Jacobi多项式和Gegenbauer多项式的三重数与SO（n），SU（n）和Sp（4）的3 j-符号的积分
Integrals involving triplets of Jacobi and Gegenbauer polynomials and some 3j-symbols of SO(n), SU(n) and Sp(4)
摘要：
正交群SO（n）在标准基和不同的半标准（树）基[SO（n）限制为SO（n '）\times SO（n“），n'+ n”=n]中的对称（最退化）不可约表示的耦合系数（3 j-符号）用Gegenbauer和Jacobi多项式的三重态积分表示.导出的通常三重超几何级数（其不揭示3 j符号的明显三角形条件）被重新排列（与math-ph/0201048相反）直接[不使用它们与互补链Sp（4）\supset SU（2）\times SU（2）的第二类半拉伸同功因子的关系]转化为求和区间极限更丰富、三角形条件更明显的公式。正交群的第一类表示和酉群的第二类表示的同素因子（当然，还有相关的积分）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）链的情况下变成了双重和，以及在更具体的条件下进入$_4F_3(1)$系列。最一般的SO（n）的isofactor的正则基耦合的两个对称的irreps的表达式也推导出来。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
