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Simple proof of a formula due to Racah

Published online by Cambridge University Press:  24 October 2008

J. S. Griffith
J. S. Griffith
Affiliation:
Department of Theoretical Chemistry, University of Cambridge
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Extract

After some manipulation from the general formula, Racah (1,2) obtained a closed formula for his V coefficient in the special case of all three mj- components zero. He also showed that the integral of the product of three Legendre polynomials followed at once from these special V coefficients. In this case the general formula ((2), eq. 10–14) becomes

where

and a, b, c are all non-negative integers. By changing the parameter of summation to (a+ b−c−p) it follows that is zero unless 2g = a+b+c is even (2).

Type
Research Notes
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 59 , Issue 2 , April 1963 , pp. 507
Copyright
Copyright © Cambridge Philosophical Society 1963

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References

REFERENCES

(1)Racah, G.Phys. Rev. 62 (1942), 438.CrossRefGoogle Scholar
(2)Fano, U., and Racah, G., Irreducible tensorial sets (Academic Press, 1959).Google Scholar
(3)Fjeldstad, J. E., Math. Scand. 2 (1954), 46.CrossRefGoogle Scholar
(4)Good, I. J., Proc. Cambridge Philos. Soc. 58 (1962), 161.CrossRefGoogle Scholar