Related papers: The search for differential equations for certain …
Sequences of orthogonal polynomials that are alternative to the Jacobi polynomials on the interval $[0,1]$ are defined and their properties are established. An $(\alpha,\beta)$-parameterized system of orthogonal polynomials of the…
New special polynomials associated with the rational solutions of analogue to the Painleve hierarchies are introduced. The Hirota relations for these special polynomials are found. Differential - difference hierarchies for finding special…
Orthogonality of the Jacobi and of Laguerre polynomials, P_n^(a,b) and L_n^(a), is established for a,b complex (a,b not negative integers and a+b different from -2,-3,...) using the Hadamard finite part of the integral which gives their…
Inspired by ideas from umbral calculus and based on the two types of integrals occurring in the defining equations for the gamma and the reciprocal gamma functions, respectively, we develop a multi-variate version of umbral calculus and of…
These notes aim to provide a classical approach to solving some conformable differential equations based on prior knowledge of how to solve ordinary differential equations. That is, using the methods of separation of variables, homogeneous…
The spectral decomposition for an explicit second-order differential operator $T$ is determined. The spectrum consists of a continuous part with multiplicity two, a continuous part with multiplicity one, and a finite discrete part with…
We study solutions of exponential polynomials over the complex field. Assuming Schanuel's conjecture we prove that certain polynomials have generic solutions in the complex field.
We discuss the construction of oscillator-like systems associated with orthogonal polynomials on the example of the Fibonacci oscillator. In addition, we consider the dimension of the corresponding lie algebras.
We study skew-orthogonal polynomials with respect to the weight function $\exp[-2V(x)]$, with $V(x)=\sum_{K=1}^{2d}(u_{K}/{K})x^{K}$, $u_{2d} > 0$, $d > 0$. A finite subsequence of such skew-orthogonal polynomials arising in the study of…
We investigate type I multiple orthogonal polynomials on $r$ intervals which have a common point at the origin and endpoints at the $r$ roots of unity $\omega^j$, $j=0,1,\ldots,r-1$, with $\omega = \exp(2\pi i/r)$. We use the weight…
This note reports on the recent advancements in the search for explicit representation, in classical special functions, of the solutions of the fourth-order ordinary differential equations named Bessel-type, Jacobi-type, Laguerre-type,…
Distributed-order PDEs are tractable mathematical models for complex multiscaling anomalous transport, where derivative orders are distributed over a range of values. We develop a fast and stable Petrov-Galerkin spectral method for such…
We have been working in many aspects of the problem of analyzing, understanding and solving ordinary differential equations (first and second order). As we have extensively mentioned, while working in the Darboux type methods, the most…
For the standard symplectic forms on Jacobi and CMV matrices, we compute Poisson brackets of OPRL and OPUC, and relate these to other basic Poisson brackets and to Jacobians of basic changes of variable.
A new method to find first integrals of nonlinear differential equations in Jacobi-type form is presented. The basic idea of our approach is to use one-parameter perturbed motions to find well-conceived nonlocal constants that are conserved…
For a long time it has been a challenging goal to identify all orthogonal polynomial systems that occur as eigenfunctions of a linear differential equation. One of the widest classes of such eigenfunctions known so far, is given by…
We discuss several numerical methods for calculating Lyapunov exponents (a quantitative measure of chaos) in systems of ordinary differential equations. We pay particular attention to constrained systems, and we introduce a variety of…
The differential systems satisfied by orthogonal polynomials with arbitrary semiclassical measures supported on contours in the complex plane are derived, as well as the compatible systems of deformation equations obtained from varying such…
Here the polynomial interpolation approach is used to introduce the main results on multivariate normal algebraic systems. Next we bring a construction which shows that any standard algebraic system, with finite set of solutions, can be…
A new approach for integration of the initial value problem for ordinary differential equations is suggested. The algorithm is based on approximation of the solution by a system of functions that contains orthogonal exponential polynomials.