qseries-sage

A Python/SageMath port of Frank Garvan's Maple qseries package (version 1.3): q-products, theta functions, product conversion (Andrews' algorithm, eta-products, Jacobi products) and the search for polynomial relations and congruences between q-series — without Maple.

It is a reimplementation written from Garvan's public documentation and tutorial, and checked function by function against the real Maple package (see Agreement with Maple). The qseries package and its documentation are by Frank Garvan: https://qseries.org/fgarvan/qmaple/qseries/.

Install

qseries-sage needs SageMath. Pick one:

With an existing SageMath (Sage 9.5 or later):

sage -pip install qseries-sage

Without SageMath, from pip alone (Linux and macOS, Python 3.10-3.13), using the passagemath wheels. This downloads a full Sage library, roughly 1 GB:

python -m pip install --upgrade pip
python -m pip install --only-binary=:all: "qseries-sage[passagemath]"

Upgrading pip first matters: the older pip that comes with some systems (e.g. Ubuntu 22.04) does not recognise passagemath's prebuilt wheels and tries to compile parts of Sage from source, which usually fails. --only-binary=:all: makes sure nothing is compiled.

Windows: passagemath supports Linux and macOS only, so install inside WSL (Windows Subsystem for Linux) with the commands above.

Check it works:

python -c "from qseries import *; print(packageversion())"   # 1.3.0

The distribution is called qseries-sage; the import name is qseries. To install from a clone of the repository instead, run pip install . in it.

Example

from sage.all import LaurentSeriesRing, QQ
from qseries import etaq, theta3, theta4, prodmake, findhom, findcong, QSeries

R = LaurentSeriesRing(QQ, 'q', default_prec=110)
q = R.gen()

# Quadratic relations among theta functions (Garvan tutorial, section 4.1)
findhom([theta3(q, 100), theta4(q, 100), theta3(q**2, 100), theta4(q**2, 100)], q, 2, 0)
# -> {X1^2 + X2^2 - 2*X3^2, X1*X2 - X4^2}

# Ramanujan's partition congruences
P = QSeries(etaq(q, 1, 1001).series.__invert__(), 1001)
findcong(P, 1000)
# -> {(4, 5, 5), (5, 7, 7), (6, 11, 11), (24, 25, 25)}

Series are QSeries objects: a Sage Laurent series with an explicit truncation order T and a rational q-power shift r (so theta2 can carry its q^(1/4) factor).

Functions (41)

Output types: QSeries, QProduct, EtaProduct, EProduct, JacProduct, PochProduct, MProduct, QFactorResult, ZQProduct.

Differences from the Maple package

• Relations are returned as Sage polynomials in X1..Xk (or X, Y for findpoly) instead of Maple's global X[i]; the SL name lists of findlincombo* are accepted but not used.
• findcong returns a set of (B, A, M) tuples.
• zqfactor returns a ZQProduct of factors (1 - a z^i q^j)^e. Like Maple's version it is best-effort: when it cannot find such a product it warns and returns None.
• findhommodp uses every coefficient, including q^0. Maple 2026 + qseries 1.3 skips the q^0 coefficient, so it reports {X[1], X[2]} for findhommodp([1+5*q, 1], 5, q, 1, 0) (claiming 1 = 0 mod 5); qseries-sage gives the correct X1 - X2.
• qfactor(f, T): Maple 2026 + qseries 1.3 returns 1 for quotients such as q^4(1-q^6)(1-q^7)/((1-q^3)(1-q^12)) when T is given (20 or 40), but factors them correctly, expanded or not, when T is left out. qseries-sage factors them either way.

Agreement with Maple

A battery of 130 cases covering every function was run in Maple 2026.2 with qseries 1.3 and compared with this package (parity/, latest report): 124 agree exactly, 1 agrees in value (we return an extra valid answer), and the other 5 are the Maple problems listed under Differences from the Maple package.

Tests

sage -python -m pytest -q

Expected values in the tests come from Garvan's documentation and tutorial, or from stated mathematical identities — never from this package's own output.

License and credits

qseries-sage 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 2 of the License, or (at your option) any later version. See LICENSE.

The qseries package for Maple, its function set and its documentation are the work of Frank Garvan (University of Florida): https://qseries.org/fgarvan/qmaple/qseries/. This port is an independent reimplementation in Python/SageMath written from that public documentation and the tutorial

F. G. Garvan, A q-product tutorial for a q-series MAPLE package, Séminaire Lotharingien de Combinatoire 42 (1999), Article B42d. arXiv:math/9812092.

One function, zqfactor, follows the algorithm of the Maple procedure as printed by Maple when the package is loaded. If you use this package in research, please cite Garvan's qseries package and tutorial.