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

We show that the complex hypergeometric function describing $6j$-symbols for the $SL(2,\mathbb C)$ 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 limit 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.