Digital Library of Mathematical Functions
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14 Legendre and Related FunctionsReal Arguments

§14.9 Connection Formulas

Contents

§14.9(i) Connections Between Pν±μ(x), P-ν-1±μ(x), Qν±μ(x), Q-ν-1μ(x)

14.9.1 πsin(μπ)2Γ(ν-μ+1)Pν-μ(x)=-1Γ(ν+μ+1)Qνμ(x)+cos(μπ)Γ(ν-μ+1)Qν-μ(x).
14.9.2 2sin(μπ)πΓ(ν-μ+1)Qν-μ(x)=1Γ(ν+μ+1)Pνμ(x)-cos(μπ)Γ(ν-μ+1)Pν-μ(x),
14.9.3 Pν-m(x)=(-1)mΓ(ν-m+1)Γ(ν+m+1)Pνm(x),
14.9.4 Qν-m(x)=(-1)mΓ(ν-m+1)Γ(ν+m+1)Qνm(x),
νm-1,m-2,.
14.9.5 P-ν-1μ(x) =Pνμ(x),
P-ν-1-μ(x) =Pν-μ(x),
14.9.6 πcos(νπ)cos(μπ)Pνμ(x)=sin((ν+μ)π)Qνμ(x)-sin((ν-μ)π)Q-ν-1μ(x).

§14.9(ii) Connections Between Pν±μ(±x), Qν-μ(±x), Qνμ(x)

14.9.7 sin((ν-μ)π)Γ(ν+μ+1)Pνμ(x)=sin(νπ)Γ(ν-μ+1)Pν-μ(x)-sin(μπ)Γ(ν-μ+1)Pν-μ(-x),
14.9.8 12πsin((ν-μ)π)Pν-μ(x)=-cos((ν-μ)π)Qν-μ(x)-Qν-μ(-x),
14.9.9 2Γ(ν+μ+1)Γ(μ-ν)Qνμ(x)=-cos(νπ)Pν-μ(x)+cos(μπ)Pν-μ(-x),
14.9.10 (2/π)sin((ν-μ)π)Qν-μ(x)=cos((ν-μ)π)Pν-μ(x)-Pν-μ(-x).

§14.9(iii) Connections Between Pν±μ(x), P-ν-1±μ(x), Qν±μ(x), Q-ν-1μ(x)

14.9.11 P-ν-1-μ(x) =Pν-μ(x),
P-ν-1μ(x) =Pνμ(x),
14.9.12 cos(νπ)Pν-μ(x)=-Qνμ(x)Γ(μ-ν)+Q-ν-1μ(x)Γ(ν+μ+1).
14.9.13 Pν-m(x)=Γ(ν-m+1)Γ(ν+m+1)Pνm(x),
νm-1,m-2,.
14.9.14 Qν-μ(x)=Qνμ(x),
14.9.15 2sin(μπ)πQνμ(x)=Pνμ(x)Γ(ν+μ+1)-Pν-μ(x)Γ(ν-μ+1).

§14.9(iv) Whipple’s Formula

14.9.16 Qνμ(x)=(12π)1/2(x2-1)-1/4P-μ-(1/2)-ν-(1/2)(x(x2-1)-1/2).

Equivalently,

14.9.17 Pνμ(x)=(2/π)1/2(x2-1)-1/4Q-μ-(1/2)ν+(1/2)(x(x2-1)-1/2).