Combinatorial rules for some useful transformations

The various transformations involved in transforming the polynomialF(z) = \Sigma_{k=0}^{n}a_{k}z^{k}into the polynomialG(s) = \Sigma_{k=0}^{n} b_{k}s^{k}are discussed. In particular, the entries of theQ-matrix of the matrix transformationb] = Qa]are generated in easily obtained combinatorial rules. The various applications of the transformations in the stability study of discrete systems in aperiodicity conditions in evaluation of total square integrals and in digital-filter design are enumerated.