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

Abstract

The $K$-type formulas of the space of $K$-finite solutions to the Heisenberg ultrahyperbolic equation ${\square }_{s}f=0$ for the non-linear 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 several properties of certain sequences $\{P_j(x;y){\}}_{j=0}^{\infty }$ and $\{Q_j(x;y){\}}_{j=0}^{\infty }$ of tridiagonal determinants, whose generating functions are given by local Heun functions. In particular, it is shown that these sequences satisfy a certain arithmetic-combinatorial property, which we refer to as a palindromic property. We further show that classical sequences of Cayley continuants $\{{\mathrm{Cay}}_j(x;y){\}}_{j=0}^{\infty }$ and Krawtchouk polynomials $\{{\mathcal{K}}_j(x;y){\}}_{j=0}^{\infty }$ also admit this property. In the end a new proof of Sylvester's formula for certain tridiagonal determinant $\mathrm{Sylv}(x;n)$ is provided from a representation theory point of view.

67 pages

References in corpus (2)

• On reducing the Heun equation to the hypergeometric equation
• Conformally invariant differential operators on Heisenberg groups and minimal representations