We present a complete algorithm that computes all hypergeometric solutions of homogeneous linear difference equations and rational solutions of parameterized linear difference equations in the setting of $\mathrm{\Pi }{\mathrm{\Sigma }}^{⁎}$-fields. More generally, we provide a flexible framework for a big class of difference fields that are built by a tower of $\mathrm{\Pi }{\mathrm{\Sigma }}^{⁎}$-field extensions over a difference field that enjoys certain algorithmic properties. As a consequence one can compute all solutions in terms of indefinite nested sums and products that arise within the components of a parameterized linear difference equation, and one can find all hypergeometric solutions of a homogeneous linear difference equation that are defined over the arising sums and products.

我们提供了一个完整的算法，该算法可以计算齐次线性差分方程的所有超几何解和参数化线性差分方程有理解。 $\mathrm{\Pi }{\mathrm{\Sigma }}^{⁎}$场。更笼统地说，我们为由$\mathrm{\Pi }{\mathrm{\Sigma }}^{⁎}$享有某些算法属性的差异字段的扩展字段扩展。结果是，可以根据在参数化线性差分方程的组成部分中出现的不确定的嵌套和和乘积来计算所有解，并且可以找到在出现的和和乘积上定义的齐次线性差分方程的所有超几何解。 。