Convolution–multiplication identities for Tutte polynomials of graphs and matroids

We give a general convolution–multiplication identity for the multivariate and bivariate rank generating polynomial of a graph or matroid. The bivariate rank generating polynomial is transformable to and from the Tutte polynomial by simple algebraic operations. Several identities, almost all already known in some form, are specializations of this identity. Combinatorial or probabilistic interpretations are given for the specialized identities.