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On q-middle convolution and generalized q-hypergeometric equation

  • Yumi Arai
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Abstract

We investigate higher-order q-difference equations obtained by successive applications of a q-analogue of addition and the q-middle convolution. We construct an n-dimensional system of linear q-difference equations and derive a single q-difference equation whose coefficients are polynomials of degree at most $1$ in the independent variable. Under certain conditions on the parameters, the resulting equation is a generalized q-hypergeometric equation of order n. We also obtain a multiple Jackson integral representation of solutions and give sufficient conditions for its convergence. By suitably choosing the functions appearing in the integrand and specializing the parameters, we obtain q-hypergeometric series solutions of type n ϕn−1.

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Cite

BibTeXRIS
BibTeX
@misc{arai2026qmiddle,
  title = {{On $q$-middle convolution and generalized $q$-hypergeometric equation}},
  author = {Yumi Arai},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.38286},
  eprint = {2610.38286},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.38286},
}
RIS
TY  - UNPB
TI  - On q-middle convolution and generalized q-hypergeometric equation
AU  - Yumi Arai
PY  - 2026
DA  - 2026/10/28
PB  - arXiv
DO  - 10.48550/arxiv.2610.38286
UR  - https://arxiv.org/abs/2610.38286
AB  - We investigate higher-order q-difference equations obtained by successive applications of a q-analogue of addition and the q-middle convolution. We construct an n-dimensional system of linear q-difference equations and derive a single q-difference equation whose coefficients are polynomials of degree at most $1$ in the independent variable. Under certain conditions on the parameters, the resulting equation is a generalized q-hypergeometric equation of order n. We also obtain a multiple Jackson integral representation of solutions and give sufficient conditions for its convergence. By suitably choosing the functions appearing in the integrand and specializing the parameters, we obtain q-hypergeometric series solutions of type _n ϕ_n−1.
ER  -