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

Question

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

Answer 1

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

Formules et propositions pour l'emploi des fonctions elliptiques d'apres les lecons et de notes manuscrites de M. K. Weierstrass redigees et publiees par M. H. A. Schwarz...Paris, 1894, Art. 26, p. 30. (available online), or

N. Akhiezer, Elements of the theory of elliptic functions, AMS, 1990. (Tables of formulas at the end, Table XII).

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.