# multivariate power series computation with recognition

I would like to do a computation in Sage involving formal power series but I believe some (all?) steps may be impossible:

define a couple of formal power series in two variables (x,y) by defining the coefficient function a(m,n) (the coefficients involve gamma functions and such, and are rational). As far as I can see there is no way to do even this step in Sage.

do a change of variables, something like (x,y) = (g(x,y),y), obtaining new power series. The function g(x,y) may be a polynomial or possibly itself an infinite power series.

multiply the resulting power series

Simplify the expression for the coefficients of the resulting power series c(m,n) in terms of classical quantities like gamma functions.

Which of these steps do you believe is possible in Sage? I could write my own software to deal with formal power series properly, but then I would not be able to recognize or simplify the resulting coefficients.

As an example:

h(x,y) is a sum of (m!)(4^n)x^m y^n with indices running from 0 to infinity, and k(x,y) is a sum of (m+n)x^m y^n,

do a substitution H(x,y) = h(x,exp(xy)y)

multiply H*k

simplify the coefficients of H*k.