Zatorsky R. Applications of the calculus of triangular matrices to combinatorial analysis and number theory

The thesis is devoted to Calculus of triangular matrices and its applications to combinatorial analysis and number theory. In particular, it laids the foundations of the theory of recurrent fractions, which are a new n-dimensional generalization of continued fractions. The positional number systems are generalized. Stanley's theorem about numerical sequences that are determined by linear recurrence relations is refined. Paradeterminant and parapermanent products of triangular matrices, for which parapermanent and paradeterminant are multiplicative functions of triangular matrices, are constructed. Properties of factorial numerical triangles, recursive combinatorial identities and polynomials of partitions are studied. General members of the results for basic operations on formal power series are expressed as parafunctions of triangular matrices.