RaduRK

RaduRK: Ramanujan-Kolberg Program

Authors

RaduRK is a Mathematica implementation of an algorithm developed by Cristian-Silviu Radu. The algorithm takes as input an arithmetic sequence a(n) generated from a large class of q-Pochhammer quotients, together with a given arithmetic progression mn+j, and the level of a given congruence subgroup. The algorithm produces expressions for the generating function of a(mn+j) in terms of Q-linear combinations of Dedekind eta quotients which are modular over the subgroup. Identities of this form include famous results by Ramanujan which demonstrate the divisibility properties of p(5n+4) and p(7n+5). The algorithm relies on certain powerful finiteness conditions imposed by the study of modular functions, and illustrates the utility of the subject to computational number theory. The package has been developed by Nicolas Allen Smoot.

Licence

This program is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with this program. If not, see https://www.gnu.org/licenses.

The package

For using the package, download the file RaduRK.wl, which can be found at

RaduRK.wl

Place this file into a directory where Mathematica will find it. For a demonstration of how to use the package see

RKSupplement1.nb.

For more ambitious examples, see:

RKSupplement2.nb.

The package requires 4ti2 and math4ti2.m, see installation instructions.

Literature

Instructions for the proper installation for these packages and RaduRK can be found in the following paper:

N. Smoot, "On the Computation of Identities Relating Partition Numbers in Arithmetic Progressions with Eta Quotients: An Implementation of Radu's Algorithm".

For details concerning the design of the algorithm, consult the following:

S. Radu, "An Algorithmic Approach to Ramanujan's Congruences," Ramanujan Journal, 20, pp. 215-251 (2009).
S. Radu, "An Algorithmic Approach to Ramanujan-Kolberg Identities," Journal of Symbolic Computation, 68, pp. 225-253 (2015).

Bugs

Please report any bugs or other suggestions to Nicolas Smoot.