# [Calculation 3] Euler’s Transformation Formula

Edited by Leun Kim

 Theorem (Euler’s Transformation Formula) $${}_2 F_1 (a,b;c;z) = (1-z)^{c-a-b} {}_2 F_1 (c-a,c-b;c;z)$$

Proof. Applying Pfaff’s Transformation Formula twice, we obtain
\begin{eqnarray*}
{}_2 F_1(a,b;c;z)
&=&
(1-z)^{-a} {}_2 F_1 \left(a,c-b;c; \frac{z}{z-1} \right)\\
&=&
(1-z)^{-a} \left(1-\frac{z}{z-1}\right)^{b-c} {}_2 F_1 \left(c-a,c-b;c;\frac{\frac{z}{z-1}}{\frac{z}{z-1} -1} \right)\\
&=&
(1-z)^{c-a-b} {}_2 F_1 (c-a,c-b;c;z),
\end{eqnarray*}
which proves the theorem.$\square$

References.
[1] aw.twi.tudelft.nl/~koekoek/onderw1112/specfunc_en.html
[2] Leun Kim, Pfaff’s Transformation Formula.

#### 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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