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32 Painlevé TranscendentsApplications

§32.15 Orthogonal Polynomials

Let pn(ξ), n=0,1,, be the orthonormal set of polynomials defined by

32.15.1 exp(14ξ4zξ2)pm(ξ)pn(ξ)dξ=δm,n,

with recurrence relation

32.15.2 an+1(z)pn+1(ξ)=ξpn(ξ)an(z)pn1(ξ),

for n=1,2,; compare §18.2. Then un(z)=(an(z))2 satisfies the nonlinear recurrence relation

32.15.3 (un+1+un+un1)un=n2zun,

for n=1,2,, and also PIV with α=12n and β=12n2.

For this result and applications see Fokas et al. (1991): in this reference, on the right-hand side of Eq. (1.10), (n+γ)2 should be replaced by n+γ at its first appearance. See also Freud (1976), Brézin et al. (1978), Fokas et al. (1992), and Magnus (1995).