2 research outputs found
Coupling coefficients of SO(n) and integrals over triplets of Jacobi and Gegenbauer polynomials
The expressions of the coupling coefficients (3j-symbols) for the most
degenerate (symmetric) representations of the orthogonal groups SO(n) in a
canonical basis (with SO(n) restricted to SO(n-1)) and different semicanonical
or tree bases [with SO(n) restricted to SO(n'})\times SO(n''), n'+n''=n] are
considered, respectively, in context of the integrals involving triplets of the
Gegenbauer and the Jacobi polynomials. Since the directly derived
triple-hypergeometric series do not reveal the apparent triangle conditions of
the 3j-symbols, they are rearranged, using their relation with the
semistretched isofactors of the second kind for the complementary chain
Sp(4)\supset SU(2)\times SU(2) and analogy with the stretched 9j coefficients
of SU(2), into formulae with more rich limits for summation intervals and
obvious triangle conditions. The isofactors of class-one representations of the
orthogonal groups or class-two representations of the unitary groups (and, of
course, the related integrals involving triplets of the Gegenbauer and the
Jacobi polynomials) turn into the double sums in the cases of the canonical
SO(n)\supset SO(n-1) or U(n)\supset U(n-1) and semicanonical SO(n)\supset
SO(n-2)\times SO(2) chains, as well as into the_4F_3(1) series under more
specific conditions. Some ambiguities of the phase choice of the complementary
group approach are adjusted, as well as the problems with alternating sign
parameter of SO(2) representations in the SO(3)\supset SO(2) and SO(n)\supset
SO(n-2)\times SO(2) chains.Comment: 26 pages, corrections of (3.6c) and (3.12); elementary proof of
(3.2e) is adde