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14 Legendre and Related FunctionsReal Arguments

§14.13 Trigonometric Expansions

When 0<θ<π, and ν+μ is not a negative integer,

14.13.1 Pνμ(cosθ) =2μ+1(sinθ)μπ1/2k=0Γ(ν+μ+k+1)Γ(ν+k+32)(μ+12)kk!sin((ν+μ+2k+1)θ),
14.13.2 Qνμ(cosθ) =π1/22μ(sinθ)μk=0Γ(ν+μ+k+1)Γ(ν+k+32)(μ+12)kk!cos((ν+μ+2k+1)θ).

These Fourier series converge absolutely when μ<0. If 0μ<12 then they converge, but, if θ12π, they do not converge absolutely.

In particular,

14.13.3 Pn(cosθ) =22n+2(n!)2π(2n+1)!k=013(2k-1)k!(n+1)(n+2)(n+k)(2n+3)(2n+5)(2n+2k+1)×sin((n+2k+1)θ),
14.13.4 Qn(cosθ) =22n+1(n!)2(2n+1)!k=013(2k-1)k!(n+1)(n+2)(n+k)(2n+3)(2n+5)(2n+2k+1)×cos((n+2k+1)θ),

with conditional convergence for each.

For other trigonometric expansions see Erdélyi et al. (1953a, pp. 146–147).