On algebraic differential equations satisfied by some elliptic functions IIArticle

• 1 University of Colorado [Boulder]
• 2 University of Cambridge [UK]

In (I) we obtained the ``implicit'' algebraic differential equation for the function defined by $Y=\sum_1^{\infty}\frac{n^a x^n}{1-x^n}$ where $a$ is an odd positive integer, and conjectured that there are no algebraic differential equations for the case when $a$ is an even integer. In this note we obtain a simple proof that (this has been known for almost 200 years)

$$Y=\sum_1^{\infty}x^{n^2}~~~~(\vert x\vert<1)$$ satisfies an algebraic differential equation, and conjecture that $Y=\sum_1^{\infty} x^{n^k}$ (where $k$ is a positive bigger than $2$) does not satisfy an algebraic differential equation.

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