Using the functions of triangular matrices (tables), we established formulas for members of high order linear homogeneous recurrence. Consequently, we obtained some connection formulas between symmetric polynomials.

In this paper, we consider determinants for some families of Toeplitz-Hessenberg matrices whose entries are Oresme numbers. These determinant formulas may also be rewritten as identities involving sums of products of Oresme numbers and multinomial coefficients. In particular, we establish a connection between the Oresme and the Fibonacci and Pell sequences via Toeplitz-Hessenberg determinants.

We consider the applications of triangular matrices and functions of them to the study of number sequences generated by infinite-order linear recurrence equations with constant coefficients. We also introduce the concept of superposition of linear recurrence equations and establish some of its properties.

Applying the apparatus of triangular matrices, this paper determines general relations between the terms of the sequences generated by linear homogeneous recurrence equations. We single out, in particular, the class of normal linear recurrence equations, for which the corresponding number sequences have some interesting number-theoretic properties.

Using the machinavy of paradeterminants and parapermanents developed in [2] we get new relations for some number-theoretical functions natural argument that were studied in [3].

Offered economical algorithm for calculation of rational shortenings of thefourth-order mixed periodic recurrence fraction

Recurrence fourth order fractions are studied. Connection with algebraic fourth order equations is established. Calculation algorithms of rational contractions of such fractions are built.

We consider relations between one class of partition polynomials, parafunctions of triangular matrices (tables), and linear recurrence relations.

New numbers, called Guinness numbers, are introduced using certain function of natural argument. Few problems related to these numbers are formulated.

The properties of parafunctions of matrices of vertical and horizontal structure are investigated. Some general formulas of inversion of polynomial sequences are built.

Робота присвячена перелiченню розбиттiв деяких класiв мультимножин, $n$-вершинних регулярних графiв другого степеня та бiтрансверсалей $n$-го порядку.

We consider new nonelementary functions such as the Fresnel integrals, generated by rising factorial powers. Graphs of such functions are plotted and some of their properties are proved. It is shown, that new integral functions are solutions of second order ordinary differential equations with variable coefficients.

Досліджуються властивості парафункцій матриць вертикальної та горизонтальної структури та будуються деякі загальні формули обернення поліноміальних послідовностей.

In this work we research the connections of Hessenberg's matrix functions with paradeterminants and parapermanents.

A new algebraic object is introduced - recurrent fractions, which is an n-dimensional generalization of continued fractions. It is used to describe an algorithm for rational approximations of algebraic irrational numbers. Some parametrization for generalized Pell's equations is constructed.

In this work we research the connections of Hessenberg’s matrix functions with paradeterminants and parapermanents.