10.46298/hrj.1984.107
https://hrj.episciences.org/107
Chowla, P
P
Chowla
Chowla, S
S
Chowla
On algebraic differential equations satisfied by some elliptic functions II
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.
episciences.org
[MATH] Mathematics [math]
2015-06-12
1984-01-01
1984-01-01
en
journal article
https://hal.archives-ouvertes.fr/hal-01104334v1
2804-7370
https://hrj.episciences.org/107/pdf
VoR
application/pdf
Hardy-Ramanujan Journal
Volume 7 - 1984
Researchers
Students