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14 Legendre and Related FunctionsReal Arguments

§14.7 Integer Degree and Order

  1. §14.7(i) μ=0
  2. §14.7(ii) Rodrigues-Type Formulas
  3. §14.7(iii) Reflection Formulas
  4. §14.7(iv) Generating Functions

§14.7(i) μ=0

For n=0,1,2,,

14.7.1 𝖯n0(x)=𝖯n(x)=Pn0(x)=Pn(x),

where Pn(x) is the Legendre polynomial of degree n. For additional properties of Pn(x) see Chapter 18.

14.7.2 𝖰n0(x)=𝖰n(x)=12Pn(x)ln(1+x1x)Wn1(x),

where W1(x)=0, and for n1,

14.7.3 Wn1(x)=s=0n1(n+s)!(ψ(n+1)ψ(s+1))2s(ns)!(s!)2(x1)s;


14.7.4 Wn1(x)=k=1n1kPk1(x)Pnk(x).
14.7.5 W0(x) =1,
W1(x) =32x,
W2(x) =52x223.

§14.7(ii) Rodrigues-Type Formulas

For m=0,1,2,, and n=0,1,2,,

14.7.8 𝖯nm(x) =(1)m(1x2)m/2dmdxm𝖯n(x),
14.7.9 𝖰nm(x) =(1)m(1x2)m/2dmdxm𝖰n(x),
14.7.10 𝖯nm(x)=(1)m+n(1x2)m/22nn!dm+ndxm+n(1x2)n.
14.7.11 Pnm(x) =(x21)m/2dmdxmPn(x),
14.7.12 Qnm(x) =(x21)m/2dmdxmQn(x),
14.7.13 Pn(x) =12nn!dndxn(x21)n,
14.7.14 Pnm(x) =(x21)m/22nn!dm+ndxm+n(x21)n,
14.7.15 Pmm(x) =(2m)!2mm!(x21)m/2.

When m is even and mn, 𝖯nm(x) and Pnm(x) are polynomials of degree n. Also,

14.7.16 𝖯nm(x)=Pnm(x)=0,

§14.7(iii) Reflection Formulas

14.7.17 𝖯nm(x) =(1)nm𝖯nm(x),
14.7.18 𝖰n±m(x) =(1)nm1𝖰n±m(x).

§14.7(iv) Generating Functions

When 1<x<1 and |h|<1,

14.7.19 n=0𝖯n(x)hn=(12xh+h2)1/2,
14.7.20 n=0𝖰n(x)hn=1(12xh+h2)1/2ln(xh+(12xh+h2)1/2(1x2)1/2).

When 1<x<1 and |h|>1,

14.7.21 n=0𝖯n(x)hn1=(12xh+h2)1/2.

When x>1, (14.7.19) applies with |h|<x(x21)1/2. Also, with the same conditions

14.7.22 n=0Qn(x)hn=1(12xh+h2)1/2ln(xh+(12xh+h2)1/2(x21)1/2).

Lastly, when x>1, (14.7.21) applies with |h|>x+(x21)1/2.

For other generating functions see Magnus et al. (1966, pp. 232–233) and Rainville (1960, pp. 163–165, 168, 170–171, 184).