In mathematics, the Gaussian or ordinary hypergeometric function 2F1(a,b;c;z) is a special function represented by the hypergeometric series, that includes many other special functions as specific or limiting cases. It is a solution of a second-order linear ordinary differential equation (ODE). ...
0
votes
0answers
21 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 ...
3
votes
1answer
34 views
An identity related to Legendre polynomials
Let $m$ be a positive integer. I believe the the following identity
$$1+\sum_{k=1}^m (-1)^k\frac{P(k,m)}{(2k)!}=(-1)^m\frac{2^{2m}(m!)^2}{(2m)!}$$
where $P(k,m)=\prod_{i=0}^{k-1} (2m-2i)(2m+2i+1)$, ...
2
votes
1answer
67 views
Prove an identity involving Gamma function and Gaussian hypergeometric function
The question asks to prove the following identity:
...
0
votes
0answers
20 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
102 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
99 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?
$$\int_0^T e^{a (T-x)} (T-x)^{1+m+n} x^k \, _1F_1\Big(1+n;2+m+n;a (x-T) \Big) \, dx$$
...
2
votes
1answer
26 views
Applications of hypergeometric continued fractions
http://en.wikipedia.org/wiki/Gauss%27s_continued_fraction
Using a technique due to Gauss a lot of special functions can be expressed as continued fractions.
What applications of this are there ...
6
votes
1answer
80 views
Understanding the solution of $\int\left(1-x^{p}\right)^{\frac{n-1}{p}}\log\left(1-x^{p}\right)dx$
I would like to solve the integral
$$\int\left(1-x^{p}\right)^{\frac{n-1}{p}}\log\left(1-x^{p}\right)dx.$$
My problem is that I arrived at a solution via wolfram alpha, but I would like to ...
2
votes
0answers
45 views
Applications for the tail bounds for the hypergeometric distribution
I am interested in the applications for the probability tail bounds (non-asymptotic) for the hypergeometric distribution. I would be very appreciative if you can name or link me to some.
4
votes
1answer
120 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 $\,_2F_1(a,b,c,z)$ ...
1
vote
1answer
62 views
Tail bound for hypergeometric distribution
I am looking for a reference (book) for the tail bound for the Hypergeometric distribution.
I know there is a nice paper by Skala (2009) but its unpublished. I am looking for a book which would be a ...
0
votes
1answer
49 views
Probability of drawing three letters out of four
We would like to organize the following prize game (simplified):
We print five letters, say, $10 \times A$, $10 \times B$, $5\times C$, $1 \times D$, $50\times E$. A customer draws a letter at each ...
1
vote
1answer
89 views
Is $ \int_0^b (a-x^m)^{1/n} dx $ solvable?
Is there a solution for the
integral
$$
\int_0^b (a-x^m)^{1/n} dx?
$$
WolframAlpha doesn't help a lot:
$$
\frac1a x(1-x^m)^{\frac1n+1}{}_2F_1\left(1,\frac1n+1+\frac1m; ...
1
vote
0answers
121 views
Two variable recurrence relations
I'm interested in solving the following type of problem...
Starting with a recurrence relation in multiple variables, for example:
$$ v_{i-1,j} + v_{i,j+1} + v_{i+1,j} + v_{i,j-1} = 4v_{i,j}$$
with ...
0
votes
1answer
114 views
simpler expression for terminating 3F2 series with negative unit argument
I am hoping to evaluate a simpler expression for the following:
$${}_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 ...
