On a convolution for $q$-normal elements
Résumé
In 2000 Carnovale and Koornwinder defined a $q$-convolution and proved that for some classes of measures it is associative and commutative. We investigate its positivity preserving properties. One of them is the notion of $q$-positivity related to $q$-moments. In this paper we describe an algebraic interpretation of $q$-positivity which leads us to the definition of $(p,q)$-convolution. It has a form similar to the $q$-convolution of Carnovale and Koornwinder coming from a braided algebra. For the new convolution we find an appropriate analogue of Fourier transform and also present a central limit theorem.