The $q$-Bannai-Ito algebra and multivariate $(-q)$-Racah and Bannai-Ito polynomials

Abstract

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 ${\mathcal{A}}_n$.

61 pages, added more details on construction of bases in section 2.3 and 2.4, various other small changes

References in corpus (5)

• The q-deformed analogue of the Onsager algebra: Beyond the Bethe ansatz approach
• The higher rank $q$-deformed Bannai-Ito and Askey-Wilson algebra
• Higher Rank Relations for the Askey-Wilson and $q$-Bannai-Ito Algebra
• Higher rank classical analogs of the Askey-Wilson algebra from the $s{l}_N$ Onsager algebra
• Bivariate Bannai-Ito polynomials

Cited by in corpus (5)

• The Askey-Wilson algebra and its avatars
• Finite-dimensional irreducible modules of the Bannai--Ito algebra at characteristic zero
• The Racah Algebra as a Subalgebra of the Bannai-Ito Algebra
• An Askey-Wilson Algebra of Rank 2
• Convolution identities for Dunkl orthogonal polynomials from the $\mathfrak{osp}(1|2)$ Lie superalgebra