The contribution of Jacques Raynal to angular-momentum theory is highly valuable. In the present article, I intend to recall the main aspects of his work related to Wigner 3j symbols. It is well known that the latter can be expressed with a hypergeometric series. The polynomial zeros of the 3j coefficients were initially characterized by the number of terms of the series minus one, which is the degree of the coefficient. A detailed study of the zeros of the 3j coefficient with respect to the degree n for \(J=a+b+c\le 240\) (a, b and c being the angular momenta in the first line of the 3j symbol) by Raynal revealed that most zeros of high degree had small magnetic quantum numbers. This led him to define the order m to improve the classification of the zeros of the 3j coefficient. Raynal did a search for the polynomial zeros of degree 1 to 7 and found that the number of zeros of degree 1 and 2 are infinite, though the number of zeros of degree larger than 3 decreases very quickly as the degree increases. Based on Whipple’s symmetries of hypergeometric \(_3F_2\) functions with unit argument, Raynal generalized the Wigner 3j symbols to any arguments and pointed out that there are twelve sets of ten formulas (twelve sets of 120 generalized 3j symbols) which are equivalent in the usual case. In this paper, we also discuss other aspects of the zeros of 3j coefficients, such as the role of Diophantine equations and powerful numbers, or the alternative approach involving Labarthe patterns.

雅克·雷纳尔（Jacques Raynal）对角动量理论的贡献非常宝贵。在本文中，我打算回顾他与Wigner 3 j符号有关的工作的主要方面。众所周知，后者可以用超几何级数表示。3 j系数的多项式零点最初由系列的项数减去1来表征，这是系数的度数。3的零点的详细研究Ĵ系数相对于所述程度Ñ为\（J = A + B + C \文件240 \） （一个，b和c ^是在3的第一行中的角动量Ĵ雷纳尔（Raynal）表示，大多数高零点的磁量子数都较小。这使他定义了阶数m来改善3 j系数零点的分类。Raynal搜索了阶数为1到7的多项式零，发现阶数为1和2的零是无限的，尽管阶数大于3的零的数量会随着阶数的增加而迅速减少。基于具有单位参数的超几何\（_ 3F_2 \）函数的Whipple对称性，Raynal将Wigner 3 j符号推广到任何参数，并指出有12套10个公式（十二套120个广义3 j符号），在通常情况下是等效的。在本文中，我们还将讨论3 j系数零点的其他方面，例如Diophantine方程和强数的作用，或涉及Labarthe模式的替代方法。