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On the product of two Legendre polynomials

Published online by Cambridge University Press:  24 October 2008

W. N. Bailey
W. N. Bailey
Affiliation:
Trinity College
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Extract

1. As long ago as 1878 Neumann gave a formula expressing the product of two Legendre polynomials as a sum of such polynomials. In the same year Adams gave an inductive proof, and obtained the result in the form

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 29 , Issue 2 , May 1933 , pp. 173 - 177
Copyright
Copyright © Cambridge Philosophical Society 1933

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References

* Neumann, F. E., Beiträge zur Theorie der Kugelfunctionen (Leipzig, 1878), Part II, p. 86Google Scholar. See also Hobson, E. W., The theory of spherical and ellipsoidal harmonics (Cambridge, 1931), p. 85.Google Scholar

† Adams, J. C., Proc. Royal Soc. 27 (1878), 63CrossRefGoogle Scholar; also Collected Scientific Papers, 1, 487.Google Scholar

‡ Hobson, loc. cit., p. 21 (17).

* Whipple, F. J. W., “On well-poised series”, Proc. London Math. Soc. (2), 24 (1925), 247–263Google Scholar, formula (7.7). See also Whippley, , “A fundamental relation between generalized hypergeometric series”, Journal London Math. Soc. 1 (1926), 138–145CrossRefGoogle Scholar. When n > q we use a limiting form of the transformation owing to the presence of a negative integer in the denominator parameters.

† Hobson, loc. cit., p. 114 (51) and (52).

* Cf. Dougall, J., “A theorem of Sonine in Bessel functions, with two extensions to Spherical Harmonics”, Proc. Edin. Math. Soc. 37 (1919), 33–47CrossRefGoogle Scholar, formula (3). Dougall's formula gives the value of a similar type of integral taken between the limits 0, 1.

† Hobson, loc. cit., p. 87.

* Hobson, loc. cit., p. 113 (49), and p. 114 (50).

* Hobson, loc. cit., p. 234 (67).

† Hobson, loc. cit., p. 234 (68).

‡ Hobson, loc. cit., p. 204 (31).