Skip to main content Accessibility help
×
×
Home

The Product of Two Legendre Polynomials

  • John Dougall (a1)
Extract

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.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      The Product of Two Legendre Polynomials
      Available formats
      ×
      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      The Product of Two Legendre Polynomials
      Available formats
      ×
      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      The Product of Two Legendre Polynomials
      Available formats
      ×
Copyright
References
Hide All
1.MacRobert, T. M., Spherical Harmonics (London), p. 95.
2.Adams, J. C., Proc. Roy. Soc., XXVII, 1878, p. 63; also Collected Scientific Papers, I, p. 187.
3.Hobson, E. W., Spherical and Ellipsoidal Harmonics (Cambridge), p. 83.
4.Bailey, W. N., “On the Product of Two Legendre Polynomials,” Proc. Camb. Phil. Soc., XXIX (1933), pp. 173177.
5.Hardy, G. H., “A Chapter from Ramanujan's Notebook,” Proc. Camb. Phil. Soc., XXI (1923), pp. 492503.
6.Dougall, J., “On Vandermonde's Theorem, and some general Expansions,” Proc Edin. Math. Soc., XXV (1907), pp. 114132.
7.Bailey, W. N., Generalized Hypergeometric Series (Cambridge University Tract, 1935), Chaps. IV, V, VI.
8.Hardy, G. H., loc. cit., p. 496.
9.Dougall, J., loc cit., equation (10).
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Glasgow Mathematical Journal
  • ISSN: 0017-0895
  • EISSN: 1469-509X
  • URL: /core/journals/glasgow-mathematical-journal
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed