The triple sum formulas for 9j coefficients of SU(2) and uq(2)

Ališauskas, Sigitas

doi:10.1063/1.1312198

Seven different triple sum formulas for $9j$ coefficients of the quantum algebra $u_{q} (2)$ are derived, using for these purposes the usual expansion of $q-9j$ coefficients in terms of $q-6j$ coefficients and recently derived summation formula of twisted $q$ -factorial series (resembling the very well-poised basic hypergeometric ${}_{5}φ_{4}$ series) as a $q$ -generalization of Dougall’s summation formula of the very well-poised hypergeometric ${}_{4}F_{3} (−1)$ series. This way for $q=1$ Rosengren’s second proof of the SU(1,1) case is adapted for the SU(2) case to derive the known triple sum formula of Ališauskas and Jucys, as well as six new independent triple sum formulas for the Wigner $9j$ coefficients of the angular momentum theory. The mutual rearrangement possibilities of the derived triple sum formulas by means of the Chu–Vandermonde summation formulas are considered and applied to derive several versions of double sum formulas for the stretched $q-9j$ coefficients, which give new rearrangement and summation formulas of special Kampé de Fériet functions and their $q$ -generalizations.

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