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arXiv · 1808.06695

The q-Heun operator of big q-Jacobi type and the q-Heun algebra

Pascal Baseilhac·Luc Vinet·Alexei Zhedanov
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Abstract

The q-Heun operator of the big q-Jacobi type on the exponential grid is defined. This operator is the most general second order q-difference operator that maps polynomials of degree $n$ to polynomials of degree $n+1$. It is tridiagonal in bases made out of either q-Pochhammer or big q-Jacobi polynomials and is bilinear in the operators of the q-Hahn algebra. The extension of this algebra that includes the q-Heun operator as generator is described. Biorthogonal Pastro polynomials are shown to satisfy a generalized eigenvalue problem or equivalently to be in the kernel of a special linear pencil made out of two q-Heun operators. The special case of the q-Heun operator associated to the little q-Jacobi polynomials is also treated.

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Pascal Baseilhac, Luc Vinet, Alexei Zhedanov. 2018-08-03. The q-Heun operator of big q-Jacobi type and the q-Heun algebra. https://arxiv.org/abs/1808.06695

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