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\(\mathbb{P}\)-species and the \(q\)-Mehler formula. (English) Zbl 1005.33006

The author presents a bijective proof of \(q\)-Mehler formula. This is done by first introducing a slight variant of the \(q\)-Hermite polynomials as \[ \overline H_n(x|q): =i^nq^{3/2}\widetilde H_n\left( \left.{-ix \over q} \right |q\right). \] (Note that these new \(q\)-Hermite polynomials are also defined combinatorially). The \(q\)-Mehler formula for \(\overline H_n (x\mid q)\) is then obtained by generalising Bessel’s result. The main result of the paper is contained in the following theorem: The polynomials \(\overline H_n(x|q)\) satisfy the following Mehler type identity \[ \begin{split} \sum^\infty_{n=0} \overline H_n(x\mid q) \overline H_n(y\mid q){t^n\over n!q}=\\ ={(q^2t^2)_\infty \over\prod^\infty_{k=0} \biggl[(1-t^2q^{2k+2})^2- t(1-q)q^k \bigl((1+t^2 q^{2k+2}) xy+tq^{k+1} (x^2+y^2)\bigr)\biggr]}. \end{split} \] The proof provided in the paper may help to find some multilinear extensions of the \(q\)-Mehler formula.

MSC:

33D45 Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)
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