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

Abstract

We investigate orthogonality under the shift-invariant resolvent delta operator $\mathscr L=D(I-D)^{-1}$. First, we prove a rigidity theorem: a monic orthogonal polynomial sequence is $\mathscr L$-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 $$\mathscr L B_{n+1}(x;\beta)=(n+1)B_n(x;\beta+1).$$ A second rigidity result shows that a single pointwise relation $P_{n+1}(c)=\theta\,\mathscr L P_{n+1}(c)$ 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.