## $$\mathbb{P}$$-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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