{"docId":107,"paperId":107,"url":"https:\/\/hrj.episciences.org\/107","doi":"10.46298\/hrj.1984.107","journalName":"Hardy-Ramanujan Journal","issn":"","eissn":"2804-7370","volume":[{"vid":18,"name":"Volume 7 - 1984"}],"section":[],"repositoryName":"Hal","repositoryIdentifier":"hal-01104334","repositoryVersion":1,"repositoryLink":"https:\/\/hal.archives-ouvertes.fr\/hal-01104334v1","dateSubmitted":"2015-03-03 16:13:41","dateAccepted":"2015-06-12 16:05:22","datePublished":"1984-01-01 08:00:00","titles":{"en":"On algebraic differential equations satisfied by some elliptic functions II"},"authors":["Chowla, P","Chowla, S"],"abstracts":{"en":"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."},"keywords":["[MATH] Mathematics [math]"]}