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14 Legendre and Related FunctionsReal Arguments

§14.18 Sums

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§14.18(i) Expansion Theorem

For expansions of arbitrary functions in series of Legendre polynomials see §18.18(i), and for expansions of arbitrary functions in series of associated Legendre functions see Schäfke (1961b).

§14.18(ii) Addition Theorems

In (14.18.1) and (14.18.2), θ1, θ2, and θ1+θ2 all lie in [0,π), and ϕ is real.

14.18.1 Pν(cosθ1cosθ2+sinθ1sinθ2cosϕ)=Pν(cosθ1)Pν(cosθ2)+2m=1(-1)mPν-m(cosθ1)Pνm(cosθ2)cos(mϕ),
14.18.2 Pn(cosθ1cosθ2+sinθ1sinθ2cosϕ)=m=-nn(-1)mPn-m(cosθ1)Pnm(cosθ2)cos(mϕ).

In (14.18.3), θ1 lies in (0,12π), θ2 and θ1+θ2 both lie in (0,π), θ1<θ2, ϕ is real, and ν-1,-2,-3,.

14.18.3 Qν(cosθ1cosθ2+sinθ1sinθ2cosϕ)=Pν(cosθ1)Qν(cosθ2)+2m=1(-1)mPν-m(cosθ1)Qνm(cosθ2)cos(mϕ).

In (14.18.4) and (14.18.5), ξ1 and ξ2 are positive, and ϕ is real; also in (14.18.5) ξ1<ξ2 and ν-1,-2,-3,.

14.18.4 Pν(coshξ1coshξ2-sinhξ1sinhξ2cosϕ)=Pν(coshξ1)Pν(coshξ2)+2m=1(-1)mPν-m(coshξ1)Pνm(coshξ2)cos(mϕ),
14.18.5 Qν(coshξ1coshξ2-sinhξ1sinhξ2cosϕ)=Pν(coshξ1)Qν(coshξ2)+2m=1(-1)mPν-m(coshξ1)Qνm(coshξ2)cos(mϕ).

§14.18(iii) Other Sums

Christoffel’s Formulas

14.18.6 (x-y)k=0n(2k+1)Pk(x)Pk(y) =(n+1)(Pn+1(x)Pn(y)-Pn(x)Pn+1(y)),
14.18.7 (x-y)k=0n(2k+1)Pk(x)Qk(y) =(n+1)(Pn+1(x)Qn(y)-Pn(x)Qn+1(y))-1.

In these formulas the Legendre functions are as in §14.3(ii) with μ=0. The formulas are also valid with the Ferrers functions as in §14.3(i) with μ=0.

Zonal Harmonic Series

14.18.8 Pν(-x)=sin(νπ)πn=02n+1(ν-n)(ν+n+1)Pn(x),
ν.

Dougall’s Expansion

14.18.9 Pν-μ(x)=sin(νπ)πn=0(-1)n2n+1(ν-n)(ν+n+1)Pn-μ(x),
-1<x1, μ0, ν.

For a series representation of the Dirac delta in terms of products of Legendre polynomials see (1.17.22).

§14.18(iv) Compendia

For collections of sums involving associated Legendre functions, see Hansen (1975, pp. 367–377, 457–460, and 475), Erdélyi et al. (1953a, §3.10), Gradshteyn and Ryzhik (2000, §8.92), Magnus et al. (1966, pp. 178–184), and Prudnikov et al. (1990, §§5.2, 6.5). See also §18.18 and (34.3.19).