On Legendre functions of imaginary degree and associated integral transforms

New integral representations, asymptotic formulas, and series expansions in powers of tanh(t/2) are obtained for the imaginary and real parts of the Legendre function Piξ (cosh t). Coefficients of these series expansions are orthogonal polynomials in the real variable ξ . A number of relations for these orthogonal polynomials are obtained on the basis of the generating function. Several inversion theorems are proven for the integral transforms involving the Legendre function of imaginary degree. In many cases it is preferable to employ these transforms, than Mehler–Fok transforms, since conditions placed on functions are less restrictive.  2003 Elsevier Science (USA). All rights reserved