Preprint Number 10–06 ON THE SEMICLASSICAL CHARACTER OF ORTHOGONAL POLYNOMIALS SATISFYING STRUCTURE RELATIONS

Abstract: We prove the semiclassical character of some sequences of orthogo-nal polynomials, say {Pn}, {Rn}, related through relations of the following type:∑N k=0 ζn,kR (α) n+i−k = ∑M k=0 ξn,kPn+j−k, where i, j,M,N,α are non-negative integers, ζn,k, ξn,k are complex numbers, and R (α) denotes the α-derivative of R. The case M = j = 0, α = 2, i = 2 is studied for a pair of orthogonal polynomials whose cor-responding orthogonality measures are coherent. The relation ∑s k=0 ξn,kPn+s−k =∑s+2 k=0 ζn,kP n+s+1−k is shown to give a characterization for the semiclassical charac-ter of {Pn}