[Calculation 13] A Simple Formula Related to Digamma Functions

Edited by Leun Kim

Theorem. The following holds:
$$
\pi \;{}_2 F_1 \left( \frac{1}{2}, \frac{1}{2}; 1; 1-x\right) =
\log \left( \frac{16}{x}\right) {}_2 F_1 \left( \frac{1}{2}, \frac{1}{2};1;x\right) – 4 \sum_{k=1}^\infty \frac{(\frac{1}{2})_k^2}{(k!)^2} \sum_{j=1}^k \frac{x^k}{(2j-1)(2j)}.
$$

 
Proof. First, we recall the Corollary in [2] with $a=b=-\frac{1}{2}$ and $n=0$:
\begin{equation}
{}_2 F_1 \left( \frac{1}{2}, \frac{1}{2}; 1; 1-x\right) =- \frac{1}{\pi} \sum_{k=0}^\infty \frac{(\frac{1}{2})_k^2}{(k!)^2} \left(2\psi\left(\frac{1}{2}+k\right) – 2\psi(k+1) + \log x \right) x^k.
\end{equation}
Thus we obtain
\begin{align*}
&\pi \; {}_2 F_1 \left( \frac{1}{2}, \frac{1}{2}; 1; 1-x\right)\\
&=
– \log x \; {}_2 F_1 \left( \frac{1}{2}, \frac{1}{2}; 1; x\right)
– 2 \sum_{k=0}^\infty \frac{(\frac{1}{2})_k^2}{(k!)^2} \left( \psi \left(k+ \frac{1}{2} \right) – \psi (k+1) \right) x^k\\
&=- \log x \; {}_2 F_1 \left( \frac{1}{2}, \frac{1}{2}; 1; x\right)
– 2 \sum_{k=0}^\infty \frac{(\frac{1}{2})_k^2}{(k!)^2} \left(
\sum_{j=1}^k \frac{2}{2j-1} – 2\log 2 – \sum_{j=1}^k \frac{1}{j} \right) x^k\\
&= \log \left( \frac{16}{x}\right) {}_2 F_1 \left( \frac{1}{2}, \frac{1}{2}; 1; x\right) – 2 \sum_{k=1}^\infty \frac{(\frac{1}{2})_k^2}{(k!)^2} \sum_{j=1}^k \frac{x^k}{j(2j-1)},
\end{align*}
where we used the expressions
$$
\psi (k+1) = \sum_{j=1}^k \frac{1}{j} – \gamma
$$ and $$
\psi \left( k + \frac{1}{2} \right) = -\gamma – 2\log 2 + \sum_{j=1}^k \frac{2}{2j-1}
$$
for the digamma functions at the second equality.$\square$
 
Taking exponential to the both sides of the equation in Theorem, we obtain the following corollary.

Corollary.
$$
\exp \left( -\pi \frac{{}_2 F_1 (\frac{1}{2}, \frac{1}{2}; 1; 1-x)}{{}_2 F_1 (\frac{1}{2}, \frac{1}{2}; 1; x)} \right)
=
\frac{x}{16} \exp \left( \frac{4 \sum_{k=1}^\infty \frac{(\frac{1}{2})_k^2}{(k!)^2} \sum_{j=1}^k \frac{x^k}{2j(2j-1)}}{{}_2 F_1 (\frac{1}{2},\frac{1}{2};1;x)}\right)
$$

 
References.
[1] Bruce C. Berndt, Ramanujan’s Notebooks, Part II, Springer-Verlag, pp. 78-79.
[2] Leun Kim, [Calculation 12] A Basic Formula for Hypergeometric Functions.
[3] Bruce C. Berndt, Ramanujan’s Notebooks, Part III, Springer-Verlag, p. 91.

 
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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