Digital Library of Mathematical Functions
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14 Legendre and Related FunctionsReal Arguments

§14.7 Integer Degree and Order

Contents

§14.7(i) μ=0

For n=0,1,2,,

14.7.1 Pn0(x)=Pn(x)=Pn0(x)=Pn(x),
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+x1-x)-Wn-1(x),

where W-1(x)=0, and for n1,

14.7.3 Wn-1(x)=s=0n-1(n+s)!(ψ(n+1)-ψ(s+1))2s(n-s)!(s!)2(x-1)s;

equivalently,

14.7.4 Wn-1(x)=k=1n1kPk-1(x)Pn-k(x).
14.7.5 W0(x) =1,
W1(x) =32x,
W2(x) =52x2-23.

§14.7(ii) Rodrigues-Type Formulas

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

14.7.8 Pnm(x) =(-1)m(1-x2)m/2mxmPn(x),
14.7.9 Qnm(x) =(-1)m(1-x2)m/2mxmQn(x),
14.7.10 Pnm(x)=(-1)m+n(1-x2)m/22nn!m+nxm+n(1-x2)n.
14.7.11 Pnm(x) =(x2-1)m/2mxmPn(x),
14.7.12 Qnm(x) =(x2-1)m/2mxmQn(x),
14.7.13 Pn(x) =12nn!nxn(x2-1)n,
14.7.14 Pnm(x) =(x2-1)m/22nn!m+nxm+n(x2-1)n,
14.7.15 Pmm(x) =(2m)!2mm!(x2-1)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,
m>n.

§14.7(iii) Reflection Formulas

§14.7(iv) Generating Functions

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

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

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

14.7.21 n=0Pn(x)h-n-1=(1-2xh+h2)-1/2.

When x>1, (14.7.19) applies with |h|<x-(x2-1)1/2. Also, with the same conditions

14.7.22 n=0Qn(x)hn=1(1-2xh+h2)1/2ln(x-h+(1-2xh+h2)1/2(x2-1)1/2).

Lastly, when x>1, (14.7.21) applies with |h|>x+(x2-1)1/2.

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