6j-symbols for symmetric representations of SO(n) as the double series

TL;DR

This paper presents new double and triple sum formulas for 6j-symbols of symmetric representations of SO(n), revealing symmetries and relating them to hypergeometric series, with implications for recoupling coefficients of symplectic groups.

Contribution

It introduces alternative double and triple sum expressions for SO(n) 6j-symbols, highlighting their symmetry properties and connecting them to hypergeometric Kampé de Fériet series, and extends results to Sp(2n).

Findings

New double sum expressions for 6j-symbols of SO(n)

Revealed Regge type symmetry in terms of Kampé de Fériet series

Derived recoupling coefficients for Sp(2n) from SO(-2n) relations

Abstract

The corrected triple sum expression of Ali\v{s}auskas (1987) for the recoupling (Racah) coefficients (6j-symbols) of the symmetric (most degenerate) representations of the orthogonal groups SO(n) (previously derived from the fourfold sum expression of Ali\v{s}auskas also related to result of Horme\ss and Junker 1999) is rearranged into three new different double sum expressions (related to the hypergeometric Kamp\'e de F\'eriet type series) and a new triple sum expression with preferable summation condition. The Regge type symmetry of special 6j-symbols of the orthogonal groups SO(n) in terms of special Kamp\'e de F\'eriet $F_{1:3}^{1:4}$ series is revealed. The recoupling coefficients for antisymmetric representations of symplectic group Sp(2n) are derived using their relation with the recoupling coefficients of the formal orthogonal group SO(-2n).