Matrix approach to polynomials 2

We continue our study of the structure initiated in [T. Arponen, A matrix approach to polynomials, Linear Algebra Appl. 359 (2003) 181-196]. Our main emphasis is exploring further structure into the formalism introduced in [T. Arponen, A matrix approach to polynomials, Linear Algebra Appl. 359 (2003) 181-196]. This formalism reveals beautiful interplay between certain elementary operators, and provides tools for example for checking, handling and generalizing combinatorial identities, as we show in examples. In addition to that we discover a group structure among Vandermonde matrices, a fascinating diophantine equation and new proofs and generalizations of recently found related results.