Dual exponential polynomials and linear differential equations

We study linear differential equations with exponential polynomial coefficients, where exactly one coefficient is of order greater than all the others. The main result shows that a nontrivial exponential polynomial solution of such an equation has a certain dual relationship with the maximum order coefficient. Several examples illustrate our results and exhibit possibilities that can occur.     

q-Laguerre polynomials and related q-partial differential equations

In this paper, we define two homogenous q-Laguerre polynomials, by introducing a modified q-differential operator, we prove that an analytic function can be expanded in terms of the q-Laguerre polynomials if and only if the function satisfies certain q-partial differential equations. Using this main result, we derive the generating functions, bilinear generating functions and mixed generating functions for the q-Laguerre polynomials and generalized q-Hahn polynomials. Cigler’s polynomials and its generating functions discussed in [J. Cao, D.-W. Niu, A note on q -difference equations for Cigler’s polynomials, J. Difference Equ. Appl. 22 (2016), 1880–1892.] are generalized. At last, we obtain an q-integral identity involving q-Laguerre polynomials. These applications indicate that the q-partial differential equation is an effective tool in studying q-Laguerre polynomials.     

Approximate solution of differential equations with the use of asymptotic polynomials

We consider an approximate solution of differential equations with initial and boundary conditions. To find a solution, we use asymptotic polynomials Q _n ^f (x) of the first kind based on Chebyshev polynomials T _n (x) of the first kind and asymptotic polynomials G _n ^f (x) of the second kind based on Chebyshev polynomials U _n (x) of the second kind. We suggest most efficient algorithms for each of these solutions. We find classes of functions for which the approximate solution converges to the exact one. The remainder is represented as an expansion in linear functionals {L _n ^f } in the first case and {M _n ^f } in the second case, whose decay rate depends on the properties of functions describing the differential equation.     

Hermite matrix polynomials and second order matrix differential equations

In this paper we introduce the class of Hermite’s matrix polynomials which appear as finite series solutions of second order matrix differential equations Y″−xAY′+BY=0. An explicit expression for the Hermite matrix polynomials, the orthogonality property and a Rodrigues’ formula are given.     

Newtonian valued differential fields with arbitrary value group

The notion of newtonianity is central to the study of the ordered differential field of logarithmic-exponential transseries done by Aschenbrenner, van den Dries, and van der Hoeven; see [Asymptotic Differential Algebra and Model Theory of Transseries, Chapter 14]. We remove the assumption of divisible value group from two of their results concerning newtonianity, namely the newtonization construction and the equivalence of newtonianity with asymptotic differential-algebraic maximality. We deduce the uniqueness of immediate differentially algebraic extensions that are asymptotically differential-algebraically maximal.     

Hilbert and Hilbert—Samuel polynomials and partial differential equations

Systems of linear partial differential equations with constant coefficients are considered. The spaces of formal and analytic solutions of such systems are described by algebraic methods. The Hilbert and Hilbert—Samuel polynomials for systems of partial differential equations are defined.Translated from Matematicheskie Zametki, vol. 77, no. 1, 2005, pp. 141–151.Original Russian Text Copyright © 2005 by A. G. Khovanskii, S. P. Chulkov.This revised version was published online in April 2005 with a corrected issue number.     

On differential equations for orthogonal polynomials on the unit circle

In this paper we characterize sequences of orthogonal polynomials on the unit circle whose corresponding Carathéodory function satisfies a Riccati differential equation with polynomial coefficients, in terms of second order matrix differential equations. In the semi-classical case, a characterization in terms of second order linear differential equations with polynomial coefficients is deduced.     

On intermediate solutions and the Wronskian for half-linear differential equations

The asymptotic behavior of nonoscillatory solutions of the half-linear differential equation is studied. In particular, two Wronskian-type functions, which have some interesting properties, similar to the one of the Wronskian in the linear case, are given. Using these properties and suitable integral inequalities, the existence of the so-called intermediate solutions is examined and an open problem is solved.     

Orthogonal polynomials and differential equations in neutron-transport and radiative-transfer theories

There is a set of orthogonal polynomials {g_n(x)} which plays a relevant role in the treatment of the case of anisotropic scattering in neutron-transport and radiative-transfer theories. They appear also in the spherical harmonics treatment of the isotropic scattering. These polynomials are orthogonal with respect to a weight function which is continuous in the interval [−1, + 1] and has a finite number of symmetric Dirac masses. Although some other structural properties of these polynomials (e.g., the three-term recurrence relation) as well as some properties of their zeros have been published, much more need to be known. In particular, neither the second-order differential equation nor the density of zeros (i.e., the number of zeros per unit of interval) of the polynomial g_n(x) have been found. Here we obtain the second-order differential equation in the case that these polynomials are hypergeometric, so leaving open the general case. Furthermore, the exact expressions of the moments around the origin of the density of zeros of g_n(x) are given in the general case. The asymptotic density of zeros is also pointed out. Finally, these polynomials are shown to belong to the Nevai's class.     

Permutation polynomials with low differential uniformity over finite fields of odd characteristic

In this paper, we propose a construction of functions with low differential uniformity based on known perfect nonlinear functions over finite fields of odd characteristic. For an odd prime power q, it is proved that the proposed functions over the finite field Fq are permutations if and only if q≡3(mod 4).     

Hilbert and Hilbert—Samuel polynomials and partial differential equations

Systems of linear partial differential equations with constant coefficients are considered. The spaces of formal and analytic solutions of such systems are described by algebraic methods. The Hilbert and Hilbert—Samuel polynomials for systems of partial differential equations are defined.     

Numerical solution of fractional differential equations with a Tau method based on Legendre and Bernstein polynomials

In this paper, we state and prove a new formula expressing explicitly the integratives of Bernstein polynomials (or B‐polynomials) of any degree and for any fractional‐order in terms of B‐polynomials themselves. We derive the transformation matrices that map the Bernstein and Legendre forms of a degree‐n polynomial on [0,1] into each other. By using their transformation matrices, we derive the operational matrices of integration and product of the Bernstein polynomials. These matrices together with the Tau method are then utilized to reduce the solution of this problem to the solution of a system of algebraic equations. The method is applied to solve linear and nonlinear fractional differential equations. Copyright © 2013 John Wiley & Sons, Ltd.     

On a class of differential equations of parabolic type with impulsive action

Existence and uniqueness theorems are obtained for semilinear differential equations of parabolic type with impulsive action in Banach spaces. These equations contain an operator, in general, noninvertible, multiplying the time derivative. The results are applied to partial differential equations with impulsive action.