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

  • Pascal Baseilhac
  • Luc Vinet
  • Alexei ZhedanovEmail author


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.


Heun operator q-orthogonal polynomials Askey–Wilson algebra 

Mathematics Subject Classification

34D45 39A13 



PB and AZ would wish to acknowledge the hospitality of the CRM during the course of this work.


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Authors and Affiliations

  1. 1.Institut Denis-Poisson CNRS/UMR 7013 – Université de Tours – Université d’Orléans Parc de GrammontToursFrance
  2. 2.Centre de Recherches MathématiquesUniversité de MontréalMontrealCanada
  3. 3.Department of Mathematics, School of InformationRenmin University of ChinaBeijingChina

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