115 views
simpler expression for terminating 3F2 series with negative unit argument
I am hoping to evaluate a simpler expression for the following:
$${}_3F_2(-n,1,1; a, (a+3)/2; -1)$$
Here $n,a \in \mathbb{N}$ and $a$ is odd. I am also interested in the asymptotics in $n \in ...
4
votes
1answer
159 views
Hypergeometric functions & integral
I'm having difficulty re-deriving a result a calculation from a paper. The integral is
$$\int_0^{2\pi} \int_0^{2\pi} ...
0
votes
0answers
71 views
Why is a factorial always present in the denominator of the hypergeometric function?
The sum definition of the generalized hypergeometric function is:
$${_p}F_q (a_1, \dots, a_p;b_1, \dots,b_q;z)=\sum\limits_{n=0}^\infty \frac{(a_1)_k (a_2)_k\cdots (a_p)_k z^k}{(b_1)_k (b_2)_k\cdots ...
