Determination of All Coherent Pairs Original Research Article

A pair of quasi-definite linear functionals {u0, u1} on the set of polynomials is called a coherent pair if their corresponding sequences of monic orthogonal polynomials {Pn} and {Tn} satisfy a relationTn=P′n+1n+1−σn P′nn, n⩾1withσnnon-zero constants. We prove that if {u0, u1} is a coherent pair, then at least one of the functionals has to be classical, i.e. Hermite, Laguerre, Jacobi, or Bessel. A similar result is derived for symmetrically coherent pairs.