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14 Legendre and Related FunctionsReal Arguments

§14.14 Continued Fractions

14.14.1 12(x2-1)1/2Pνμ(x)Pνμ-1(x)=x0y0+x1y1+x2y2+,

where

14.14.2 xk =14(ν-μ-k+1)(ν+μ+k)(x2-1),
yk =(μ+k)x,

provided that xk+1 and yk do not vanish simultaneously for any k=0,1,2,.

14.14.3 (ν-μ)Qνμ(x)Qν-1μ(x)=x0y0-x1y1-x2y2-,
νμ,

where now

14.14.4 xk =(ν+μ+k)(ν-μ+k),
yk =(2ν+2k+1)x,

again provided xk+1 and yk do not vanish simultaneously for any k=0,1,2,.