  It was recently shown by the authors that deformations of hypergroup
convolutions w.r.t. positive semicharacters can be used to explain
probabilistic connections between the Gelfand pairs (SL(d,C), SU(d)) and
Hermitian matrices. We here study connections between general convolution
semigroups on commutative hypergroups and their deformations. We are able to
develop a satisfying theory, if the underlying positive semicharacter has some
growth property. We present several examples which indicate that this growth
condition holds in many interesting cases.
