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In this paper, we give the generating functions of binary product between 2-orthogonal Chebyshev polynomials and kFibonacci, k-Pell, k-Jacobsthal numbers and the other orthogonal Chebyshev polynomials.

В настоящей работе для вероятности продолжения критических ветвящихся случайных процессов получено асимптотическое разложение в предположении о существовании факториальных моментов λk при k = 5, 6, . . . , m, m < ∞.

The method of integral representation and calculation of the combinatorial sums of various type (the method of coefficients) using the formal Laurent power series over $\mathbbm C$, the theory of analitical functions and the theory of multiple residues in $\mathbbm C^n$ were proposed by the author in the late seventies. This method was applied in various fields of mathematics. The method of coefficients is important for a difficult problem of calculation of the multiple sums with linear constraints on summation indices. Various combinatorial problems can be formulated in terms of such constraints. The calculation of the multiple sum with $q$-binomial coefficients and linear recurrent constraints on summation indices was published by the author in \guillemotleft The Bulletin of Irkutsk State University. Series Mathematics.\guillemotright in 2016. This problem appears at the enumeration of all own $t$-dimensional subspaces of the space $V_m$ over field $GF(q)$. V.P. Krivokolesko and E.K. Leinartas in \guillemotleft The Bulletin of Irkutsk State University. Series Mathematics.\guillemotright\, in 2012, using the Hadamard composition have proved the multiple identity with polynomial coefficients and various constraints on the limits of summation, containing the family of free parameters. This identity is generalisation of the identities studied earlier by several authors, since constructions of the Deubechies filters in the wavelets theory. Using the author's method of coefficients the short and simple calculation of Krivokolesko--Leinartas sum is carried out. These calculations also automatically provides an equivalent way of calculation of the specified sum by means of a traditional method of generation functions, using only the well-known operations over corresponding multiple formal Laurent power series.

Equations of motion in Newtonian and Hamiltonian forms are used for classical molecular dynamics simulation of particle system time evolution. When Newton equations of motion are used for finding of particle coordinates and velocities in $N$-particle system it takes to solve $3N$ ordinary differential equations of second order at every time step. Traditionally numerical schemes of Verlet method are used for solving Newtonian equations of motion of molecular dynamics. A step of integration is necessary to decrease for Verlet numerical schemes steadiness conservation on sufficiently large time intervals. It leads to a significant increase of the volume of calculations. Numerical schemes of Verlet method with Hamiltonian conservation control (the energy of the system) at every time moment are used in the most software packages of molecular dynamics for numerical integration of equations of motion. It can be used two complement each other approaches to decrease of computational time in molecular dynamics calculations. The first of these approaches is based on enhancement and software optimization of existing software packages of molecular dynamics by using of vectorization, parallelization and special processor construction. The second one is based on the elaboration of efficient methods for numerical integration for equations of motion. A procedure for constructing of explicit, implicit and symmetric symplectic numerical schemes with given approximation accuracy in relation to integration step for solving of molecular dynamic equations of motion in Hamiltonian form is proposed in this work. The approach for construction of proposed in this work procedure is based on the following points: Hamiltonian formulation of equations of motion; usage of Taylor expansion of exact solution; usage of generating functions, for geometrical properties of exact solution conservation, in derivation of numerical schemes. Numerical experiments show that obtained in this work symmetric symplectic third-order accuracy scheme conserves basic properties of the exact solution in the approximate solution. It is more stable for approximation step and conserves Hamiltonian of the system with more accuracy at a large integration interval then second order Verlet numerical schemes.

The Bernoulli polynomials for natural x were first considered by J.Berno\-ulli (1713) in connection with the problem of summation of the powers of consecutive positive integers. For arbitrary $x$ these polynomials were studied by L.Euler. The term ''Bernoulli polynomials'' was introduced by Raabe (J.L. Raabe, 1851). The Bernoulli numbers and polynomials are well studied, and are widely used in various fields of theoretical and applied mathematics. The article is devoted to some generalizations of the Bernoulli numbers and polynomials to the case of several variables. The concept of Bernoulli numbers associated to a rational cone generated by vectors with integer coordinates is defined. Using the Bernoulli numbers, we introduce the Bernoulli polynomials of several variables. Next we construct a difference operator acting on functions defined in a rational cone, and by methods of the theory of generating functions we prove a multidimensional analogue of the main property, which is the fact that the Bernoulli polynomials satisfy a difference equation. Also, we calculate the values of the integrals of the Bernoulli polynomials over shifts of the fundamental parallelotope, and for the sum of monomials over integer points of a rational parallelotope we find a multidimensional analogue of the Bernoulli formula, where the sum above is expressed in terms of the integral of the Bernoulli polynomial over a parallelotope with variable "top" vertex.

This paper discusses the substitution isomers on vertices, edges and faces of the [C60] fullerene. The derivation of the isomers based on Polya theorem. Installed formula symmetry Z, generating functions for determining the number of chiral and non-chiral isomers substitution distribution isomers families ρ(m), and depending on the number m of seats substitution. By additive scheme calculated thermodynamic properties (ΔfH°298, S°298, C°p298, ΔHsubl, ΔfG°298, lgK298) gaseous [C60]-[C100] -fullerens.

The laws of single-particle migration of atoms in the one-layer hexagonal structure are considered. It is accepted that impurity atom contacts with one of structure cell and can execute the single-order jumps. The exact solution of the infinite collection of microscopic equation of migration is written in the technique of the generating function and appropriate macroscopic characteristics are found. It is determined that the general pattern of transfer quality coincides with the law of particle diffuse propagation and rate of process propagation in the graphene structure exceeds the analogous value for the comparable two-dimensional square array. The method for finding the exact solution of migration problem in the limited medium is formulated on the basis of results for infinite medium. The migration pattern in the frontier area of the graphene sheet and in the structure which is shaping the carbon nanotubes is defined. The probable variations of the observed characteristics which are connected with the impurity availability which is the corollary of migration are discussed. [ABSTRACT FROM AUTHOR]