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14 Legendre and Related FunctionsReal Arguments

§14.3 Definitions and Hypergeometric Representations

  1. §14.3(i) Interval 1<x<1
  2. §14.3(ii) Interval 1<x<
  3. §14.3(iii) Alternative Hypergeometric Representations
  4. §14.3(iv) Relations to Other Functions

§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+x1x)μ/2𝐅(ν+1,ν;1μ;1212x).

Ferrers Function of the Second Kind

14.3.2 𝖰νμ(x)=π2sin(μπ)(cos(μπ)(1+x1x)μ/2𝐅(ν+1,ν;1μ;1212x)Γ(ν+μ+1)Γ(νμ+1)(1x1+x)μ/2𝐅(ν+1,ν;1+μ;1212x)).

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(νm+1)(1x2)m/2𝐅(ν+m+1,mν;m+1;1212x);


14.3.5 𝖯νm(x)=(1)mΓ(ν+m+1)Γ(νm+1)(1x1+x)m/2𝐅(ν+1,ν;m+1;1212x).

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)(x21)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(νm+1)(x21)m/2𝐅(ν+m+1,mν;m+1;1212x).

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

14.3.9 Pνμ(x)=(x1x+1)μ/2𝐅(ν+1,ν;μ+1;1212x),


14.3.10 𝑸νμ(x)=eμπiQνμ(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(12ν12μ+1)(1x2)μ/2𝐅(12ν12μ,12ν12μ+12;12;x2),
14.3.14 w2(ν,μ,x) =2μΓ(12ν+12μ+1)Γ(12ν12μ+12)x(1x2)μ/2𝐅(1212ν12μ,12ν12μ+1;32;x2).
14.3.15 Pνμ(x)=2μ(x21)μ/2𝐅(μν,ν+μ+1;μ+1;1212x),
14.3.16 cos(νπ)Pνμ(x)=2νπ1/2xνμ(x21)μ/2Γ(ν+μ+1)𝐅(12μ12ν,12μ12ν+12;12ν;1x2)π1/2(x21)μ/22ν+1Γ(μν)xν+μ+1𝐅(12ν+12μ+1,12ν+12μ+12;ν+32;1x2),
14.3.17 Pνμ(x)=π(x21)μ/22μ(𝐅(12μ12ν,12ν+12μ+12;12;x2)Γ(12μ12ν+12)Γ(12ν+12μ+1)x𝐅(12μ12ν+12,12ν+12μ+1;32;x2)Γ(12μ12ν)Γ(12ν+12μ+12)),
14.3.18 Pνμ(x) =2μxνμ(x21)μ/2𝐅(12μ12ν,12μ12ν+12;μ+1;11x2),
14.3.19 𝑸νμ(x) =2νΓ(ν+1)(x+1)μ/2(x1)(μ/2)+ν+1𝐅(ν+1,ν+μ+1;2ν+2;21x),
14.3.20 2sin(μπ)π𝑸νμ(x)=(x+1)μ/2Γ(ν+μ+1)(x1)μ/2𝐅(ν+1,ν;1μ;1212x)(x1)μ/2Γ(νμ+1)(x+1)μ/2𝐅(ν+1,ν;μ+1;1212x).

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(12μ)Γ(ν+μ+1)Γ(νμ+1)Γ(1μ)(1x2)μ/2Cν+μ(12μ)(x).
14.3.22 Pνμ(x) =2μΓ(12μ)Γ(ν+μ+1)Γ(νμ+1)Γ(1μ)(x21)μ/2Cν+μ(12μ)(x).
14.3.23 Pνμ(x) =1Γ(1μ)(x+1x1)μ/2ϕi(2ν+1)(μ,μ)(arcsinh((12x12)1/2)).

Compare also (18.11.1). From (15.9.15) it follows that 12μ=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).