Coupling coefficients of SO(n) and integrals involving Jacobi and Gegenbauer polynomials

Abstract

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 ′ )×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)⊃SU(2)×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 classtwo 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)⊃SO(n − 1) or U(n)⊃U(n − 1) and semicanonical SO(n) ⊃SO(n − 2)×SO(2) chains, as well as into the 4 F 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)⊃SO(2) and SO(n) ⊃SO(n − 2)×SO(2) chains.

Coupling coefficients of SO(n) and integrals involving Jacobi and Gegenbauer polynomials

“…Recently in section 3 of [1], the integrals involving triplets of the Gegenbauer and the Jacobi polynomials and corresponding to special coupling coefficients of SO(n) have been rearranged, using their relation with the semistretched isofactors of the second kind for the complementary chain Sp(4) ⊃ SU (2) × SU (2).…”

“…In contrast to the possible proof of identities (3.2c)-(3.2e) of [1] by a direct but not obvious transformation procedure discussed in the concluding remarks (section 7) of [1], expression (3.2d) of [1] for the integrals involving the product of three Jacobi polynomials may be rearranged straightforwardly, without any allusion to the special isofactors of Sp (4). For this purpose we apply the symmetry relation (3.2a)-(3.2b) of [1] (i.e.…”

“…For this purpose we apply the symmetry relation (3.2a)-(3.2b) of [1] (i.e. interchange α a and β a , a = 0, 1, 2, 3) to (3.2d).…”

“…interchange α a and β a , a = 0, 1, 2, 3) to (3.2d). When α 0 and β 0 are integers, the 3 F 2 (1) type sums over z i in the (modified) expressions (3.2d) and (3.2e) of [1] correspond to the Clebsch-Gordan coefficients of SU(2) with the equivalent Regge 3 × 3 symbols…”

“…Note that the proof of relation between the corresponding finite 3 F 2 (1) series in (3.2d) and (3.2e) based on the composition of Thomae's transformation formulae or their Whipple's specifications for single restricting parameter (see [4,5]) is rather complicated. The denominator factor (l i + n/2) z i in (3.6c) of [1] should be corrected to (l i + n/2) z i , and the denominator factor (α i + 3/2) z i in (3.12) of [1] should be corrected to (α i + k i + 3/2) z i . The factor C l +n/2−1 l i −l (cos θ) in (3.8a) of [1] should be replaced (twice) by C l +n/2−1 l 1 −l (cos θ).…”