Algebraic properties of codimension series of PI-algebras

Let cn(R), n = 0, 1, 2, …, be the codimension sequence of the PI-algebra R over a field of characteristic 0 with T-ideal T(R) and let c(R, t) = c0(R) + c1(R)t + c2(R)t2 + … be the codimension series of R (i.e., the generating function of the codimension sequence of R). Let R1,R2 and R be PI-algebras such that T(R) = T(R1)T(R2). We show that if c(R1, t) and c(R2, t) are rational functions, then c(R, t) is also rational. If c(R1, t) is rational and c(R2, t) is algebraic, then c(R, t) is also algebraic. The proof is based on the fact that the product of two exponential generating functions behaves as the exponential generating function of the sequence of the degrees of the outer tensor products of two sequences of representations of the symmetric groups Sn.