  We introduce a finite version of free probability for rectangular matrices
that amounts to operations on singular values of polynomials. We show that we
can replicate the transforms from free probability, and that asymptotically
there is convergence from rectangular finite free probability to rectangular
free probability. Lastly, we show that classical distribution results such as a
law of large numbers or a central limit theorem can be made explicit in this
new framework where random variables are replaced by polynomials.
