论文标题
Wigner $ 3J $系数的某些属性：非平凡的零和连接到高几幅功能
Some properties of Wigner $3j$ coefficients: non-trivial zeros and connections to hypergeometric functions
论文作者
论文摘要
雅克·雷纳尔（Jacques Raynal）对角度摩肌理论的贡献非常有价值。在本文中，我打算回忆起他与Wigner $ 3J $符号有关的作品的主要方面。众所周知，后者可以通过超几何序列表示。 $ 3J $系数的多项式零最初的特征是串联术语减去一个系数，这是系数的程度。 $ j = a + j = a + b + b + c \ leq 240 $（$ a $，$ b $和$ c $是$ 3J $符号的第一行）的$ 3J $系数的零研究的详细研究。这使他定义了$ m $的订单，以改善$ 3J $系数的零分类。雷纳尔（Raynal）对1至7度的多项式零进行了搜索，发现1和2度的零的数量是无限的，尽管随着程度的增加，程度大于3的零的零数会很快减少。基于Whipple的超几何$ _3F_2 $与单位参数的函数的转换，Raynal将Wigner $ 3J $符号推广到任何参数，并指出十二套十个公式（120套120个广义$ 3J $符号）在惯用情况下是等效的。在本文中，我们还讨论了$ 3J $系数的零的其他方面，例如Diophantine方程和强大数字的作用，或涉及Labarthe模式的替代方法。
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 $J = a + b + c \leq 240$ ($a$, $b$ 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 transformations 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.
