Elliptic hypergeometric function and $6j$-symbols for the SL(2,$\mathbb{C}$) group

TL;DR

This paper demonstrates that the complex hypergeometric function for $6j$-symbols of the $SL(2,bC)$ group is a degeneration of the elliptic $V$-function, revealing new relations and symmetries.

Contribution

It establishes a connection between $6j$-symbols for $SL(2,bC)$ and elliptic hypergeometric functions, introducing new difference relations and symmetry transformations.

Findings

The $6j$-symbols are special degenerations of the elliptic $V$-function.

Derived mixed difference-recurrence relations for the hypergeometric function.

Identified functions related to the Faddeev modular double and their symmetries.

Abstract

We show that the complex hypergeometric function describing $6j$-symbols for $SL(2,\mathbb{C})$ group is a special degeneration of the $V$-function -- an elliptic analogue of the Euler-Gauss $_2F_1$ hypergeometric function. For this function, we derive mixed difference-recurrence relations as limiting forms of the elliptic hypergeometric equation and some symmetry transformations. At the intermediate steps of computations, there emerge a function describing the $6j$-symbols for the Faddeev modular double and the corresponding difference equations and symmetry transformations.