 |
Free and accessible knowledge |
Journal articles
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.
https://hal.archives-ouvertes.fr/hal-01104334
Contributor : Ariane Rolland Connect in order to contact the contributor
Submitted on : Friday, January 16, 2015 - 3:25:49 PM
Last modification on : Monday, March 28, 2022 - 8:14:08 AM
Long-term archiving on: : Friday, September 11, 2015 - 6:58:42 AM
Contributor : Ariane Rolland Connect in order to contact the contributor
Submitted on : Friday, January 16, 2015 - 3:25:49 PM
Last modification on : Monday, March 28, 2022 - 8:14:08 AM
Long-term archiving on: : Friday, September 11, 2015 - 6:58:42 AM
File
7Article3.pdf
Explicit agreement for this submission