# Where does the Weierstrass expansion of $\operatorname{sn}$ come from?

In Table of Integrals, Series and Products (p. 869) by Gradshteyn and Ryzhik, I found the identity (called 'the Weierstrass expansion of $$\operatorname{sn}$$') $$\operatorname{sn}u=\frac{B}{A}$$ where $$\operatorname{sn}$$ is the Jacobi elliptic function $$\operatorname{sn}$$, $$B=\sum_{n=0}^\infty (-1)^n b_n\frac{u^{2n+1}}{(2n+1)!},$$ $$A=1-\sum_{n=1}^\infty (-1)^{n+1}a_{n+1}\frac{u^{2n+2}}{(2n+2)!},$$ $$b_0=1,\, b_1=1+k^2,\, b_2=1+k^4+4k^2,\,\ldots ,$$ and $$a_2=2k^2,\, a_3=8(k^2+k^4),\ldots ,$$ ($$k$$ is the elliptic modulus). Note that Gradshteyn and Ryzhik write only the first few terms and don't write the general term.

My question

I know the Weierstrass factorization theorem. But it is unclear to me how the sequences $$\{a_n\}_n$$ and $$\{b_n\}_n$$ are generated. How can one come up with them? What is the "general term" of the sequences?

This question has also been posted on MSE: https://math.stackexchange.com/questions/4683898/where-does-the-weierstrass-expansion-of-operatornamesn-come-from

This is the representation of Jacobi sine in terms of theta-functions. Your $$A$$ and $$B$$ are $$\theta_1$$ and $$\theta_3$$ up to constant factors. See, for example
The reference in Gradshtein-Ryzhyk is on another handbook on elliptic functions, by A. M. Zhuravskiy (1941, in Russian). I checked it, it gives expressions for coefficients for $$n\leq 6$$. The general pattern does not seem to have any closed form expression.
These coefficients are even polynomials in $$k$$, with integer coefficients, but entering their coefficients into OEIS does not give a result.
• Table XII gives the elliptic functions as a ratio of theta functions; that doesn't give the coefficients of the power series of $A$ and $B$. Commented Apr 23, 2023 at 14:40