In the book Higher Transcendental Functions, Volume I. McGraw-Hill. by Bateman, P.95, Bateman mentioned the two linearly independent solutions of a hypergeometric ODE, namely $u_1(z) = \enspace _2F_1(1/2, 1/2; 1; z)$ and $u_2(z) = i \enspace _2F_1(1/2, 1/2; 1; 1-z)$, can be analytically continued around a (counterclockwise) loop centred at $0$ and $1$ respectively.
They stated the result as follows:
$$C_{0}: u_1 \rightarrow u_1; \quad u_2\rightarrow 2 u_1+u_2,$$
$$C_{1}: u_1 \rightarrow u_1-2 u_2; \quad u_2 \rightarrow u_2 $$
where $C_0, C_1$ stands for the loop centred at the respective points.
Therefore the monodromy matrix for this two functions around 0 and 1 are: $M_0 = \begin{bmatrix}1&0 \\ 2&1\end{bmatrix}$ and $M_1 = \begin{bmatrix}1 & -2 \\ 0 & 1 \end{bmatrix}$
The book also mentioned the result follows from this identity:
$$ \dfrac\pi2 F(1 / 2,1 / 2 ; 1 ; z)+ \dfrac12 \log (1-z) F(1 / 2,1 / 2 ; 1 ; 1-z)=\sum_{n=0}^{\infty} \frac{(1 / 2)_n(1 / 2)_n}{n ! n !}[\psi(n+1)-\psi(n+1 / 2)](1-z)^n $$
where $\psi(z) = \dfrac{d (\log \Gamma(z))}{dz}$.
My Questions:
- The book states that the continuation follows from section 2.1.4, I don't understand the main argument of going through the proof, can anyone briefly outline the idea to me?
- In the same vein, I need to attain the monodromy matrix $M_0, M_1$ for $u_1(z) = \enspace _2F_1\left(\dfrac16, \dfrac56; 1; \dfrac{27}{4}z\right)$ and $u_2(z) = i \enspace _2F_1\left(\dfrac16, \dfrac56; 1; 1-\dfrac{27}{4}z\right)$ along $C_0$ and $C_1$, do anyone know how to do it, or point to me good references outlining the details?
Thank you so much.
In general $_2F_1\left(a,1-a;1|z\right)$ and $_2F_1\left(a,1-a;1|1-z\right)$ satisfy the same differential equations and have monodromy of the form $$ M_0=\begin{pmatrix}1&0\\1/u-u&1\end{pmatrix} \qquad M_1=\begin{pmatrix}1&u-1/u\\0&1\end{pmatrix} $$ where $u=\exp(i\pi a)$. The cases $a=\frac16,\frac14,\frac13,\frac12$ are of particular interest because the group they generate is (conjugate to) a finite index subgroup of $SL(2,\mathbb Z)$, see e.g. https://en.wikipedia.org/wiki/J-invariant#Inverse_function