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14 Legendre and Related FunctionsReal Arguments

§14.3 Definitions and Hypergeometric Representations


§14.3(i) Interval -1<x<1

The following are real-valued solutions of (14.2.2) when μ, ν and x(-1,1).

Ferrers Function of the First Kind

14.3.1 Pνμ(x)=(1+x1-x)μ/2F(ν+1,-ν;1-μ;12-12x).

Ferrers Function of the Second Kind

14.3.2 Qνμ(x)=π2sin(μπ)(cos(μπ)(1+x1-x)μ/2F(ν+1,-ν;1-μ;12-12x)-Γ(ν+μ+1)Γ(ν-μ+1)(1-x1+x)μ/2F(ν+1,-ν;1+μ;12-12x)).

Here and elsewhere in this chapter

14.3.3 F(a,b;c;x)=1Γ(c)F(a,b;c;x)

is Olver’s hypergeometric function (§15.1).

Pνμ(x) exists for all values of μ and ν. Qνμ(x) is undefined when μ+ν=-1,-2,-3,.

When μ=m=0,1,2,, (14.3.1) reduces to

14.3.4 Pνm(x)=(-1)mΓ(ν+m+1)2mΓ(ν-m+1)(1-x2)m/2F(ν+m+1,m-ν;m+1;12-12x);


14.3.5 Pνm(x)=(-1)mΓ(ν+m+1)Γ(ν-m+1)(1-x1+x)m/2F(ν+1,-ν;m+1;12-12x).

When μ=m () (14.3.2) is replaced by its limiting value; see Hobson (1931, §132) for details. See also (14.3.12)–(14.3.14) for this case.

§14.3(ii) Interval 1<x<

The following are solutions of (14.2.2) when μ, ν and x>1.

Associated Legendre Function of the First Kind

Associated Legendre Function of the Second Kind

14.3.7 Qνμ(x)=eμπiπ1/2Γ(ν+μ+1)(x2-1)μ/22ν+1xν+μ+1F(12ν+12μ+1,12ν+12μ+12;ν+32;1x2),

When μ=m=1,2,3,, (14.3.6) reduces to

14.3.8 Pνm(x)=Γ(ν+m+1)2mΓ(ν-m+1)(x2-1)m/2F(ν+m+1,m-ν;m+1;12-12x).

As standard solutions of (14.2.2) we take the pair Pν-μ(x) and Qνμ(x), where

14.3.9 Pν-μ(x)=(x-1x+1)μ/2F(ν+1,-ν;μ+1;12-12x),


14.3.10 Qνμ(x)=e-μπiQνμ(x)Γ(ν+μ+1).

Like Pνμ(x), but unlike Qνμ(x), Qνμ(x) is real-valued when ν, μ and x(1,), and is defined for all values of ν and μ. The notation Qνμ(x) is due to Olver (1997b, pp. 170 and 178).

§14.3(iii) Alternative Hypergeometric Representations

14.3.11 Pνμ(x) =cos(12(ν+μ)π)w1(ν,μ,x)+sin(12(ν+μ)π)w2(ν,μ,x),
14.3.12 Qνμ(x) =-12πsin(12(ν+μ)π)w1(ν,μ,x)+12πcos(12(ν+μ)π)w2(ν,μ,x),


14.3.13 w1(ν,μ,x) =2μΓ(12ν+12μ+12)Γ(12ν-12μ+1)(1-x2)-μ/2F(-12ν-12μ,12ν-12μ+12;12;x2),
14.3.14 w2(ν,μ,x) =2μΓ(12ν+12μ+1)Γ(12ν-12μ+12)x(1-x2)-μ/2F(12-12ν-12μ,12ν-12μ+1;32;x2).
14.3.15 Pν-μ(x)=2-μ(x2-1)μ/2F(μ-ν,ν+μ+1;μ+1;12-12x),
14.3.16 cos(νπ)Pν-μ(x)=2νπ1/2xν-μ(x2-1)μ/2Γ(ν+μ+1)F(12μ-12ν,12μ-12ν+12;12-ν;1x2)-π1/2(x2-1)μ/22ν+1Γ(μ-ν)xν+μ+1F(12ν+12μ+1,12ν+12μ+12;ν+32;1x2),
14.3.17 Pν-μ(x)=π(x2-1)μ/22μ(F(12μ-12ν,12ν+12μ+12;12;x2)Γ(12μ-12ν+12)Γ(12ν+12μ+1)-xF(12μ-12ν+12,12ν+12μ+1;32;x2)Γ(12μ-12ν)Γ(12ν+12μ+12)),
14.3.18 Pν-μ(x) =2-μxν-μ(x2-1)μ/2F(12μ-12ν,12μ-12ν+12;μ+1;1-1x2),
14.3.19 Qνμ(x) =2νΓ(ν+1)(x+1)μ/2(x-1)(μ/2)+ν+1F(ν+1,ν+μ+1;2ν+2;21-x),
14.3.20 2sin(μπ)πQνμ(x)=(x+1)μ/2Γ(ν+μ+1)(x-1)μ/2F(ν+1,-ν;1-μ;12-12x)-(x-1)μ/2Γ(ν-μ+1)(x+1)μ/2F(ν+1,-ν;μ+1;12-12x).

For further hypergeometric representations of Pνμ(x) and Qνμ(x) see Erdélyi et al. (1953a, pp. 123–139), Andrews et al. (1999, §3.1), Magnus et al. (1966, pp. 153–163), and §15.8(iii). For further hypergeometric representations of Qνμ(x) see Cohl et al. (2021).

§14.3(iv) Relations to Other Functions

In terms of the Gegenbauer function Cα(β)(x) and the Jacobi function ϕλ(α,β)(t) (§§15.9(iii), 15.9(ii)):

14.3.21 Pνμ(x) =2μΓ(1-2μ)Γ(ν+μ+1)Γ(ν-μ+1)Γ(1-μ)(1-x2)μ/2Cν+μ(12-μ)(x).
14.3.22 Pνμ(x) =2μΓ(1-2μ)Γ(ν+μ+1)Γ(ν-μ+1)Γ(1-μ)(x2-1)μ/2Cν+μ(12-μ)(x).
14.3.23 Pνμ(x) =1Γ(1-μ)(x+1x-1)μ/2ϕ-i(2ν+1)(-μ,μ)(arcsinh((12x-12)1/2)).

Compare also (18.11.1). From (15.9.15) it follows that 1-2μ=0,-1,-2, and ν+μ+1=0,-1,-2, are removable singularities of the right-hand sides of (14.3.21) and (14.3.22).