The multiple sum formulas for 9j and 12j coefficients of SU(2) and uq(2)
Abstract
Seven different triple sum formulas for 9j coefficients of the quantum algebra uq(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 5φ4 series) as a q-generalization of Dougall's summation formula of the very well-poised hypergeometric 4F3(-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 Kamp\'e de F\'eriet functions and their q-generalizations. Several fourfold sum formulas [with each sum of the 5F4(1) or 5φ4 type] for the 12j coefficients of the second kind (without braiding) of the SU(2) and uq(2) are proposed, as well as expressions with five sums [of the 4F3(1) and 3F2(1) or 4φ3 and 3φ2 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 4F3(1) and 5F4(1) or 4φ3 and 5φ4 series, respectively] are considered.
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