Some properties of Wigner $3j$ coefficients: non-trivial zeros and connections to hypergeometric functions

TL;DR

This paper reviews properties of Wigner 3j coefficients, focusing on their polynomial zeros, connections to hypergeometric functions, and generalizations, highlighting Raynal's contributions and exploring zeros' classifications and related mathematical structures.

Contribution

It revisits Raynal's work on Wigner 3j symbols, extends the analysis of their zeros, and discusses their generalizations via hypergeometric functions and related mathematical frameworks.

Findings

Most zeros of high-degree 3j coefficients have small magnetic quantum numbers.

Number of zeros of degree 1 and 2 are infinite, but higher degrees have fewer zeros.

Generalizations of 3j symbols involve twelve sets of formulas based on hypergeometric transformations.

Abstract

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.