Determine the number of auxiliary series required by the functional-equation search

Determine how many auxiliary series of the form S_{C_1,C_2}(x) are required to uniquely define a selected Andrews–Gordon type double series through a system of functional equations, and determine whether the selected series belongs to a system involving two, three, or more mutually related series.

Background

The search algorithm generates q-difference equations relating the double series S_{C_1,C_2}(x) for different values of C_1 and C_2. To obtain a system that uniquely determines the target series, the authors must know how many related auxiliary series are needed. They explicitly state that their procedure does not determine in advance whether the relevant system contains two, three, or more series, leaving the required system size unresolved.

References

Unlike fneq_A_PNAS, we do not know how many other series are needed to uniquely define all of those series at once. In other words, we do not know if the series we chose to study is one of the two, or three, or more similar series all related in that many functional equations.

— Finding systems of functional equations for Andrews-Gordon type series  (2501.03189 - Kılıç et al., 6 Jan 2025) in Section 2, The search algorithm