2 × 2 hypergeometric operators with diagonal eigenvalues

Abstract

In this work we give all the order-two hypergeometric operators D, symmetric with respect to some 2 × 2 irreducible matrix-weight W on (0,1) such that DPn=Pnλn00μn with no repetition among the eigenvalues {λn,μn}n∈N0 , where {Pn}n∈N0 is the (unique) sequence of monic orthogonal polynomials with respect to W. We obtain a new family of such operators and weights, depending on three parameters, generalizing some older examples. We also give, in a very explicit way, the corresponding monic orthogonal polynomials, their three term recurrence relation and their squared matrix-norms.