FW: Hypergeometric functions

["Dan Lozier" <lozier@nist.gov>]

-----Original Message-----
From: opsftalk@nist.gov [mailto:opsftalk@nist.gov]On Behalf
Of Nico
Temme
Sent: Tuesday, October 07, 2003 7:52 AM
To: dlozier@nist.gov
Subject: Hypergeometric functions




Dear colleagues,

Eleutherius Symeonidis <e.symeonidis@ku-eichstaett.de>
(Germany)
wrote me a letter. I cannot help him,  and a first try to
approach
the OPSF group is opsftalk. If you want a pdf file of the
letter,
please let me know (I attach this file to this submission,
but I
don't know if it works).

I will submit also to the Newsletter, when no answers or
suggestions come through.

The formulas are in LaTeX style.

Greetings,

	Nico.


Eleutherius wrote:

in my research work on the Poisson integral for a ball in
non-euclidean spaces I have obtained the following
identities for the
hypergeometric function:

$$ \displaylines{ \sum_{k=0}^\infty
\frac{\Gamma(\frac{n+k}{2})\Gamma(\frac{k+1}{2})}
{\Gamma(\frac{n}{2}+k-1)}t^k\cdot
F\left(k,k+n-1;\frac{n}{2}+k;
\frac{1-\sqrt{1-t^2}}{2}\right)C_k^\frac{n-2}{2}(x)\equiv\hf
ill\cr
\hfill\equiv\frac{\Gamma(\frac{n+1}{2})(n-2)}
{2\Gamma(\frac{n}{2}+1)}
\sqrt{1-t^2}\cdot
F\left(n,1;\frac{n}{2}+1;\frac{1+xt}{2}\right)\cr}
$$
and
$$
\frac{1}{2}+\sum_{k=1}^\infty \frac{{2k-1 \choose
k-1}{{\frac{n}{2}
+k-2}
\choose k-1}}{{\frac{n}{2}+2k-2 \choose k-1}}F\left(k,
\frac{n-1}{2}+k;\frac{n}{2}+2k;x\right)\left(-\frac{x}{4}\ri
ght)^k
\equiv\frac{\sqrt{1-x}}{2}\,,
$$
if $|x|,|t|<1$ and $n\in N$, $n\ge 3$, where
$C_k^\frac{n-2}{2}(x)$ denotes the Gegenbauer polynomial
$$ {k+n-3 \choose
k}F\left(-k,k+n-2;\frac{n-1}{2};\frac{1-x}{2} \right)\,.
$$

Since I am not a specialist in the field of special
functions, I would
like to ask you whether these identities  could be directly
proven by
other techniques.

sym.pdf