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14 Legendre and Related FunctionsReal Arguments

§14.10 Recurrence Relations and Derivatives

14.10.1 Pνμ+2(x)+2(μ+1)x(1-x2)-1/2Pνμ+1(x)+(ν-μ)(ν+μ+1)Pνμ(x)=0,
14.10.2 (1-x2)1/2Pνμ+1(x)-(ν-μ+1)Pν+1μ(x)+(ν+μ+1)xPνμ(x)=0,
14.10.3 (ν-μ+2)Pν+2μ(x)-(2ν+3)xPν+1μ(x)+(ν+μ+1)Pνμ(x)=0,
14.10.4 (1-x2)dPνμ(x)dx=(μ-ν-1)Pν+1μ(x)+(ν+1)xPνμ(x),
14.10.5 (1-x2)dPνμ(x)dx=(ν+μ)Pν-1μ(x)-νxPνμ(x).

Qνμ(x) also satisfies (14.10.1)–(14.10.5).

14.10.6 Pνμ+2(x)+2(μ+1)x(x2-1)-1/2Pνμ+1(x)-(ν-μ)(ν+μ+1)Pνμ(x)=0,
14.10.7 (x2-1)1/2Pνμ+1(x)-(ν-μ+1)Pν+1μ(x)+(ν+μ+1)xPνμ(x)=0.

Qνμ(x) also satisfies (14.10.6) and (14.10.7). In addition, Pνμ(x) and Qνμ(x) satisfy (14.10.3)–(14.10.5).