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The Product of Two Legendre Polynomials

Published online by Cambridge University Press:  18 May 2009

John Dougall
John Dougall
Affiliation:
Glasgow.
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1. It is known that any polynomial in μ. can be expanded as a linear function of Legendre polynomials [1]. In particular, we have

The earlier coefficients, say A0, A2, A4 may easily be found by equating the coefficients of μp+q, μp+q-2, μp+q-4 on the two sides of (1). The general coefficient A2k might then be surmised, and the value verified by induction. This may have been the method followed by Ferrers, who stated the result as an exercise in his Spherical Harmonics (1877). A proof was published by J. C. Adams [2]. The proof now to be given follows different lines from his.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 1 , Issue 3 , September 1953 , pp. 121 - 125
Copyright
Copyright © Glasgow Mathematical Journal Trust 1953

References

REFERENCES

1.MacRobert, T. M., Spherical Harmonics (London), p. 95.Google Scholar
2.Adams, J. C., Proc. Roy. Soc., XXVII, 1878, p. 63; also Collected Scientific Papers, I, p. 187.Google Scholar
3.Hobson, E. W., Spherical and Ellipsoidal Harmonics (Cambridge), p. 83.Google Scholar
4.Bailey, W. N., “On the Product of Two Legendre Polynomials,” Proc. Camb. Phil. Soc., XXIX (1933), pp. 173–177.CrossRefGoogle Scholar
5.Hardy, G. H., “A Chapter from Ramanujan's Notebook,” Proc. Camb. Phil. Soc., XXI (1923), pp. 492–503.Google Scholar
6.Dougall, J., “On Vandermonde's Theorem, and some general Expansions,” Proc Edin. Math. Soc., XXV (1907), pp. 114–132.Google Scholar
7.Bailey, W. N., Generalized Hypergeometric Series (Cambridge University Tract, 1935), Chaps. IV, V, VI.Google Scholar
8.Hardy, G. H., loc. cit., p. 496.Google Scholar
9.Dougall, J., loc cit., equation (10).Google Scholar