EXPANSIONS IN SERIES OF LEGENDRE FUNCTIONS

The classical Laplace expansion of an 'arbitrary ' function / in a series of Legendre polynomials is generalized, resulting in a (bilateral) series /(*) = £ "nKM for-Kx<l, for suitable functions / and a range of real or complex parameters n and v in the Legendre functions. There is also a related expansion of zero, 0 = £ bn?t+n(x) for-K*<l, where bn = (-l)"an. Combining these gives /(*) = £ c^K+Tnix) for-l<jc<l. n = — oo The third of these expansions was suggested by a previously unpublished biorthogonality relation, but it is more easily approached via the first two expansions. A key role is played by Love's much earlier work on a generalized Neumann-type integral [3]. 1