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

Abstract

The corrected triple sum expression of Ališauskas (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 Ališauskas also related to result of Hormeßand Junker 1999) is rearranged into three new different double sum expressions (related to the hypergeometric Kampé de Fériet 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 Kampé de Fériet $F_{1:3}^{1:4}$ 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).