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From the realization of $q-$oscillator algebra in terms of generalizedderivative, we compute the matrix elements from deformed exponential functionsand deduce generating functions associated with Rogers-Szeg\H{o} polynomials aswell as their relevant properties. We also compute the matrix elementsassociated to the $(p,q)-$oscillator algebra (a generalization of the $q-$one)and perform the Fourier-Gauss transform of a generalization of the deformedexponential functions.

$(q,\mu)$ and $(p,q,\zeta)-$exponential functions: Rogers-Szeg\H{o} polynomials and Fourier-Gauss transform