Transformation of characteristic equations under bilinear mappings

The paper is concerned with calculation of a polynomical resulting from the characteristic polynomial due to fractional-linear mapping of the left-hand half-plane into the circle. A transition matrix is introduced which performs linear transformation of the coefficients and depending only on the order $n$ of the initial system. The properties of that matrix are studied and a simple recurrence relation is proposed for calculating the coefficients leads to significant saving of time both in manual and computer calculation.