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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+x1x)μ/2k=0(1212x)kΓ(kν)Γ(k+ν+1)k!Γ(kμ+1)(ψ(k+ν+1)ψ(kν)).
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)/(1x))μ/2 in (14.11.3) is replaced by ((x+1)/(x1))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).