Expansions in Series of Legendre Functions

The classical Laplace expansion of an ‘arbitrary’ function f in a series of Legendre polynomials is generalized, resulting in a (bilateral) series f ( x ) = ∑ n = ‐ ∞ ∞a n P v + n μ ( x )for −1 < x < 1, for suitable functions f and a range of real or complex parameters μ and ν in the Legendre functions. There is also a related expansion of zero, 0 = ∑ n = ‐ ∞ ∞b n P v + n μ ( x )for −1 < x < 1, where b n = (−1) n a n . Combining these gives f ( x ) = ∑ n = ‐ ∞ ∞c 2 nP v + 2 n μ ( x )for −1 < x < 1. The third of these expansions was suggested by a previously unpublished biorthogonality relation, but it is more easily approached via the first two expansions.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 ].