SO（n）的耦合系数与Jacobi和Gegenbauer多项式的三重积分
Coupling coefficients of SO(n) and integrals over triplets of Jacobi and Gegenbauer polynomials
摘要：
耦合系数的表达式（3 j-符号）对于最简并的正交群SO（n）在标准基上的（对称）表示（SO（n）限制于SO（n-1））和不同的半正则基或树基[SO（n）限制于SO（n '}）\times SO（n“），n'+ n”=n]，在上下文中的积分涉及三元组的盖根鲍尔和雅可比多项式。由于直接导出的三重超几何级数没有揭示3 j符号的表观三角形条件，利用它们与补链Sp（4）\supset SU（2）\times SU（2）的第二类半拉伸同因式的关系，并与SU（2）的拉伸9 j系数类比，将它们重新排列成求和区间有更丰富限制和明显三角形条件的公式。正交群或酉群的第一类表示的等因子（当然，涉及Gegenbauer多项式和Jacobi多项式的三元组的相关积分）在正则SO（n）\supset SO（n-1）或U（n）\supset U（n-1）和半正则SO（n）\supset SO（n-2）\times SO（2）链的情况下变成双重和，在更具体的条件下，也可以引入_4F_3（1）系列。修正了补群方法中相位选择的一些模糊性，以及SO（3）\supset SO（2）和SO（n）\supset SO（n-2）\times SO（2）链中SO（2）表示的符号参数交替的问题.
英文摘要：
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.
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