Digital Library of Mathematical Functions
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14 Legendre and Related FunctionsReal Arguments

§14.6 Integer Order

Contents

§14.6(i) Nonnegative Integer Orders

For m=0,1,2,,

14.6.1 Pνm(x) =(-1)m(1-x2)m/2mPν(x)xm,
14.6.2 Qνm(x) =(-1)m(1-x2)m/2mQν(x)xm.
14.6.3 Pνm(x) =(x2-1)m/2mPν(x)xm,
14.6.4 Qνm(x) =(x2-1)m/2mQν(x)xm,
14.6.5 (ν+1)mQνm(x)=(-1)m(x2-1)m/2mQν(x)xm.

§14.6(ii) Negative Integer Orders

For m=1,2,3,,

14.6.6 Pν-m(x) =(1-x2)-m/2x1x1Pν(x)(x)m.
14.6.7 Pν-m(x) =(x2-1)-m/21x1xPν(x)(x)m,
14.6.8 Qν-m(x) =(-1)m(x2-1)-m/2xxQν(x)(x)m.

For connections between positive and negative integer orders see (14.9.3), (14.9.4), and (14.9.13).