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14 Legendre and Related FunctionsReal Arguments

§14.6 Integer Order

Contents
  1. §14.6(i) Nonnegative Integer Orders
  2. §14.6(ii) Negative Integer Orders

§14.6(i) Nonnegative Integer Orders

For m=0,1,2,,

14.6.1 𝖯νm(x) =(1)m(1x2)m/2dm𝖯ν(x)dxm,
14.6.2 𝖰νm(x) =(1)m(1x2)m/2dm𝖰ν(x)dxm.
14.6.3 Pνm(x) =(x21)m/2dmPν(x)dxm,
14.6.4 Qνm(x) =(x21)m/2dmQν(x)dxm,
14.6.5 (ν+1)m𝑸νm(x)=(1)m(x21)m/2dm𝑸ν(x)dxm.

§14.6(ii) Negative Integer Orders

For m=1,2,3,,

14.6.6 𝖯νm(x) =(1x2)m/2x1x1𝖯ν(x)(dx)m.
14.6.7 Pνm(x) =(x21)m/21x1xPν(x)(dx)m,
14.6.8 Qνm(x) =(1)m(x21)m/2xxQν(x)(dx)m.

For connections between positive and negative integer orders see (14.9.3), (14.9.4), and (14.9.13). For generalizations see Cohl and Costas-Santos (2020).