**56**: 296

https://doi.org/10.1140/epja/s10050-020-00303-9

Regular Article - Theoretical Physics

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

### A tribute to Jacques Raynal

^{1}
CEA, DAM, DIF, 91297, Arpajon, France

^{2}
Université Paris-Saclay, CEA, Laboratoire Matière sous Conditions Extrêmes, 91680, Bruyères-le-Châtel, France

^{a}
jean-christophe.pain@cea.fr

Received:
1
October
2020

Accepted:
6
November
2020

Published online:
23
November
2020

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 3*j* symbols. It is well known that the latter can be expressed with a hypergeometric series. The polynomial zeros of the 3*j* 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 3*j* coefficient with respect to the degree *n* for (*a*, *b* and *c* being the angular momenta in the first line of the 3*j* 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 3*j* 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 functions with unit argument, Raynal generalized the Wigner 3*j* symbols to any arguments and pointed out that there are twelve sets of ten formulas (twelve sets of 120 generalized 3*j* symbols) which are equivalent in the usual case. In this paper, we also discuss other aspects of the zeros of 3*j* coefficients, such as the role of Diophantine equations and powerful numbers, or the alternative approach involving Labarthe patterns.

*© Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2020*