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.
