To Nicolás Andruskiewitsch on his 60th birthday, with admirationWe introduce bivariate versions of the continuous [Formula: see text]-Hermite polynomials. We obtain algebraic properties for them (generating function, explicit expressions in terms of the univariate ones, backward difference equations and recurrence relations) and analytic properties (determining the orthogonality measure). We find a direct link between bivariate continuous [Formula: see text]-Hermite polynomials and the star product method of [S. Kolb and M. Yakimov, Symmetric pairs for Nichols algebras of diagonal type via star products, Adv. Math. 365 (2020), Article ID: 107042, 69 pp.] for quantum symmetric pairs to establish deformed quantum Serre relations for quasi-split quantum symmetric pairs of Kac–Moody type. We prove that these defining relations are obtained from the usual quantum Serre relations by replacing all monomials by multivariate orthogonal polynomials.

中文翻译：

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双变量连续 q-Hermite 多项式和变形的量子 Serre 关系

在 Nicolás Andruskiewitsch 60 岁生日之际，怀着钦佩之情，我们介绍了连续 [公式：见正文]-Hermite 多项式的双变量版本。我们获得了它们的代数性质（生成函数、单变量的显式表达式、后向差分方程和递推关系）和分析性质（确定正交性度量）。我们发现双变量连续 [公式：见正文]-Hermite 多项式与 [S. Kolb 和 M. Yakimov，通过星积的对角类型 Nichols 代数的对称对，Adv。数学。365 (2020), 文章 ID: 107042, 69 pp.] 用于量子对称对建立 Kac-Moody 型准分裂量子对称对的变形量子 Serre 关系。