Limit of orthogonal polynomials for big n

1. Mar 30, 2010

zetafunction

given the 'normalized' Chebyshev and Legendre Polynomials

$$\frac{L_{2n}(x)}{L_{2n}(0)}$$ and $$\frac{T_{2n}(x)}{T_{2n}(0)}$$

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

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

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

$$1+ \sum_{n\ge 1}a(2n)(-1)^{n}x^{2n}$$ with ALL the a(2n) being positive numbers.