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On algebraic differential equations satisfied by some elliptic functions II

Abstract : 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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Submitted on : Friday, January 16, 2015 - 3:25:49 PM
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P Chowla, S Chowla. On algebraic differential equations satisfied by some elliptic functions II. Hardy-Ramanujan Journal, Hardy-Ramanujan Society, 1984, Volume 7 - 1984, pp.13 - 16. ⟨10.46298/hrj.1984.107⟩. ⟨hal-01104334⟩



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