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

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 $J=a+b+c\le 240$ (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.

The Whipple parameters

Whipple [68] introduced six parameters $r_i$ ($i{=}0{,}1{,}2{,}3{,}4{,}5$) such that

$$\begin{aligned} \sum _{i=0}^5r_i=0. \end{aligned}$$

(81)

Setting $\alpha _{\ell mn}=1/2+r_{\ell }+r_m+r_n$ and $\beta _{mn}=1+r_m-r_n$, he defined the function

$$\begin{aligned} F_p(\ell ;mn)=\frac{1}{\varGamma (\alpha _{ijk},\beta _{m\ell },\beta _{n\ell })}~_3F_2\left[ \begin{array}{clll} \alpha _{imn},\alpha _{jmn},\alpha _{kmn}\\ \beta _{m\ell },\beta _{n\ell } \end{array};1 \right] , \end{aligned}$$

(82)

where i, j, k are used to represent those three numbers out of the six integers 0, 1, 2, 3, 4, 5 not already represented by $\ell$, m and n. By changing the signs of all the $r_i$ parameters and using the constraint (81), Whipple defined another function [68]:

$$\begin{aligned} F_n(\ell ;mn)=\frac{1}{\varGamma \left( \alpha _{\ell mn},\beta _{\ell m}, \gamma _{\ell n}\right) }. \end{aligned}$$

(83)

By permutation of the suffixes $\ell$, m, n over the six integers 0, 1, 2, 3, 4, 5, sixty $F_p$ functions and sixty $F_n$ functions can be written down. If there is no negative integer in the numerator parameters, these series converge only if the real parts of $\alpha _{ijk}$ in (82) and $\alpha _{\ell mn}$ in (83) are positive.