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Let M and N be fixed non-negative integer numbers and let π N be a polynomial of degree N . Suppose that (Pn) n≥0 and (Qn) n≥0 are two orthogonal polynomial sequences such thatwhere r n,j are complex number independent of x. It is shown that under natural constraints, (Pn) n≥0 and (Qn) n≥0 are semiclassical orthogonal polynomial sequences. Moreover, their corresponding moment linear functionals are related by a rational modification in the distributional sense. This leads to the concept of π N −coherent pair with index M and order (m, k).

Let u be a nonzero linear functional acting on the space of polynomials. Let Dq,ω be a Hahn operator acting on the dual space of polynomials. Suppose that there exist polynomials φ and ψ, with deg φ ≤ 2 and deg ψ ≤ 1, so that the functional equationDq,ω(φu) = ψu holds, where the involved operations are defined in a distributional sense. In this note we state necessary and sufficient conditions, involving only the coefficients of φ and ψ, such that u is regular, that is, there exists a sequence of orthogonal polynomials with respect to u. A key step in the proof relies upon the fact that a distributional Rodrigues-type formula holds without assuming that u is regular.

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