Let a
np
n(x)=xp
n−1(x)−a
n−1p
n−2(x) be the recurrence relation of the orthonormal polynomials related to the weight function |x|
ρ exp(−|x|
α), ρ < −1, α < 0, on the whole real line. Freud's conjecture states that
The proof for an even integer α=2m uses nonlinear equations F
n(a)=n + ρ odd(n), considered by Freud himself. It is shown that F
n(a*)−n=o(n) when n → ∞, where a*
n is the expected asymptotically valid estimate [n/C(α)]
1/α. Bounds on a
n−a*
n are obtained through the invertibility properties of the matrix [a
k ∂F
n (a)/∂a
k], shown to be symmetric and positive definite. The numerical computation of the solution by Newton's method is considered.
![$$\left( l \right) \mathop {lim}\limits_{n \to \infty } \frac{{^a n}}{{\left[ {n/C\left( \alpha \right)} \right]^{1/\alpha } }} = 1, C\left( \alpha \right) = \frac{{2\Gamma \left( \alpha \right)}}{{\left( {\Gamma \left( {\alpha /2} \right)} \right)^2 }} = \frac{{2^\alpha \Gamma \left( {\left( {\alpha + 1} \right)/2} \right)}}{{\sqrt \pi \Gamma \left( {\alpha /2} \right)}}.$$](/content/345532gt71117k83/304_3-540-16059-0_Chapter_38_TeX2GIFE1.gif)
