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PreprintCC BY-NC-ND 4.0HAL

Hahn-Type Orthogonality for the Resolvent Delta Operator ℒ=D(I−D)−1

  • Youssef Esstafa
  • Ridha Sfaxi
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Abstract

We investigate orthogonality under the shift-invariant resolvent delta operator ℒ=D(I−D)−1. First, we prove a rigidity theorem: a monic orthogonal polynomial sequence is 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 𝒮β=β/(β+1)ℒ0+1/(β+1)δ satisfies the exact Hahn relation ℒ Bn+1(x;β)=(n+1)Bn(x;β+1). A second rigidity result shows that a single pointwise relation Pn+1(c)=θ ℒ Pn+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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Cite

BibTeXRIS
BibTeX
@misc{esstafa2026hahntype,
  title = {{Hahn-Type Orthogonality for the Resolvent Delta Operator $\mathscr L=D(I-D)^-1$}},
  author = {Youssef Esstafa and Ridha Sfaxi},
  howpublished = {HAL},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.31805},
  eprint = {2609.31805},
  archivePrefix = {arXiv},
  url = {https://hal.science/hal-05764115v1},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - Hahn-Type Orthogonality for the Resolvent Delta Operator ℒ=D(I−D)^−1
AU  - Youssef Esstafa
AU  - Ridha Sfaxi
PY  - 2026
DA  - 2026/09/25
PB  - HAL
DO  - 10.48550/arxiv.2609.31805
UR  - https://hal.science/hal-05764115v1
AB  - We investigate orthogonality under the shift-invariant resolvent delta operator ℒ=D(I−D)^−1. First, we prove a rigidity theorem: a monic orthogonal polynomial sequence is 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 𝒮_β=β/(β+1)ℒ^0+1/(β+1)δ satisfies the exact Hahn relation 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 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.
ER  -