An Integral Equation Involving Legendre Functions

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.