The multiple sum formulas for 9j and 12j coefficients of SU(2) and $u_q$(2)

Abstract

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 recent summation formula of twisted $q$-factorial series (resembling the very well-poised basic hypergeometric $_5ϕ_4$ series) as a $q$-generalization of Dougall's summation formula of the very well-poised hypergeometric $_4F_3(-1)$ series. This way for $q=1$ the new proof of the known triple sum formula is proposed, as well as six new triple sum formulas for $9j$ coefficients of the SU(2) group, in 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. Several fourfold sum formulas [with each sum of the $_5F_4(1)$ or $_5ϕ_4$ type] for the $12j$ coefficients of the second kind (without braiding) of the SU(2) and $u_q(2)$ are proposed, as well as expressions with five sums [of the $_4F_3(1)$ and $_3F_2(1)$ or $_4ϕ_3$ and $_3ϕ_2$ type] for the $12j$ coefficients of the first kind (with braiding) instead of the usual expansion in terms of $q$-$6j$ coefficients. Stretched and doubly stretched $q$-$12j$ coefficients [as triple, double or single sums, related to composed or separate hypergeometric $_4F_3(1)$ and $_5F_4(1)$ or $_4ϕ_3$ and $_5ϕ_4$ series, respectively] are considered.