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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PB and AZ would wish to acknowledge the hospitality of the CRM during the course of this work.
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PB is supported by the CNRS. The research of LV is funded in part by a discovery Grant from the Natural Sciences and Engineering Research Council (NSERC) of Canada. Work of AZ is supported by the National Science Foundation of China (Grant No. 11771015).
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Baseilhac, P., Vinet, L. & Zhedanov, A. The q-Heun operator of big q-Jacobi type and the q-Heun algebra. Ramanujan J 52, 367–380 (2020). https://doi.org/10.1007/s11139-018-0106-8
- Heun operator
- q-orthogonal polynomials
- Askey–Wilson algebra
Mathematics Subject Classification