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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 a generalization in terms of Gegenbauer polynonials see Theorem 2.1 in Cohl (2013).

§14.28(iii) Other Sums

See §14.18(iv).