Application of a composition of generating functions for obtaining explicit formulas of polynomials

The study introduces a fractional mathematical model in the Caputo sense for hematopoietic stem cell-based therapy, utilizing generalized Bernoulli polynomials (GBPs) and operational matrices to solve a system of nonlinear equations. The significance of the study lies in the potential therapeutic applications of hematopoietic stem cells (HSCs), particularly in the context of HIV infection treatment, and the innovative use of GBPs and Lagrange multipliers in solving the fractional hematopoietic stem cells model (FHSCM). The aim of the study is to introduce an optimization algorithm for approximating the solution of the FHSCM using GBPs and Lagrange multipliers and to provide a comprehensive exploration of the mathematical techniques employed in this context. The research methodology involves formulating operational matrices for fractional derivatives of GBPs, conducting a convergence analysis of the proposed method, and demonstrating the accuracy of the method through numerical simulations. The major conclusion is the successful introduction of GBPs in the context of the FHSCM, featuring innovative control parameters and a novel optimization technique. The study also highlights the significance of the proposed method in providing accurate solutions for the FHSCM, thus contributing to the field of mathematical modeling in biological and medical research.

Moment representation of Bernoulli polynomial, Euler polynomial and Gegenbauer polynomials

This work develops an optimization method based on a new class of basis function, namely the generalized Bernoulli polynomials (GBP), to solve a class of nonlinear 2-dim fractional optimal control problems. The problem is generated by nonlinear fractional dynamical systems with fractional derivative in the Caputo type and the Goursat–Darboux conditions. First, we use the GBP to approximate the state and control variables with unknown coefficients and parameters. Afterwards, we substitute the obtained values for the variables and parameters in the objective function, nonlinear fractional dynamical system and Goursat–Darboux conditions. The 2-dim Gauss–Legendre quadrature rule together with a fractional operational matrix construct a constrained problem, that is solved by the Lagrange multipliers method. The convergence of the GBP method is proved and its efficiency is demonstrated by several examples.

В основе работы лежит формула бинома Ньютона и её обобщения на последовательности многочленов биномиального типа. Даны применения к обобщённой проблеме Варинга (Хуа Ло-кен) и проблеме Гильберта – Камке (Г.И.Архипов). Доказана формула Тейлора – Маклорена для многочленов и гладких функций и даны её приложения в численном анализе (решение уравнений методом касательных Ньютона, лемма Гензеля в полныхнеархимедовских полях, приближенное вычисление значений гладких функций в точке). Даётся аналог формулы бинома Ньютона для многочленов Бернулли и доказывается формула Эйлера — Маклорена суммирования значений функции по целым точкам, выведена формула Пуассона суммирования значений функции. Рассмотрены примеры последовательностей многочленов биномиального типа (степени, нижние и верхние факториальныестепени, многочлены Абеля и Лагерра). Найдены биномиальные свойства многочленов Аппеля и Эйлера. Для многочленов и гладких функций от нескольких переменных доказана формула Тейлора, получены многомерные аналоги формул Эйлера – Маклорена и Пуассона суммирования значений функции по решётке. Рассмотрен многомерный аналог этих формул для решётки в многомерном комплексном пространстве. Доказаны ряд свойствпоследовательности многочленов биномиального типа от нескольких переменных.

We have considered a proposed method to efficiently handle the piecewise formulation in the generation of approximate solutions for gas dynamic equation via the Legendre and Chebychev basis functions. Two cases of the gas dynamic equation were considered for numerical illustrations aided by MAPLE 18 software. We did obtained some fascinating results. To be precise, applying the proposed method to the homogeneous gas dynamic via the Legendre polynomial, we attained a maximum error of order 10−4 as against that of VIM with a maximum error of order 10−3 at t = 0.0001. Also, the proposed method via Chebyshev polynomials at t = 0.0001 attained a maximum error of order 10−4 as against the VIM which has a maximum error of order 10−2. In like manner, applying the proposed method to the nonhomogeneous gas dynamic via the Legendre polynomials we attained a maximum error of order 10−4 at t = 0.0001 as against that of VIM, which does not show convergent. Also, proposed method via Chebyshev polynomials at t = 0.0001 attained a maximum error of order 10−4 as against the VIM which does converge at every grid points. Thus, it is obvious that the proposed method is a better convergent iterative scheme than the in as much as t decreases.

Thermal equilibrium of a two-dimensional model of mobile point-like charges with pair logarithmic interactions, immersed in a fixed neutralising background, is exactly solvable at a special coupling constant. Assuming gauge invariance of the statistical mechanics and the requirement of local charge neutrality in the bulk regime of the system, the exact solution implies explicit expressions for inverse elements of an infinite interaction matrix. This result is equivalent to new summation formulas over upper index for bilinear products of Jacobi and Gegenbauer polynomials which are complementary to standard completeness relations involving lower indices for these polynomials. We go beyond the physical input and derive multi-variable generalizations of the summation formulas.

A spectral method based on the Legendre polynomials for solving Helmholz equations was proposed. With an explicit formula for the Legendre polynomials in terms of arbitrary order of their derivatives, the successive integration of the Legendre polynomials was represented by the Legendre polynomials. Then the method was formulized for second-order differential equations in one dimension and two dimensions. Numerical results indicate that the suggested method is significantly accurate and in satisfactory agreement with the exact solution.

Linear recurrent sequences and irrationality measures

By using a symbolic method, known in the literature as the classical umbral calculus, a symbolic representation of Levy processes is given and a new family of time-space harmonic polynomials with respect to such processes, which includes and generalizes the exponential complete Bell polynomials, is introduced. The usefulness of time-space harmonic polynomials with respect to Levy processes is that it is a martingale the stochastic process obtained by replacing the indeterminate x of the polynomials with a Levy process, whereas the Levy process does not necessarily have this property. Therefore to find such polynomials could be particularly meaningful for applications. This new family includes Hermite polynomials, time-space harmonic with respect to Brownian motion, Poisson-Charlier polynomials with respect to Poisson processes, Laguerre and actuarial polynomials with respect to Gamma processes , Meixner polynomials of the first kind with respect to Pascal processes, Euler, Bernoulli, Krawtchuk, and pseudo-Narumi polynomials with respect to suitable random walks. The role played by cumulants is stressed and brought to the light, either in the symbolic representation of Levy processes and their infinite divisibility property, either in the generalization, via umbral Kailath-Segall formula, of the well-known formulae giving elementary symmetric polynomials in terms of power sum symmetric polynomials. The expression of the family of time-space harmonic polynomials here introduced has some connections with the so-called moment representation of various families of multivariate polynomials. Such moment representation has been studied here for the first time in connection with the time-space harmonic property with respect to suitable symbolic multivariate Levy processes. In particular, multivariate Hermite polynomials and their properties have been studied in connection with a symbolic version of the multivariate Brownian motion, while multivariate Bernoulli and Euler polynomials are represented as powers of multivariate polynomials which are time-space harmonic with respect to suitable multivariate Levy processes.

Generalized Bernoulli polynomials were introduced by Shintani in 1976 in order to express the special values at non-positive integers of Dedekind zeta functions for totally real numbers. The coefficients of such polynomials are finite combinations of products of Bernoulli numbers which are difficult to get hold of. On the other hand, Zagier was able to get the explicit formula for the special values in cases of real quadratic number fields. In this paper, we shall improve Shintani’s formula by proving that the special values can be determined by a finite set of polynomials. This provides a convenient way to evaluate the special values of various types of Dedekind functions. Indeed, a much broader class of zeta functions considered by the author [Minking Eie, The special values at negative integers of Dirichlet series associated with polynomials of several variables, Proceedings of A. M. S. 119 (1993), 51–61] admits a similar formula for its special values. As a consequence, we are able to find infinitely many identities among Bernoulli numbers through identities among zeta functions. All these identities are difficult to prove otherwise.

The main objective in this paper is to study on special numbers and polynomials that contain finite sums of powers of binomial coefficients. By using generating function methods, some formulas and relations related to these numbers and the Apostol-Bernoulli and Apostol-Euler numbers of negative higher order, the Bernoulli and Euler numbers, the Stirling type numbers, the combinatorial numbers, the Bell polynomials, the Fubini type polynomials, and the Legendre polynomials are presented. Moreover, asymptotic expressions of the finite sums of powers of binomial coefficients for these numbers are given. Some numeric values of these asymptotic expressions are illustrated by the tables. Finally, some inequalities for these numbers are given.

Asymptotic expansions of certain generalized Bernoulli polynomials are obtained, some in terms of elementary functions and others in terms of gamma functions and their derivatives. The latter results can also be written in terms of elementary functions, by using the known asymptotic expansions of the gamma functions, and the leading terms are obtained in this way. The results obtained give the asymptotic form of the coefficients occurring in all the usual central-difference formulae of the calculus of finite differences.

Asymptotic behaviour of Bernoulli, Euler, and generalized Bernoulli polynomials

In this paper, we perform a further investigation of the Gegenbauer polynomials, the Chebyshev polynomials of the first and second kinds and the Legendre polynomials. By making use of some analytic and combinatorial methods, we establish some new expressions for sums of products of arbitrary numbers of Chebyshev polynomials of the first and second kinds and Legendre polynomials.

The main aim of this paper is that for any second-order linear recurrence sequence, the generating function of which is f ( t ) = 1 1 + a t + b t 2 , we can give the exact coefficient expression of the power series expansion of f x ( t ) for x ∈ R with elementary methods and symmetry properties. On the other hand, if we take some special values for a and b, not only can we obtain the convolution formula of some important polynomials, but also we can establish the relationship between polynomials and themselves. For example, we can find relationship between the Chebyshev polynomials and Legendre polynomials.