[Calculation 1] Fundamentals of Hypergeometric Functions

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Edited by Leun Kim

The classical hypergeometric function ${}_{2}F_1$ is defined by
$$
{}_{2}F_1(a,b;c;z)
=
\sum_{k=0}^\infty \frac{(a)_k(b)_k}{(c)_k} \frac{z^k}{k!}
$$
where $(\cdot)_k$ is Pochhammer symbol, that is,
$$
(q)_k = \frac{\Gamma(q+k)}{\Gamma(q)}
$$
provided that $q+k$ is not a negative integer, with the convention $1/{\pm\infty} = 0$. Note that we may naturally generalize the classical hypergeometric function to the generalized hypergeometric series:
$$
{}_p F_q \begin{bmatrix}a_1, a_2, \cdots, a_p \\ b_1, b_2, \cdots, b_q;z \end{bmatrix} = \sum_{k=0}^\infty \frac{(a_1)_k (a_2)_k \cdots (a_p)_k}{(b_1)_k (b_2)_k \cdots (b_q)_k} \frac{z^k}{k!}
$$
for $p,q\in \mathbb N$. Note that we usually set $p=q+1$ so that the series converges when $|z|<1$ for all possible choices of parameters. Of course, there are lots of formulas, which expand the domain of the function through the analytic continuation. For the simplicity, we frequently use the notation
$$
{}_p F_q (a_1, \cdots, a_p; b_1, \cdots, b_q; z) = {}_p F_q \begin{bmatrix}a_1, \cdots, a_p \\ b_1, \cdots, b_q;z \end{bmatrix}.
$$
 

Theorem 1. (Euler’s Integral Representation)
Let $\text{Re} c> \text{Re} b>0$, then
$$
_2 F_1 (a,b;c;z)
=
\frac{\Gamma(c)}{\Gamma(b)\Gamma(c-b)}
\int_0^1 \frac{t^{b-1} (1-t)^{c-b-1}}{(1-tz)^a}\,dt
$$
holds in the $z$ plane cut along the real axis from $1$ to $\infty$.

 
Proof. By the binomial theorem, we have
$$
\frac{1}{(1-tz)^{a}} = \sum_{k=0}^\infty \frac{(a)_k}{k!}(tz)^k
$$
so that
\begin{eqnarray*}
\int_0^1 \frac{t^{b-1}(1-t)^{c-b-1}}{(1-tz)^a}\,dt
&=&
\int_0^1 t^{b-1}(1-t)^{c-b-1}\sum_{k=0}^\infty \frac{(a)_k}{k!}(tz)^k\,dt\\
&=&
\sum_{k=0}^\infty \frac{(a)_k}{k!} z^k \int_0^1 t^{k+b-1} (1-t)^{c-b-1}\,dt\\
&=&
\sum_{k=0}^\infty \frac{(a)_k}{k!} z^k \frac{\Gamma(k+b)\Gamma(c-b)}{\Gamma(k+c)}.
\end{eqnarray*}
Therefore we obtain
\begin{eqnarray*}
\frac{\Gamma(c)}{\Gamma(b)\Gamma(c-b)}
\int_0^1 \frac{t^{b-1}(1-t)^{c-b-1}}{(1-tz)^a}\,dt
&=&
\sum_{k=0}^\infty \frac{(a)_k}{k!} z^k \frac{\Gamma(k+b)}{\Gamma(b)} \frac{\Gamma(c)}{\Gamma(k+c)}\\
&=&
\sum_{k=0}^\infty \frac{(a)_k (b)_k}{(c)_k k!} z^k\\
&=&
_2 F_1 (a,b;c;z)
\end{eqnarray*}
which proves Theorem 1 for $|z|<1$. Since the integral is analytic in the cut plane, the theorem holds.$\square$
 
References.
[1] en.wikipedia.org/wiki/Hypergeometric_function
[2] aw.twi.tudelft.nl/~koekoek/onderw1112/specfunc_en.html

 
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Leun Kim

Ph.D Candidate at The University of Tokyo
I was born and raised in Daegu, S. Korea. I majored in electronics and math in Seoul from 2007 to 2012. I've had a great interest in math since freshman year, and I studied PDE in Osaka, Japan from 2012-2014. I worked at a science museum and HUFS from 2014 in Seoul. Now I'm studying PDE in Tokyo, Japan. I also developed an interest in music, as I met a great piano teacher Oh in 2001, and joined an indie metal band in 2008. In my spare time, I enjoy various things, such as listening music, blogging, traveling, taking photos, and playing Go and Holdem. Please do not hesitate to contact me with comments, email, guestbook, and social medias.



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