About the Project
14 Legendre and Related FunctionsReal Arguments

§14.9 Connection Formulas

Contents
  1. §14.9(i) Connections Between 𝖯ν±μ(x), 𝖯ν1±μ(x), 𝖰ν±μ(x), 𝖰ν1μ(x)
  2. §14.9(ii) Connections Between 𝖯ν±μ(±x), 𝖰νμ(±x), 𝖰νμ(x)
  3. §14.9(iii) Connections Between Pν±μ(x), Pν1±μ(x), 𝑸ν±μ(x), 𝑸ν1μ(x)
  4. §14.9(iv) Whipple’s Formula

§14.9(i) Connections Between 𝖯ν±μ(x), 𝖯ν1±μ(x), 𝖰ν±μ(x), 𝖰ν1μ(x)

14.9.1 πsin(μπ)2Γ(νμ+1)𝖯νμ(x)=1Γ(ν+μ+1)𝖰νμ(x)+cos(μπ)Γ(νμ+1)𝖰νμ(x).
14.9.2 2sin(μπ)πΓ(νμ+1)𝖰νμ(x)=1Γ(ν+μ+1)𝖯νμ(x)cos(μπ)Γ(νμ+1)𝖯νμ(x),
14.9.3 𝖯νm(x)=(1)mΓ(νm+1)Γ(ν+m+1)𝖯νm(x),
14.9.4 𝖰νm(x)=(1)mΓ(νm+1)Γ(ν+m+1)𝖰νm(x),
νm1,m2,.
14.9.5 𝖯ν1μ(x) =𝖯νμ(x),
𝖯ν1μ(x) =𝖯νμ(x),
14.9.6 πcos(νπ)cos(μπ)𝖯νμ(x)=sin((ν+μ)π)𝖰νμ(x)sin((νμ)π)𝖰ν1μ(x).

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

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

§14.9(iii) Connections Between Pν±μ(x), Pν1±μ(x), 𝑸ν±μ(x), 𝑸ν1μ(x)

14.9.11 Pν1μ(x) =Pνμ(x),
Pν1μ(x) =Pνμ(x),
14.9.12 cos(νπ)Pνμ(x)=𝑸νμ(x)Γ(μν)+𝑸ν1μ(x)Γ(ν+μ+1).
14.9.13 Pνm(x)=Γ(νm+1)Γ(ν+m+1)Pνm(x),
νm1,m2,.
14.9.14 𝑸νμ(x)=𝑸νμ(x),
14.9.15 2sin(μπ)π𝑸νμ(x)=Pνμ(x)Γ(ν+μ+1)Pνμ(x)Γ(νμ+1).

§14.9(iv) Whipple’s Formula

14.9.16 𝑸νμ(x)=(12π)1/2(x21)1/4Pμ(1/2)ν(1/2)(x(x21)1/2).

Equivalently,

14.9.17 Pνμ(x)=(2/π)1/2(x21)1/4𝑸μ(1/2)ν+(1/2)(x(x21)1/2).