[Calculation 9] Simple Corollaries from Gauss and Bailey Formula

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Edited by Leun Kim

Corollary 1. For $\frac{1}{2} < z < 2$,
$$
{}_2 F_1 \left(\frac{1}{2}, \frac{1}{2};1;1-\frac{1}{z}\right)
=
\sqrt{z} {}_2 F_1 \left(\frac{1}{2}, \frac{1}{2};1;1-z\right).
$$

 
Proof. We recall Bailey’s Formula ((i) in [2]) for $w\in\mathbb R$:
\begin{equation}\tag{1}
(1-w)^{-a} {}_2 F_1 \left( a,b;c; – \frac{w}{1-w}\right) = {}_2 F_1 (a,c-b;c;w), \qquad -1< w <\frac{1}{2} \end{equation} Setting $a=b=\frac{1}{2}$, $c=1$ and $z=1-w$ at (1), we obtain $$ z^{-1/2} {}_2 F_1 \left(\frac{1}{2}, \frac{1}{2};1;1-\frac{1}{z}\right) = {}_2 F_1 \left(\frac{1}{2}, \frac{1}{2};1;1-z\right), \qquad \frac{1}{2} < z < 2. $$ $\square$

Corollary 2.
 $$
{}_2 F_1 \left( \frac{1}{2}, \frac{1}{2}; 1; 1 – \left( \frac{1-z}{1+z}\right)^2\right)
=
(1+z) {}_2 F_1 \left(\frac{1}{2}, \frac{1}{2}; 1; z^2\right)
$$

 
Proof. Replacing
$$
z \to \left(\frac{1-z}{1+z}\right)^2
$$
in Corollary 1 above, we have
\begin{equation}
{}_2 F_1 \left( \frac{1}{2}, \frac{1}{2}; 1; 1 – \left( \frac{1-z}{1+z}\right)^2\right)
=
\frac{1+z}{1-z} {}_2 F_1\left( \frac{1}{2}, \frac{1}{2};1; – \frac{4z}{(1-z)^2}\right).
\end{equation}
We remember that Gauss’s Quadratic Transformation ([3]) with $a=b=\frac{1}{2}$ and $z \to -z$ is
\begin{equation}
{}_2 F_1 \left( \frac{1}{2}, \frac{1}{2}; 1; – \frac{4z}{(1-z)^2}\right)
=
(1-z) {}_2 F_1 \left( \frac{1}{2}, \frac{1}{2}; 1; z^2 \right).
\end{equation}
Piecing (2) and (3) together, we obtain Corollary 2.$\square$
 
References.
[1] Bruce C. Berndt, Ramanujan’s Notebooks, Part II, Springer-Verlag, pp.92-93.
[2] Leun Kim, [Calculation 7] Bailey’s Formulas for Hypergeometric Functions.
[3] Leun Kim, [Calculation 8] Gauss’s Quadratic Trasformation.

 
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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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