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We give identifications of the $q$-deformed Segal-Bargmann transform anddefine the Segal-Bargmann transform on mixed $q$-Gaussian variables. We provethat, when defined on the random matrix model of \'Sniady for the $q$-Gaussianvariable, the classical Segal-Bargmann transform converges to the $q$-deformedSegal-Bargmann transform in the large $N$ limit. We also show that the$q$-deformed Segal-Bargmann transform can be recovered as a limit of a mixtureof classical and free Segal-Bargmann transform.

Segal-Bargmann transform: the $q$-deformation