090 Maurice Kenfack Nangho

Maurice Kenfack Nangho (University of Dschang, Cameroon)// A characterization of Askey-Wilson polynomials: proof of a conjecture by Mourad Ismail // Abstract: We show that the only monic orthogonal polynomials {P_n}_{n=0}^{\infty} that satisfy \pi(x)\mathcal{D}_{q}^2P_{n}(x)=\sum_{j=-2}^{2}a_{n,n+j}P_{n+j}(x),\; x=\cos\theta,\;\quad a_{n,n-2}\neq 0,\quad n=2,3,\dots (1) where \pi(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. This completes and proves a conjecture by Ismail concerning a structure relation satisfied by Askey-Wilson polynomials. We study a limiting case of these polynomials when one of the parameters goes to infinity, namely continuous dual q-Hahn polynomials when q>1. Solutions to the associated indeterminate moment problem by general theory are found and an orthogonality relation is established.