The symmetries of the 2phi1

TL;DR

This paper proves that within a broad class of transformations, the only symmetries of the basic hypergeometric function 2phi1 are Heine's transformations, using q-difference equations to establish the result.

Contribution

It demonstrates that Heine's transformations are the only symmetries of 2phi1 within a specified class of transformations, extending understanding of its invariance properties.

Findings

Heine's transformations are the only symmetries of 2phi1 in the considered class.

Results on q-difference equations satisfied by 2phi1 support the main proof.

The paper characterizes the symmetry group of 2phi1 within a broad transformation class.

Abstract

We show that the only symmetries of the 2phi1 within a large class of possible transformations are Heine's transformations. The class of transformations considered consists of equation of the form 2phi1(a,b;c;q,z)= f(a,b,c,z) 2phi1(L(a,b,c,q,z)), where f is a q-hypergeometric term and L a linear operator on the logarithms of the parameters. We moreover prove some results on q-difference equations satisfied by 2phi1, which are used to prove the main result.