Properties of Certain Classes of Semiclassical Orthogonal Polynomials

Abstract

In this lecture we discuss properties of orthogonal polynomials for weights which are semiclassical perturbations of classical orthogonality weights. We use the moments, together with the connection between orthogonal polynomials and Painlevé equations to obtain explicit expressions for the recurrence coefficients of polynomials associated with a semiclassical Laguerre and a generalized Freud weight. We analyze the asymptotic behavior of generalized Freud polynomials in two different contexts. We show that unique, positive solutions of the nonlinear difference equation satisfied by the recurrence coefficients exist for all real values of the parameter involved in the semiclassical perturbation but that these solutions are very sensitive to the initial conditions. We prove properties of the zeros of semiclassical Laguerre and generalized Freud polynomials and determine the coefficients a _n,n+j in the differential-difference equation

$$\displaystyle x\frac {d}{dx}P_n(x)=\sum _{k=-1}^{0}a_{n,n+k}P_{n+k}(x), $$

where P _n(x) are the generalized Freud polynomials. Finally, we show that the only monic orthogonal polynomials $\{P_n\}_{n=0}^{\infty }$ that satisfy

$$\displaystyle \pi (x)\mathcal {D}_{q}^2P_{n}(x)=\sum _{j=-2}^{2}a_{n,n+j}P_{n+j}(x),\; x=\cos \theta ,\;~ a_{n,n-2}\neq 0,~ n=2,3,\dots , $$

where π(x) is a polynomial of degree at most 4 and $\mathcal {D}_{q}$ is the Askey–Wilson operator, are Askey–Wilson polynomials and their special or limiting cases, using this relation to derive bounds for the extreme zeros of Askey–Wilson polynomials.