# [Calculation 5] Gauss’s Summation Formula

Edited by Leun Kim

 Theorem. (Gauss’s Summation Formula) For $\text{Re}c>\text{Re}b>0$, $${}_2 F_1 (a,b;c;1) = \frac{\Gamma(c) \Gamma(c-a-b)}{\Gamma(c-a)\Gamma(c-b)}$$ holds.

Proof. We remember that the Euler Integral Representation for the hypergeometric function is
$$_2 F_1 (a,b;c;z) = \frac{\Gamma(c)}{\Gamma(b)\Gamma(c-b)} \int_0^1 \frac{t^{b-1} (1-t)^{c-b-1}}{(1-tz)^a}\,dt.$$
Taking the limit $z\to 1$ both sides, we obtain
\begin{eqnarray*}
_2 F_1 (a,b;c;1)
&=&
\frac{\Gamma(c)}{\Gamma(b)\Gamma(c-b)}
\int_0^1 t^{b-1} (1-t)^{c-a-b-1}\,dt\\
&=&
\frac{\Gamma(c)}{\Gamma(b)\Gamma(c-b)}
\frac{\Gamma(b)\Gamma(c-b-a)}{\Gamma(c-a)}\\
&=&
\frac{\Gamma(c) \Gamma(c-a-b)}{\Gamma(c-a)\Gamma(c-b)},
\end{eqnarray*}
which proves the theorem.$\square$

 Corollary. (Chu–Vandermonde Identity) $${}_2 F_1 (-n,b;c;1) = \frac{(c-b)_n}{(c)_n}$$

Proof. Taking $a = -n$ in Gauss’s Summation Formula, we have
$${}_2 F_1 (-n,b;c;1) = \frac{\Gamma(c)\Gamma(c-b+n)}{\Gamma(c+n)\Gamma(c-b)} = \frac{(c-b)_n}{(c)_n}.$$

References.
[1] en.wikipedia.org/wiki/Hypergeometric_function
[2] Leun Kim, Fundamentals of Hypergeometric Functions.

#### 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.

#### Latest posts by Leun Kim (see all)

• […] ← Previous […]

• […] the first equality, and the Chu-Vandermonde identity (in [2]) for the second equality. Finally we […]

• […] Proof. First, we recall the Gauss Summation Formula ([2]) and Kummer’s Theorem ([3]) as […]

• […] , where are constants. The constant can be found by limiting and with Gauss Summation Formula [2]. Then we […]

• Dear Leun Kim: I am happy to see that you are doing some calculations related to hypergeoemtric and generalized hypergeometric series. I am working in this area for the last several years. To know more about the results you mentioned, send me your e-mail address so t hat I may send you some of mynelementary results.
All best wishes

• author

Thanks for the comment! I’m just a beginner to these kind of topics, but you can send me your works if you don’t mind : )