$q$ -Ito代数与多元$(-q)$ -Racah和Bannai-Ito多项式
The $q$-Bannai-Ito algebra and multivariate $(-q)$-Racah and Bannai-Ito polynomials
摘要：
Gasper和Rahman多元$(-q)$ -Racah多项式作为最近定义的高阶$q$ -Bannai-Ito代数$\mathcal{A}_n^q$的对角化不同交换子代数的基之间的连接系数出现。将代数的作用提升到联系系数上，利用差分算子得到了$\mathcal{A}_n^q$的一种实现。这为多元$(-q)$ -Racah多项式的双谱性提供了代数解释，如在[Iliev，Trans.Amer.Math.Soc.363（3）（2011），1577-1598]中建立的。此外，我们将Bannai-Ito正交多项式扩展到多个变量，并使用这些来表示$q = 1$更高秩Bannai-Ito代数$\mathcal{A}_n$的连接系数，从而证明了[De Bie等人，303（2016），390-414]。我们推导出这些多元Bannai-Ito多项式的正交关系，并提供了一个离散实现的$\mathcal{A}_n$。
英文摘要：
The Gasper and Rahman multivariate $(-q)$-Racah polynomials appear as connection coefficients between bases diagonalizing different abelian subalgebras of the recently defined higher rank $q$-Bannai-Ito algebra $\mathcal{A}_n^q$. Lifting the action of the algebra to the connection coefficients, we find a realization of $\mathcal{A}_n^q$ by means of difference operators. This provides an algebraic interpretation for the bispectrality of the multivariate $(-q)$-Racah polynomials, as was established in [Iliev, Trans. Amer. Math. Soc. 363 (3) (2011), 1577-1598]. Furthermore, we extend the Bannai-Ito orthogonal polynomials to multiple variables and use these to express the connection coefficients for the $q = 1$ higher rank Bannai-Ito algebra $\mathcal{A}_n$, thereby proving a conjecture from [De Bie et al., Adv. Math. 303 (2016), 390-414]. We derive the orthogonality relation of these multivariate Bannai-Ito polynomials and provide a discrete realization for $\mathcal{A}_n$.
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