[Calculation 2] Pfaff’s Transformation Formula

Edited by Leun Kim

 Theorem. (Pfaff’s Transformation Formula) $${}_2 F_1(a,b;c;z) = (1-z)^{-a} {}_2 F_1 \left(a,c-b;c; \frac{z}{z-1} \right)$$

Proof. We remember the Euler Integral Representation for the hypergeometric function:
$${}_2 F_1 (a,b;c;z) = \frac{\Gamma(c)}{\Gamma(b)\Gamma(c-b)} \int_0^1 t^{b-1}(1-t)^{c-b-1} (1-tz)^{-a}\,dt.$$
Substitution $t=1-s$ yields
\begin{eqnarray*}
{}_2 F_1 (a,b;c;z)
&=&
\frac{\Gamma(c)}{\Gamma(b)\Gamma(c-b)}
\int_0^1 (1-s)^{b-1} s^{c-b-1} (1-z+sz)^{-a}\,ds\\
&=&
(1-z)^{-a}
\frac{\Gamma(c)}{\Gamma(b)\Gamma(c-b)}
\int_0^1 s^{c-b-1} (1-s)^{b-1} \left( 1- \frac{sz}{z-1}\right)^{-a}\,ds\\
&=&
(1-z)^{-a} {}_2 F_1 \left(a,c-b;c; \frac{z}{z-1} \right)
\end{eqnarray*}
which proves the theorem for $\text{Re}c>\text{Re}b>0$. This condition can be removed by continuation of $b$ and $c.$$\square$

References.
[1] aw.twi.tudelft.nl/~koekoek/onderw1112/specfunc_en.html
[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.

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