二元差分域上一阶差分方程的多项式解

Polynomial solutions of the first order difference equationsin the bivariate difference field

本文研究了在二元域F(α,β)(α、β是域F上代数无关的超越元)上一阶线性差分方程aσ(f)+bf=g的多项式解f,式中a、b、g是二元域F(α,β)上的已知多项式,σ是域F(α,β)上满足σ(α)=β,σ(β)=uα+vβ的域同构,其中u,v≠0.通过多项式次数分解得到多项式解f存在的性质,然后根据待定系数法求得多项式f,并给出了这类差分方程多项式解在符号求和方面的应用.

This paper study polynomial solutions of the difference equation aσ(f)+bf=g in the bivariate difference field F(α,β),whereα,βare two algebraically independent transcendental elements,σis a transformation that satisfiesσ(α)=β,σ(β)=uα+vβ,u,v≠0,a,b,g are known binary polynomials in F(α,β).The nature of the polynomial solution f is obtained by decomposition of the polynomial degree,and then the polynomial f is obtained according to the undetermined coefficient method.And some examples are given to illustrate the applications of this class of difference equations in symbolic summation.