On Ordinary, Linear q-Difference Equations, with Applications to q-Sato Theory

Thomas Ernst

doi:10.1155/2015/824549

Abstract

The purpose of this paper is to develop the theory of ordinary, linear q-difference equations, in particular the homogeneous case; we show that there are many similarities to differential equations. In the second part we study the applications to a q-analogue of Sato theory. The q-Schur polynomials act as basis function, similar to q-Appell polynomials. The Ward q-addition plays a crucial role as operation for the function argument in the matrix q-exponential and for the q-Schur polynomials.

Highlights

We begin this paper with an introduction to q-difference equations
In the second part we study the applications to a q-analogue of Sato theory
In a previous article [1] we introduced the concept q-analogues of matrix formulas

Summary

Introduction

We begin this paper with an introduction to q-difference equations. Since there is a well-known parallel approach to this theme, we quote some of the historical facts about this. We show an example of solutions to a q-difference equation with constant coefficients; the multiple root case can be solved in a similar way. Equation (16) has been studied by Adams [11], who generalized the results of Carmichael and Mason He assumed the coefficient functions aj(x) to be analytic or to have poles of finite order at the origin. (1) the general solution of the homogeneous equation involving n arbitrary constants and known as the complementary function,. It is obvious that we can continue this process to find qanalogues of any homogeneous, linear differential equation with constant coefficients, which has an exact solution in terms of sums of exponential functions. We can find particular solutions very similar to the ordinary case q = 1 by replacing integers by q-integers and solving the resulting system of equations. ∑s (n, k)q θqk, k=1 where S(n, k)q and s(n, k)q are q-Stirling numbers, inverse to each other

First Matrix Calculations

Time Evolution