Analogs of recurrent Newton formulas for systems of transcendental equations

We consider a general system of transcendental equations in the $n$-dimensional complex space $\Bbb C^n$ and introduce the concept of $\sigma$-power sum of the roots of a system $\sigma_\alpha$. Multidimensional residue theory yields the formulas that relate the $\sigma$-power sums of various orders to each other. We establish a connection between the $\sigma$-power sums and the power sums of the roots of the system.