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An Integral Equation Involving Legendre Functions

Erdélyi, A. (1964) An Integral Equation Involving Legendre Functions. Journal of the Society for Industrial and Applied Mathematics, 12 (1). pp. 15-30. ISSN 0368-4245.

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Rodrigues’s formula can be applied also to (1.1) and (1.3) but here the situation is slightly more involved in that the integrals with respect to σ^2 are of fractional order and their inversion requires the knowledge of differentiation and integration of fractional order. In spite of this complication the method has its merits and seems more direct than that employed in [1] and [3]. Moreover, once differentiation and integration of fractional order are used, it seems appropriate to allow a derivative of fractional order with respect to σ^-1 to appear so that the ultraspherical polynomial in (1.3) may be replaced by an (associated) Legendre function. This will be done in the present paper.

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Additional Information:© 1964 Society for Industrial and Applied Mathematics. Received by the editors March 15, 1963. The preparation of this paper was partly supported by the National Science Foundation under Grant No. GP-213 to the California Institute of Technology. The author is indebted to Dr. Higgins for permission to utilize the typescript of the latter’s paper before publication, and also for obtaining access to further unpublished work.
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National Science FoundationGP-213
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ID Code:12095
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Deposited On:22 Oct 2008 22:55
Last Modified:03 Oct 2019 00:25

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