Degenerate Sklyanin algebras, Askey-Wilson polynomials and Heun operators

Abstract

The $q$-difference equation, the shift and the contiguity relations of the Askey-Wilson polynomials are cast in the framework of the three and four-dimensional degenerate Sklyanin algebras ${\mathfrak{ska}}_3$ and ${\mathfrak{ska}}_4$. It is shown that the $q$-para Racah polynomials corresponding to a non-conventional truncation of the Askey-Wilson polynomials form a basis for a finite-dimensional representation of ${\mathfrak{ska}}_4$. The first order Heun operators defined by a degree raising condition on polynomials are shown to form a five-dimensional vector space that encompasses ${\mathfrak{ska}}_4$. The most general quadratic expression in the five basis operators and such that it raises degrees by no more than one is identified with the Heun-Askey-Wilson operator.

arXiv comment19 pages, 32 references

References in corpus (11)

• The Relationship between Zhedanov's Algebra AW(3) and the Double Affine Hecke Algebra in the Rank One Case
• Algebraic Heun operator and band-time limiting
• The structure relation for Askey-Wilson polynomials
• Algebraization of difference eigenvalue equations related to $U_q(s{l}_2)$
• Degenerations of Ruijsenaars-van Diejen operator and q-Painleve equations
• Tridiagonalization and the Heun equation
• The Heun-Askey-Wilson algebra and the Heun operator of Askey-Wilson type
• The Heun operator as a Hamiltonian
• Degenerations of Sklyanin algebra and Askey-Wilson polynomials
• The Heun-Racah and Heun-Bannai-Ito algebras
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Cited by in corpus (6)

• The Askey-Wilson algebra and its avatars
• Bispectrality and biorthogonality of the rational functions of $q$-Hahn type
• The rational Sklyanin algebra and the Wilson and para-Racah polynomials
• Sklyanin-like algebras for ($q$-)linear grids and ($q$-)para-Krawtchouk polynomials
• Generalized quadratic commutator algebras of PBW-type
• Orthogonal polynomials and the deformed Jordan plane