Zonal harmonic series expansions of Legendre functions and associated Legendre functions

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Abstract
The Legendre functions pnu m(+or-cos theta ) of complex degree nu and integral order m may be expanded in terms of Legendre functions of integral degree and order, the latter being the zonal harmonic functions Pnm(cos theta ). Two methods for improving the convergence of a standard series expansion are discussed for the cases m=0 and 1. The first method involves repeated application of the relations between contiguous Legendre functions and the second employs integral relations between Legendre functions of different order. The formulae derived are suitable for computation and easily programmed.