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

TL;DR

This paper derives explicit formulas for coupling coefficients of SO(n) and integrals over triplets of Jacobi and Gegenbauer polynomials, revealing triangle conditions and sum limits, with applications to group representation theory.

Contribution

It provides new rearranged formulas for 3j-symbols of SO(n) using hypergeometric series and group isofactors, clarifying triangle conditions and sum limits.

Findings

Explicit formulas for SO(n) coupling coefficients are derived.

Rearranged hypergeometric series reveal triangle conditions.

Integrals over polynomial triplets are expressed as double sums and hypergeometric series.

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'})\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.