# Origin of the Recurrence Relation for Clebsch-Gordan Coefficients

The Clebsch-Gordan coefficients $$C_{\pm }(J,M)$$ arise in quantum mechanics in the problem of addition of angular momentum. They also arise in mathematics in the more theoretical (but related) problem of the decomposition of the tensor product of irreducible representations of a group into irreducible representations. Originally, Clebsch and Gordan discovered the coefficients while working on a problem in invariant theory (which in modern language is equivalent to the decomposition problem of representations mentioned above).

It is well known, as can be found in any quantum mechanics textbook, that the Clebsch-Gordan coefficients satisfy the recurrence relation:

$$C_{\pm }(J,M)\langle j_{1}\,m_{1}\,j_{2}\,m_{2}|J\,(M\pm 1)\rangle =C_{\pm }(j_{1},m_{1}\mp 1)\langle j_{1}\,(m_{1}\mp 1)\,j_{2}\,m_{2}|J\,M\rangle +C_{\pm }(j_{2},m_{2}\mp 1)\langle j_{1}\,m_{1}\,j_{2}\,(m_{2}\mp 1)|J\,M\rangle$$

This relation (when supplemented by a rather trivial initial condition) allows us to calculate all of the Clebsch-Gordan coefficients, a problem which is very difficult to solve without this relation.

I'm interested in who first discovered this recurrence relation, and what line of thought led them to it (was it considerations in the representation theory of Lie algebras?) Wikipedia says it was first discovered by Giulio Racah in 1941, but this is clearly not true, as the relation appears explicitly in Condon and Shortley's classic "The Theory of Atomic Spectra" from 1935 (in chapter 3, section 14). In fact Racah himself cites Condon and Shortley for proof of the relation, in his 1942 paper "Theory of Complex Spectra II".

Unfortunately, Condon and Shortley in their preface explain that they deliberately hide the group-theoretic arguments and methods that underlie their treatment, in an attempt to make the book more readable to the physicist, for whom Lie group theory was an obscure part of mathematics at the time. For this reason I'm interested in where the relation first appeared? I could not find an earlier reference to them, however I did find a similar recurrence relation for the related Lande-g factor in Weyl's 1929 "The Theory of Group and Quantum Mechanics" (in chapter IV, section A.4).