episciences.org_107_1653157564
1653157564
episciences.org
raphael.tournoy+crossrefapi@ccsd.cnrs.fr
episciences.org
HardyRamanujan Journal
28047370
10.46298/journals/hrj
https://hrj.episciences.org
01
01
1984
Volume 7  1984
On algebraic differential equations satisfied by some elliptic functions II
P
Chowla
S
Chowla
In (I) we obtained the ``implicit'' algebraic differential equation for the function defined by $Y=\sum_1^{\infty}\frac{n^a x^n}{1x^n}$ 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.
01
01
1984
107
https://hal.archivesouvertes.fr/hal01104334v1
10.46298/hrj.1984.107
https://hrj.episciences.org/107

https://hrj.episciences.org/107/pdf