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

Pascal Baseilhac (1) , Luc Vinet (2) , Alexei Zhedanov (3)

1 IDP - Institut Denis Poisson
2 CRM - Centre de Recherches Mathématiques [Montréal]
3 Department of Mathematics [Renmin]

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.

Mots clés: Heun operator; Askey–Wilson algebra; q-orthogonal polynomials

Domaines: Analyse classique [math.CA]; Physique mathématique [math-ph]; Algèbres quantiques [math.QA]

Pascal Baseilhac, Luc Vinet, Alexei Zhedanov. The q-Heun operator of big q-Jacobi type and the q-Heun algebra. Ramanujan Journal, 2019, ⟨10.1007/s11139-018-0106-8⟩. ⟨hal-02143736⟩