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14 Legendre and Related FunctionsReal Arguments

§14.16 Zeros

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§14.16(i) Notation

Throughout this section we assume that μ and ν are real, and when they are not integers we write

14.16.1 μ =m+δμ,
ν =n+δν,

where m, n and δμ, δν(0,1). For all cases concerning Pνμ(x) and Pνμ(x) we assume that ν-12 without loss of generality (see (14.9.5) and (14.9.11)).

§14.16(ii) Interval -1<x<1

The number of zeros of Pνμ(x) in the interval (-1,1) is max(ν-|μ|,0) if any of the following sets of conditions hold:

  • (a)

    μ0.

  • (b)

    μ>0, nm, and δν>δμ.

  • (c)

    μ>0, n<m, and m-n is odd.

  • (d)

    ν=0,1,2,3,.

The number of zeros of Pνμ(x) in the interval (-1,1) is max(ν-|μ|,0)+1 if either of the following sets of conditions holds:

  • (a)

    μ>0, n>m, and δνδμ.

  • (b)

    μ>0, n<m, and m-n is even.

The zeros of Qνμ(x) in the interval (-1,1) interlace those of Pνμ(x). Qνμ(x) has max(ν-|μ|,0)+k zeros in the interval (-1,1), where k can take one of the values -1, 0, 1, 2, subject to max(ν-|μ|,0)+k being even or odd according as cos(νπ) and cos(μπ) have opposite signs or the same sign. In the special case μ=0 and ν=n=0,1,2,3,, Qn(x) has n+1 zeros in the interval -1<x<1.

For uniform asymptotic approximations for the zeros of Pn-m(x) in the interval -1<x<1 when n with m (0) fixed, see Olver (1997b, p. 469).

§14.16(iii) Interval 1<x<

Pνμ(x) has exactly one zero in the interval (1,) if either of the following sets of conditions holds:

  • (a)

    μ>0, μ>ν, μ, and sin((μ-ν)π) and sin(μπ) have opposite signs.

  • (b)

    μν, μ, and μ is odd.

For all other values of μ and ν (with ν-12) Pνμ(x) has no zeros in the interval (1,).

Qνμ(x) has no zeros in the interval (1,) when ν>-1, and at most one zero in the interval (1,) when ν<-1.