# §32.15 Orthogonal Polynomials

Let $p_{n}(\xi)$, $n=0,1,\dots$, be the orthonormal set of polynomials defined by

 32.15.1 $\int_{-\infty}^{\infty}\exp\left(-\tfrac{1}{4}\xi^{4}-z\xi^{2}\right)p_{m}(\xi% )p_{n}(\xi)\mathrm{d}\xi=\delta_{m,n},$

with recurrence relation

 32.15.2 $a_{n+1}(z)p_{n+1}(\xi)=\xi p_{n}(\xi)-a_{n}(z)p_{n-1}(\xi),$ ⓘ Symbols: $n$: integer, $z$: real and $p_{n}(\xi)$: polynomials Permalink: http://dlmf.nist.gov/32.15.E2 Encodings: TeX, pMML, png See also: Annotations for 32.15 and 32

for $n=1,2,\dots$; compare §18.2. Then $u_{n}(z)=(a_{n}(z))^{2}$ satisfies the nonlinear recurrence relation

 32.15.3 $(u_{n+1}+u_{n}+u_{n-1})u_{n}=n-2zu_{n},$ ⓘ Symbols: $n$: integer, $z$: real and $u_{n}(z)$: solution Permalink: http://dlmf.nist.gov/32.15.E3 Encodings: TeX, pMML, png See also: Annotations for 32.15 and 32

for $n=1,2,\dots$, and also $\mbox{P}_{\mbox{\scriptsize IV}}$ with $\alpha=-\tfrac{1}{2}n$ and $\beta=-\tfrac{1}{2}n^{2}$.

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