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I am reading "Differential equations for characters of Virasoro and affine Lie algebra" written by T. Eguchi and H. Ooguri (Nucl. Phys. B313, 1989). In page 497, an ordinary differential equation is introduced: \begin{eqnarray} (2\pi i)^3f'''+12\eta_1(2\pi i)^2f''+(24\eta_1^2-\frac{25}{64}g_2)2\pi if'-\frac{23}{4\times 64}g_3f=0 \end{eqnarray} where $\eta_1(\tau)=2\pi i (\mathrm{log}\ \eta(\tau))'$, $g_2$ and $g_3$ are the Laurent expansion coefficeints of the Weierstrass $P$-function $P(z)=\frac{1}{z^2}+\frac{1}{20}g_2+\frac{1}{28}g_3+\cdots$.

The authors claim that the following three functions are solutions to the ODE

\begin{eqnarray} &&\lambda^{-1/24}(1-\lambda)^{-1/24}+\lambda^{-1/24}(1-\lambda)^{1/12} \\ &&\lambda^{-1/24}(1-\lambda)^{-1/24}-\lambda^{1/24}(1-\lambda)^{1/12} \\ &&\lambda^{1/12}(1-\lambda)^{-1/24} \end{eqnarray} where $\lambda$ is a quotient of theta constans, $$ \lambda(\tau)=\theta_2(0,\tau)^4/\theta_3(0,\tau)^4=16q^{1/2}\prod_{n\ge1}\frac{(1+q^n)^8}{(1+q^{n-1/2})^8}, $$ but I cannot verify the claim. Maybe we need some formula for theta constans to verify it. Please tell me how to verify the claim.

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    The Weierstrass function $\mathscr{P}(z;g_2,g_3)$ takes two parameters $g_2$ and $g_3$ associated to a period lattice which we can choose. If the functions you specify are solutions to the differential equation, they are solutions for a specific period lattice chosen for $\mathscr{P}$. Are they given in the paper? – 2017-01-18
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    This question appears to be pure mathematics. I do not see where physics comes into it. (I appreciate that it might be useful in a physics context, but this is true of the whole of mathematics.) It also appears to be an exercise ("How do I do this calculation?") within the scope of the [homework-and-exercises policy](http://meta.physics.stackexchange.com/questions/714/how-do-i-ask-homework-questions-on-physics-stack-exchange) ; the policy requires you to **ask about a specific physics concept**. – 2017-01-18

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