Integrals involving triplets of Jacobi and Gegenbauer polynomials and some 3j-symbols of SO(n), SU(n) and Sp(4)
The coupling coefficients (3j-symbols) for the symmetric (most degenerate) irreducible representations of the orthogonal groups SO(n) in a canonical basis and different semicanonical (tree) bases [with SO(n) restricted to SO(n')\times SO(n''), n'+n''=n] are expressed in terms of the integrals involving triplets of the Gegenbauer and the Jacobi polynomials. The derived usual triple-hypergeometric series (which do not reveal the apparent triangle conditions of the 3j-symbols) are rearranged (in contrast with math-ph/0201048) directly [without using their relation with the semistretched isofactors of the second kind for the complementary chain Sp(4)\supset SU(2)\times SU(2)] into formulas with more rich limits for summation intervals and obvious triangle conditions. The isofactors for the class-one representations of the orthogonal groups and for the class-two representations of the unitary groups (and, of course, the related integrals) 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. Expressions for the most general isofactors of SO(n) for coupling of the two symmetric irreps in the canonical basis are also derived.