Associated Askey-Wilson polynomials as Laguerre-Hahn orthogonal polynomials

Abstract

One looks for [formal] orthogonal polynomials satisfying interesting differential or difference relations and equations (Laguerre-Hahn theory). The divided difference operator used here is essentially the Askey-Wilson operator

$$Df\left( x \right) = \frac{{E_2 f\left( x \right) - E_1 f\left( x \right)}}{{E_2 x - E_1 x}} = \frac{{f\left( {y_2 \left( x \right)} \right) - f\left( {y_1 \left( x \right)} \right)}}{{y_2 \left( x \right) - y_1 \left( x \right)}}$$

where y₁(x) and y₂(x) are the two roots of Ay²+2Bxy+Cx²++2Dy+2Ex+f=0.

The related Laguerre-Hahn orthogonal polynomials are then introduced as the denominators P_o,P₁,… of the successive approximants Q_n/P_n of the Gauss-Heine-like continued fraction f(x)=1/(x−r_o−s₁/(x−r₁−s₂/…)) satisfying the Riccati equation a(x)Df(x)=b(x)E₁f(x)E₂f(x)+c(x)Mf(x)+d(x) where a,b,c,d are polynomials and Mf(x)=(E₁f(x)+E₂f(x))/2.

In the classical case (degrees a,b,c,d≤2,0,1,0), closed-forms for the recurrence coefficients r_n and s_n are obtained and show that we are dealing essentially with the associated Askey-Wilson polynomials.

One finds for P_n difference relations (a_n+a)DP_n=(c_n-c)MP_n++2s_nd_nMP_n−1−2bMQ_n and a writing in terms of solutions of linear second order difference equations P_n=(X_nY₋₁−Y_nX₋₁)/(X_oY₋₁−Y_oX₋₁).