32 Painlevé Transcendents Applications 32.14 Combinatorics 32.16 Physical

§32.15 Orthogonal Polynomials

Keywords:: Painlevé transcendents, orthogonal polynomials
Permalink:: http://dlmf.nist.gov/32.15

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

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

Symbols:: $\delta_{{j,k}}$ : Kronecker delta, : differential of , $\mathop{\exp\/}\nolimits z$ : exponential function, $\int$ : integral, : integer, : integer, : real and $p_{n}(\xi)$ : polynomials
Permalink:: http://dlmf.nist.gov/32.15.E1
Encodings:: TeX, pMML, png

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:: : integer, : real and $p_{n}(\xi)$ : polynomials
Permalink:: http://dlmf.nist.gov/32.15.E2
Encodings:: TeX, pMML, png

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:: : integer, : real and $u_{n}(z)$ : solution
Permalink:: http://dlmf.nist.gov/32.15.E3
Encodings:: TeX, pMML, png

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).