Generalized Askey functions and their walks through dimensions

We call ΦdΦd the class of continuous functions φ:[0,∞)→[0,∞)φ:[0,∞)→[0,∞) such that the radial function ψ(x):=φ(‖x‖),x∈Rd, is positive definite on RdRd, for dd a positive integer. We then introduce the generalized Askey class   of functions φn,k,m(⋅):[0,∞)→[0,∞)φn,k,m(⋅):[0,∞)→[0,∞) and show for which values of n,kn,k and mm such a class belongs to the class ΦdΦd. We then show walks through dimensions for scale mixtures of members of the class ΦdΦd with respect to nonnegative bounded measures; in particular, we show that, for a given member of ΦdΦd, there exist some classes of measures whose associated scale mixture does not preserve the same isotropy index dd and allows us to jump into another dimension d′d′ for the class ΦdΦd. These facts open surprising connections with the celebrated class of multiply monotone functions.