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Differential systems with a Fuchsian linear part are studied in regions including all the singularities in the complex plane of these equations. Such systems are not necessarily analytically equivalent to their linear part (they are not…
We consider here a particular quadratic equation linking two elements of a C-Algebra. By analysing powers of the unknowns, it appears a double sequence of polynomials related to classical Bernoulli polynomials. We get the generating…
The linearization of complex ordinary differential equations is studied by extending Lie's criteria for linearizability to complex functions of complex variables. It is shown that the linearization of complex ordinary differential equations…
A nonlinear inequality is formulated in the paper. An estimate of the rate of decay of solutions to this inequality is obtained. This inequality is of interest in a study of dynamical systems and nonlinear evolution equations. It can be…
We compute the graded polynomial identities of the infinite dimensional upper triangular matrix algebra over an arbitrary field. If the grading group is finite, we prove that the set of graded polynomial identities admits a finite basis. We…
We consider higher symmetries and operator symmetries of linear partial differential equations. The higher symmetries form a Lie algebra, and operator ones form an associative algebra. The relationship between these symmetries is…
In this note we will present how Euler's investigations on various different subjects lead to certain properties of the Legendre polynomials. More precisely, we will show that the generating function and the difference equation for the…
A general formalism to solve nonlinear differential equations is given. Solutions are found and reduced to those of second order nonlinear differential equations in one variable. The approach is uniformized in the geometry and solves…
In this article, a new approach based on linear algebra is adopted to study a hybrid Sheffer polynomial sequences. The recurrence relations and differential equation for these polynomials are derived by using the properties and…
We derive some q-analogs of Euler-Cassini-type identities and of recurrence formulas for powers of Fibonacci polynomials.
Can we effectively learn a nonlinear representation in time comparable to linear learning? We describe a new algorithm that explicitly and adaptively expands higher-order interaction features over base linear representations. The algorithm…
We derive two new identities involving the Bernoulli numbers, the Euler numbers, and the Stirling numbers of the first kind using analytic continuation of a well known identity for the Stirling numbers of the first kind.
This paper deals with the solution of large classes of systems of nonlinear partial differential equations (PDEs) in spaces of generalized functions that are constructed as the completion of uniform convergence spaces. The existence result…
We describe a general method to determine the Apery limits of a differential equation that have a modular-function origin. As a by-product of our analysis, we discover a family of identities involving the special values of L-functions…
Using the theory of exponential Riordan arrays and orthogonal polynomials, we demonstrate that the "descending power" Eulerian polynomials, and their once shifted sequence, are moment sequences for simple families of orthogonal polynomials,…
Recently, Kim-Kim Introduced some interesting identities of symmetry for q-Bernoulli polynomials under symmetry group of degree n. In this paper, we study the degenerate q-Euler polynomials and derive some identities of symmetry for these…
We show that a wide range of overdetermined boundary problems for semilinear equations with position-dependent nonlinearities admits nontrivial solutions. The result holds true both on the Euclidean space and on compact Riemannian…
The article describes interesting nonlinear differential identities satisfied by standard exponential and trigonometric functions, which appeared as byproducts of medical imaging research. They look like some kind of non-commutative…
In this paper, we consider the problem of representing any polynomial in terms of the degenerate Bernoulli polynomials and more generally of the higher-order degenerate Bernoulli polynomials. We derive explicit formulas with the help of…
Using matrix identities, we construct explicit pseudo-exponential-type solutions of linear Dirac, Loewner and Schr\"odinger equations depending on two variables and of nonlinear wave equations depending on three variables.