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Limit of orthogonal polynomials for big n

  1. Mar 30, 2010 #1
    given the 'normalized' Chebyshev and Legendre Polynomials

    [tex] \frac{L_{2n}(x)}{L_{2n}(0)} [/tex] and [tex] \frac{T_{2n}(x)}{T_{2n}(0)} [/tex]

    for n even and BIG 2n--->oo

    then it would be true that (in this limit) [tex] \frac{L_{2n}(x)}{L_{2n}(0)}=\frac{sin(x)}{2x} [/tex] and [tex] \frac{T_{2n}(x)}{T_{2n}(0)}=J_{0}(2x) [/tex]

    here 'T' is used for Chebyshev polynomials and 'L' is used for Legendre ones , J0 is the zeroth order Bessel function.

    Curiously enough the sign of the polynomials and the Taylor series representation for the functions follow both a sequence

    [tex] 1+ \sum_{n\ge 1}a(2n)(-1)^{n}x^{2n} [/tex] with ALL the a(2n) being positive numbers.
     
  2. jcsd
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