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The Al-Salam-Chihara polynomials are an important family of orthogonalpolynomials in one variable $x$ depending on 3 parameters $\alpha$, $\beta$ and$q$. They are closely connected to a model from statistical mechanics calledthe partially asymmetric simple exclusion process (PASEP) and they can beobtained as a specialization of the Askey-Wilson polynomials. We give twodifferent combinatorial formulas for the coefficients of the (transformed)Al-Salam-Chihara polynomials. Our formulas make manifest the fact that thecoefficients are polynomials in $\alpha$, $\beta$ and $q$ with positivecoefficients.

Combinatorial formulas for the coefficients of the Al-Salam-Chihara polynomials