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14 Legendre and Related FunctionsComplex Arguments

§14.28 Sums

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§14.28(i) Addition Theorem

When z1>0, z2>0, |ph(z1-1)|<π, and |ph(z2-1)|<π,

14.28.1 Pν(z1z2-(z12-1)1/2(z22-1)1/2cosϕ)=Pν(z1)Pν(z2)+2m=1(-1)mΓ(ν-m+1)Γ(ν+m+1)Pνm(z1)Pνm(z2)cos(mϕ),

where the branches of the square roots have their principal values when z1,z2(1,) and are continuous when z1,z2\(0,1]. For this and similar results see Erdélyi et al. (1953a, §3.11).

§14.28(ii) Heine’s Formula

14.28.2 n=0(2n+1)Qn(z1)Pn(z2)=1z1-z2,
z11, z22,

where 1 and 2 are ellipses with foci at ±1, 2 being properly interior to 1. The series converges uniformly for z1 outside or on 1, and z2 within or on 2.

For generalizations in terms of Gegenbauer and Jacobi polynomials, see Theorem 2.1 in Cohl (2013b) and Theorem 1 in Cohl (2013a) respectively.

§14.28(iii) Other Sums

See §14.18(iv).