Advancing the Theory of Sheffer Sequences via Dual Padé Rational Approximants
Abstract
Sheffer sequences offer an operational framework that unifies a wide range of classical polynomial families. In this article, a rational approximation is developed for both the amplitude and the delta-operator symbol in the exponential generating function. The resulting Padé Sheffer polynomials are shown, within the umbral calculus framework, to retain the Sheffer structure. Explicit operational representations are obtained, together with finite-order differential recurrences and closed-form connection constants characterizing the deviation from the underlying polynomial family. An important consequence of the construction is that rational generating indicators reduce the usual transcendental Lagrange inversion problem to a finite multinomial computation. The framework also suggests an interpretation in terms of weighted finite automata, linking polynomial constructions with state-space dynamics for unbounded combinatorial partitions.
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BibTeX
@misc{ahmad2026advancing,
title = {{Advancing the Theory of Sheffer Sequences via Dual Padé Rational Approximants}},
author = {Ujair Ahmad and Subuhi Khan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.25457},
eprint = {2610.25457},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.25457},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Advancing the Theory of Sheffer Sequences via Dual Padé Rational Approximants
AU - Ujair Ahmad
AU - Subuhi Khan
PY - 2026
DA - 2026/10/18
PB - arXiv
DO - 10.48550/arxiv.2610.25457
UR - https://arxiv.org/abs/2610.25457
AB - Sheffer sequences offer an operational framework that unifies a wide range of classical polynomial families. In this article, a rational approximation is developed for both the amplitude and the delta-operator symbol in the exponential generating function. The resulting Padé Sheffer polynomials are shown, within the umbral calculus framework, to retain the Sheffer structure. Explicit operational representations are obtained, together with finite-order differential recurrences and closed-form connection constants characterizing the deviation from the underlying polynomial family. An important consequence of the construction is that rational generating indicators reduce the usual transcendental Lagrange inversion problem to a finite multinomial computation. The framework also suggests an interpretation in terms of weighted finite automata, linking polynomial constructions with state-space dynamics for unbounded combinatorial partitions.
ER -