Difference integrability conditions for parameterized linear difference and differential equations

Abstract

This paper is devoted to integrability conditions for systems of linear difference and differential equations with difference parameters. It is shown that such a system is difference isomonodromic if and only if it is difference isomonodromic with respect to each parameter separately. Due to this result, it is no longer necessary to solve non-linear difference equations to verify isomonodromicity, which will improve efficiency of computation with these systems.

Isomonodromic differential equations and differential categories

“…See also a survey [47] of Bolibrukh's results on isomonodromicity and the references given there. An explicit computational approach to testing whether a system of difference equations with differential or difference parameters is isomonodromic can be found in [5,44].…”

“…See also a survey [47] of Bolibrukh's results on isomonodromicity and the references given there. An explicit computational approach to testing whether a system of difference equations with differential or difference parameters is isomonodromic can be found in [5,44].The paper is organized as follows. We start by recalling the basic definitions and properties of differential algebraic groups, differential Tannakian categories, and the PPV theory in Section 2.…”

Tannakian Categories With Semigroup Actions

“…The authors expect that the results of the present paper on actions of semigroups (instead of Lie rings) on Tannakian categories will lead to a construction of Picard-Vessiot rings with semigroup actions (that is, with several difference parameters, not necessarily commuting) with immediate practical applications in the nearest future. This includes the problem of difference isomonodromy [22], which awaits the full development of the Picard-Vessiot theory with semigroup actions.…”

“…In addition, we show in Example 3.10 how the classical contiguity relations for the hypergeometric functions are reflected in our Tannkian approach. The q-difference analogue of the hypergeometric functions studied in this framework can be found in [22]. Moreover, such recurrence relations are not only of interest from the point of view of analysis and special functions, but, as emphasized in [27,13], they also appear in the representation theory of Lie groups: they are encoded in the properties of tensor products of representations, including decompositions of tensor products into the irreducible components (e.g., Clebsch-Gordan coefficients).…”

Laguerre Collocation Method for Solving Fredholm Integro-Differential Equations with Functional Arguments

“…Orthogonal polynomials occur often as solutions of mathematical and physical problems. They play an important role in the study of wave mechanics, heat conduction, electromagnetic theory, quantum mechanics, medicine and mathematical statistics, and so forth [1][2][3][4][5][6][7][8][9][10][11]. They provide a natural way to solve, expand, and interpret solutions to many types of important equations.…”