32 Painlevé Transcendents Applications 32.14 Combinatorics 32.16 Physical Applications

§32.15 Orthogonal Polynomials

ⓘ

Keywords:: Painlevé transcendents, applications, orthogonal polynomials
Referenced by:: §18.32, §18.38(iii), Table 18.39.1, §18.39(iii)
Permalink:: http://dlmf.nist.gov/32.15
See also:: Annotations for Ch.32

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

32.15.1		$\int_{-\infty}^{\infty}\exp\left(-\tfrac{1}{4}\xi^{4}-z\xi^{2}\right)p_{m}(\xi% )p_{n}(\xi)\,\mathrm{d}\xi=\delta_{m,n},$
ⓘ Symbols: $\delta_{\NVar{j},\NVar{k}}$ : Kronecker delta, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\exp\NVar{z}$ : exponential function, $\int$ : integral, $m$ : integer, $n$ : integer, $z$ : real and $p_{n}(\xi)$ : polynomials Permalink: http://dlmf.nist.gov/32.15.E1 Encodings: TeX, pMML, png See also: Annotations for §32.15 and Ch.32

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 Ch.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 Ch.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).

32.14 Combinatorics 32.16 Physical Applications

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