[Calculation 8] Gauss’s Quadratic Trasformation

Edited by Leun Kim

 Theorem. (Gauss’s Quadratic Transformation) \tag{1} (1+z)^{-2a} {}_2 F_1 \left(a,b; 2b; \frac{4z}{(1+z)^2}\right) = {}_2 F_1 \left(a, 1+\frac{1}{2}-b;b+\frac{1}{2};z^2\right).

Proof. The proof is almost similar to that of [2]. Note that the left hand side of (1) can be expanded in powers of $z$, since it is analytic in a certain neighborhood of $z=0$. Thus by the binomial theorem, we have
\begin{eqnarray*}
(1+z)^{-2a} {}_2 F_1 \left(a,b; 2b; \frac{4z}{(1+z)^2}\right)
&=&
\sum_{k=0}^\infty \frac{(a)_k (b)_k 4^k}{(2b)_k k!} z^k (1+z)^{-2a-2k}\\
&=&
\sum_{k=0}^\infty \sum_{r=0}^\infty \frac{(a)_k (b)_k (-4)^k}{(2b)_k k!} \frac{(2a+2k)_r}{r!} z^{k+r}.
\end{eqnarray*}
Calculating the coefficient of $z^n$, we get
\tag{2}
\sum_{k=0}^n \frac{(a)_k (b)_k 4^k (-1)^k}{(2b)_k k!} \frac{(2a+2k)_{n-k}}{(n-k)!} = \frac{(2a)_n}{n!}\sum_{k=0}^n \frac{(b)_k (2a+n)_k (-n)_k}{(2b)_k (\frac{1}{2} + a)_k k!}

where we used the identities
$$(2a+2k)_{n-k} = \frac{(2a)_n (2a+n)_k}{4^k (a)_k (\frac{1}{2} +a)_k}, \qquad \frac{(-1)^k}{(n-k)!} = \frac{(-n)_k}{n!}.$$
Note that the right hand side of (2) vanishes when $n$ is odd. Also we can easily verify that
\tag{3}
\frac{(2a)_n}{n!}\sum_{k=0}^n \frac{(b)_k (2a+n)_k (-n)_k}{(2b)_k (\frac{1}{2} + a)_k k!}
=
\frac{(a)_{n/2} (a+\frac{1}{2}-b)_{n/2}}{(b+\frac{1}{2})_{n/2} n!}

for all even $n$. But the right hand side of (3) is exactly the coefficients of $z^n$, in the right hand side of (1). This proves the theorem. For the another proof, I recommend Erdelyi [1].$\square$

References.
[1] A. Erdelyi, Higher Transcendental Functions, Vol. 1, McGraw-Hill, New York, 1953, pp.64-68.
[2] Leun Kim, [Calculation 6] Kummer’s Theorem.
[3] Bruce C. Berndt, Ramanujan’s Notebooks, Part II, Springer-Verlag, p.63.

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.