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

We show that the complex hypergeometric function describing 6j-symbols for SL(2,C) group is a special degeneration of the V -function — an elliptic analogue of the Euler-Gauss 2F1 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 there emerge the function describing 6j-symbols for the Faddeev modular double and the corresponding difference equations and symmetry transformations.

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