pages.uoregon.edu/posts/2026-02-16-wigner.md

title	author	published	references
Computing the Wigner 9j Symbols Efficiently	Thomas (Tom) C. Gorordo	false	type id author title title-short book GrahamKnuthPatashnik1994ConcreteMathematics2e family given Graham Ronald L. family given Knuth Donald E. family given Patashnik Oren Concrete Mathematics: A Foundation for Computer Science Concrete Mathematics

As a test post, here's a transcription of some notes I have laying around from writing an add9j PR for Jutho/WignerSymbols.jl.

Wigner Symbols Formulae

The Wigner 3x\text{j} symbols are all exactly representable in the form

W_{3x} = \frac{n\sqrt{s}}{q}\hspace{1cm} n,s,q\in\mathbb{Z}

So, if one can use exact integer arithmetic in computing the three components, a computer implementation of the symbols should be able to incur at-worst typical floating point root and ratio errors.

This requires the usage of formulae that are amenible to such integer manipulations.

3j Formula

Let \delta(j_1, j_2, j_3) = (j_1 \leq j_2 + j_3)\bigwedge (j_2 \leq j_1 + j_3)\bigwedge (j_3 \leq j_1 + j_2)\bigwedge(j_1 + j_2 + j_3\in\mathbb{Z}) for j_i\in\frac{1}{2}\mathbb{Z} a half-integer be called the "triangle condition". If not satisfied, a symbol is zero.

The 3j symbol can be computed according to:

6j Formula

The 6j symbol, similarly, can be computed as:

Define the Wei square brackets purely in terms of binomial coefficients by

9j Formula

The 9j formula has a few additional pieces.

One might have some fun manipulating some of these forms using the tricks in [@GrahamKnuthPatashnik1994ConcreteMathematics2e].

Calculation and Tabulation

For fast, exact evaluation, the main integer-arithmetic idea is to simplify the evaluation of the various sums of ratios of (potentially large) factorials by using prime factorizations of those factorials to reduce ratios by an LCD ad extract common factors across the sums. Remaining arithmetic can be done via a big integer implementation.

The core of this approach uses that prime factorizations of factorials are conveniently obtained recursively (and thus, to tabulate). The crux is:

Euclid's Lemma

If p | ab for prime p, then p|a or p|b.

\implies If prime p|n! then p|k for k\leq n one of the factors.

More specifically, Legendre's Theorem says:

\sharp_p(n!) = \sum_{k=1} \sharp_p(k) = \sum_{k=1}^{\lfloor\log_p(n)\rfloor} \lfloor\frac{n}{p^k}\rfloor

where \sharp_p(k) gives the number of times p divides k.