________ this document in[* DVI.....]format. _________________________________________________________________ 90d:33008 (21:05) 33A65 39A10 39A70 Magnus, Alphonse P. (B-UCL) Associated Askey-Wilson polynomials as Laguerre-Hahn orthogonal polynomials. (English) Orthogonal polynomials and their applications (Segovia, 1986), 261--278, Lecture Notes in Math., 1329, Springer, Berlin-New York, 1988. _________________________________________________________________ In what may be called the Laguerre-Hahn theory one looks for orthogonal polynomials which satisfy differential and difference relations and equations. The difference operator used in this paper is essentially the Askey-Wilson operator $$ (Df)(x)=\frac{f(y\sb 2(x))-f(y\sb 1(x))}{y\sb 2(x)-y\sb 1(x)}. $$ Here $y\sb 1(x)$ and $y\sb 2(x)$ are the two roots of the quadratic equation $Ay\sp 2+2 Bxy+Cx\sp 2+2Dy+2Ex+F=0$. Given polynomials $a,b,c,d$, let $f$ satisfy the Riccati equation $$\multline a(x)Df(x)=\\ b(x) f(y\sb 1(x)) f(y\sb 2(x))+c(x) \frac{f(y\sb 1(x))+f(y\sb 2(x))}{2}+d(x).\endmultline $$ Then $f$ can be written as a continued fraction $f(x)=1/x-r\sb 0-s\sb 1/ x-r\sb 1-s\sb 2/\cdots$ of Gauss-Heine type. The related Laguerre-Hahn polynomials (the author's terminology) are introduced as the denominators $P\sb n$ of the successive approximants $Q\sb n/P\sb n$ of $f$. They satisfy the three-term recurrence relations $P\sb {n+1}(x)=(x-r\sb n)P\sb n(x)-s\sb n P\sb {n-1}(x)$, $P\sb 0=1$, $P\sb {-1}=0$. In the classical case, degrees $a,b,c,d\le 2,0,1,0$, closed forms for the coefficients $r\sb n$ and $s\sb n$ are obtained showing the correspondence with the associated Askey-Wilson polynomials. In this paper also the more general case, degrees $a,b,c,d\le m+2$, $m$, $m+1$, $m$ is considered. Further difference relations for the $P\sb n$ are given and they are written in terms of linear second order difference equations. \{For the entire collection see MR 89f:00027\}. Reviewed by van Eijndhoven, S. J. L. (NL-EIND) © Copyright American Mathematical Society 1990, 1995