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The K-type formulas of the space of K-finite solutions to the Heisenberg ultrahyperbolic equation \(\square _sf=0\) for the nonlinear group \(\widetilde {SL}(3,\mathbb {R})\) are classified. This completes a previous study of Kable for the linear group \(SL(m,\mathbb {R})\) in the case of \(m=3\) , as well as generalizes our earlier results on a certain second-order differential operator. As a by-product we also show “functional equations” of certain sequences \(\{P_k(x;y)\}_{k=0}^\infty \) and \(\{Q_k(x;y)\}_{k=0}^\infty \) of tridiagonal determinants, whose generating functions are given by local Heun functions.

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Classification of K-type Formulas for the Heisenberg Ultrahyperbolic Operator \(\square _s\) for \( \widetilde {SL}(3,\mathbb {R})\) and Tridiagonal Determinants for Local Heun Functions

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  • 发表日期：

    2025年

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  • Toshihisa Kubo,
  • Bent Ørsted

摘要

The K-type formulas of the space of K-finite solutions to the Heisenberg ultrahyperbolic equation \(\square _sf=0\) for the nonlinear group \(\widetilde {SL}(3,\mathbb {R})\) are classified. This completes a previous study of Kable for the linear group \(SL(m,\mathbb {R})\) in the case of \(m=3\) , as well as generalizes our earlier results on a certain second-order differential operator. As a by-product we also show “functional equations” of certain sequences \(\{P_k(x;y)\}_{k=0}^\infty \) and \(\{Q_k(x;y)\}_{k=0}^\infty \) of tridiagonal determinants, whose generating functions are given by local Heun functions.