二元Rogers-Szegö多项式
The Bivariate Rogers-Szegö Polynomials
摘要：
本文给出了一种算子方法来推导二元Rogers-Szegö多项式$h_n(x,y|q)$的Mehler公式和Rogers公式。Mehler公式的证明可以看作是Askey，Rahman和Suslov提出的连续大$q$ -Hermite多项式$H_n(x;a|q)$的非对称泊松核公式的一种新的证明方法. $h_n(x,y|q)$的Mehler公式涉及${}_3ϕ_2$和，Rogers公式涉及${}_2ϕ_1$和。这些结果的证明是基于参数增广的$q$ -指数算子和齐次$q$ -移位算子在两个变量。推广了Hou，Lascoux和Mu关于Rogers-Szegö多项式$h_n(x|q)$的最新结果，得到了$h_n(x,y|q)$的另一个Rogers型公式.最后给出了$H_n(x;a|q)$的一个基变换公式，该公式可用于Askey-Wilson积分的计算.
英文摘要：
We present an operator approach to deriving Mehler's formula and the Rogers formula for the bivariate Rogers-Szegö polynomials $h_n(x,y|q)$. The proof of Mehler's formula can be considered as a new approach to the nonsymmetric Poisson kernel formula for the continuous big $q$-Hermite polynomials $H_n(x;a|q)$ due to Askey, Rahman and Suslov. Mehler's formula for $h_n(x,y|q)$ involves a ${}_3ϕ_2$ sum and the Rogers formula involves a ${}_2ϕ_1$ sum. The proofs of these results are based on parameter augmentation with respect to the $q$-exponential operator and the homogeneous $q$-shift operator in two variables. By extending recent results on the Rogers-Szegö polynomials $h_n(x|q)$ due to Hou, Lascoux and Mu, we obtain another Rogers-type formula for $h_n(x,y|q)$. Finally, we give a change of base formula for $H_n(x;a|q)$ which can be used to evaluate some integrals by using the Askey-Wilson integral.
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