Other special functions with SL(3,Z) modularity

Determine whether families of special functions other than the elliptic Gamma function obey analogous SL(3,Z) modular transformation laws, possibly with different phase factors, and whether the elliptic Gamma function is isolated or belongs to a broader class of modular special functions.

Background

The paper presents the elliptic Gamma function as the prototypical special function satisfying an SL(3,Z) modular relation together with related identities. Unlike the systematic construction of SL(2,Z)-invariant combinations of conformal characters, no comparably general construction principle is known for SL(3,Z) modularity. The authors therefore identify the existence and classification of additional modular special functions as an unresolved structural problem with potential applications to quantum-field-theory partition functions.

References

This naturally raises the question of whether there exist other families of special functions obeying similar SL$(3,\mathbb Z)$ modular transformation laws, possibly with different phase factors.

— Mellin space reflections, modularity of elliptic Gamma functions and beyond  (2608.24083 - Lei et al., 25 Aug 2026) in Section 1, Introduction

Determining whether such a realization can be expressed in terms of a recognizable special function, and understanding its associated SL$(3,\mathbb Z)$ transformation law, are interesting questions that we leave for future work.

— Mellin space reflections, modularity of elliptic Gamma functions and beyond  (2608.24083 - Lei et al., 25 Aug 2026) in Section 3, Subsection “Transposed trilinear reflection”