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arXiv · 2609.31805

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

Youssef Esstafa·Ridha Sfaxi
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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 $$\mathcal S_β=\fracβ{β+1}\mathcal L^0+\frac{1}{β+1}δ$$ satisfies the exact Hahn relation $$\mathscr L B_{n+1}(x;β)=(n+1)B_n(x;β+1).$$ A second rigidity result shows that a single pointwise relation $P_{n+1}(c)=θ\,\mathscr L P_{n+1}(c)$ 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.

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Youssef Esstafa, Ridha Sfaxi. 2026-09-25. Hahn-Type Orthogonality for the Resolvent Delta Operator $\mathscr L=D(I-D)^{-1}$. https://arxiv.org/abs/2609.31805

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