Hahn-Type Orthogonality for the Resolvent Delta Operator $\mathcal{L}=D(I-D{)}^{-1}$

Abstract

We investigate orthogonality under the shift-invariant resolvent delta operator $\mathcal{L}=D(I-D{)}^{-1}$. First, we prove a rigidity theorem: a monic orthogonal polynomial sequence is $\mathcal{L}$-Appell if and only if it is a translated Laguerre sequence. We then move beyond the Appell setting and show that the diagonal family associated with $${\mathcal{S}}_{\beta }=\frac{\beta }{\beta +1}{\mathcal{L}}^0+\frac{1}{\beta +1}\delta$$ satisfies the exact Hahn relation $$\mathcal{L}{B}_{n+1}(x;\beta )=(n+1)B_n(x;\beta +1).$$ A second rigidity result shows that a single pointwise relation $P_{n+1}(c)=\theta \mathcal{L}{P}_{n+1}(c)$ already forces the orthogonality functional to be a translated Laguerre functional or a translated member of the diagonal family. Connection and structure relations, a semiclassical Pearson equation, a differential characterization, and a generating function complete the description. Thus the resolvent operator provides an explicit setting in which delta-operator lowering and semiclassical orthogonality are linked by exact structural identities.