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Sklyanin-like algebras for (q-)linear grids and (q-)para-Krawtchouk polynomials
Journal of Mathematical Physics ( IF 1.3 ) Pub Date : 2021-01-01 , DOI: 10.1063/5.0024444
Geoffroy Bergeron 1 , Julien Gaboriaud 1 , Luc Vinet 1 , Alexei Zhedanov 2
Affiliation  

S-Heun operators on linear and $q$-linear grids are introduced. These operators are special cases of Heun operators and are related to Sklyanin-like algebras. The Continuous Hahn and Big $q$-Jacobi polynomials are functions on which these S-Heun operators have natural actions. We show that the S-Heun operators encompass both the bispectral operators and Kalnins and Miller's structure operators. These four structure operators realize special limit cases of the trigonometric degeneration of the original Sklyanin algebra. Finite-dimensional representations of these algebras are obtained from a truncation condition. The corresponding representation bases are finite families of polynomials: the para-Krawtchouk and $q$-para-Krawtchouk ones. A natural algebraic interpretation of these polynomials that had been missing is thus obtained. We also recover the Heun operators attached to the corresponding bispectral problems as quadratic combinations of the S-Heun operators

中文翻译:

用于 (q-) 线性网格和 (q-)para-Krawtchouk 多项式的类 Sklyanin 代数

引入了线性和 $q$-线性网格上的 S-Heun 算子。这些算子是 Heun 算子的特例,与类 Sklyanin 代数有关。连续 Hahn 和 Big $q$-Jacobi 多项式是这些 S-Heun 算子对其具有自然作用的函数。我们表明 S-Heun 算子包含双谱算子和 Kalnins 和 Miller 结构算子。这四个结构算子实现了原始 Sklyanin 代数三角退化的特殊极限情况。这些代数的有限维表示是从截断条件中获得的。相应的表示基础是多项式的有限族:para-Krawtchouk 和 $q$-para-Krawtchouk 的多项式。这样就得到了这些缺失多项式的自然代数解释。
更新日期:2021-01-01
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