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PreprintCC BY 4.0arXiv

Umbral calculus over a vector space

  • Abdullah Alharthi
  • Eugene Lytvynov
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Abstract

Let V be a vector space over 𝔽=ℝ or β„‚. We develop a basis-free umbral calculus over V. We define the vector space of polynomials over V, and polynomial sequences in it. We discuss shift-invariant operators acting in polynomials over V. We define polynomial sequences of binomial type and Sheffer sequences over V. We provide equivalent characterizations of these polynomial sequences. We prove two recurrence formulas for Sheffer sequences. With each Sheffer sequence, we associate a linear operator acting in polynomials over V, which we call a Sheffer operator. We prove that the set of Sheffer operators is a group for the usual product of linear operators, which is isomorphic to the Riordan group of pairs of formal tensor power series in a variable from V. Under the assumption that V is an algebra, we lift every Sheffer sequence over 𝔽 to a Sheffer sequence over V. We provide examples of such lifting.

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Cite

BibTeXRIS
BibTeX
@misc{alharthi2026umbral,
  title = {{Umbral calculus over a vector space}},
  author = {Abdullah Alharthi and Eugene Lytvynov},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.10030},
  eprint = {2609.10030},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.10030},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Umbral calculus over a vector space
AU  - Abdullah Alharthi
AU  - Eugene Lytvynov
PY  - 2026
DA  - 2026/09/09
PB  - arXiv
DO  - 10.48550/arxiv.2609.10030
UR  - https://arxiv.org/abs/2609.10030
AB  - Let V be a vector space over 𝔽=ℝ or β„‚. We develop a basis-free umbral calculus over V. We define the vector space of polynomials over V, and polynomial sequences in it. We discuss shift-invariant operators acting in polynomials over V. We define polynomial sequences of binomial type and Sheffer sequences over V. We provide equivalent characterizations of these polynomial sequences. We prove two recurrence formulas for Sheffer sequences. With each Sheffer sequence, we associate a linear operator acting in polynomials over V, which we call a Sheffer operator. We prove that the set of Sheffer operators is a group for the usual product of linear operators, which is isomorphic to the Riordan group of pairs of formal tensor power series in a variable from V. Under the assumption that V is an algebra, we lift every Sheffer sequence over 𝔽 to a Sheffer sequence over V. We provide examples of such lifting.
ER  -