Note onSU 3 Wigner coefficients

On the Symmetric Tensor Operators of the Unitary Groups

“…It is evident that the sum in (13) in this new form has a much wider summation region, because it involves n -1 new regions. However, this procedure does not change the value of the sum (13), because nonvanishing terms in these new regions are compensated by a set of terms equal in absolute value and opposite in sign to the first ones.…”

“…of Un-I' one obtains i=l (13) The number of the summation parameters in (13) is n. The terms of this sum depend on QI(O ~ QI ~ P -q). However, the final result must be independent of this parameter.…”

“…However, the final result must be independent of this parameter. It turns out that in expression (13) the summation with respect to one of parameters rin can be carried out by the use of the summation formula (14) a, b, c being nonnegative integers. Equation (14) can be proved by induction starting from Eq.…”

“…10. In order to use this summation formula for the purpose indicated in Eq. (13), we transform the factorials depending on rin (i fixed) into the quasipowers (c.f. Ref.…”

“…However, this procedure does not change the value of the sum (13), because nonvanishing terms in these new regions are compensated by a set of terms equal in absolute value and opposite in sign to the first ones. These terms can be found by renumerating the summation parameters r in -j E-7 rin -i, j ;>t i labeling the newly appearing regions.…”