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14 Legendre and Related FunctionsReal Arguments

§14.11 Derivatives with Respect to Degree or Order

14.11.1 νPνμ(x)=πcot(νπ)Pνμ(x)-1πAνμ(x),
14.11.2 νQνμ(x)=-12π2Pνμ(x)+πsin(μπ)sin(νπ)sin((ν+μ)π)Qνμ(x)-12cot((ν+μ)π)Aνμ(x)+12csc((ν+μ)π)Aνμ(-x),

where

14.11.3 Aνμ(x)=sin(νπ)(1+x1-x)μ/2k=0(12-12x)kΓ(k-ν)Γ(k+ν+1)k!Γ(k-μ+1)(ψ(k+ν+1)-ψ(k-ν)).
14.11.4 μPνμ(x)|μ=0 =(ψ(-ν)-πcot(νπ))Pν(x)+Qν(x),
14.11.5 μQνμ(x)|μ=0 =-14π2Pν(x)+(ψ(-ν)-πcot(νπ))Qν(x).

(14.11.1) holds if Pνμ(x) is replaced by Pνμ(x), provided that the factor ((1+x)/(1-x))μ/2 in (14.11.3) is replaced by ((x+1)/(x-1))μ/2. (14.11.4) holds if Pνμ(x), Pν(x), and Qν(x) are replaced by Pνμ(x), Pν(x), and Qν(x), respectively.

See also Szmytkowski (2006, 2009, 2011, 2012), Cohl (2010, 2011) and Magnus et al. (1966, pp. 177–178).