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 $\Pi\Sigma^*$-fields. More generally, we provide a flexible framework for a big class of difference fields that is built by a tower of $\Pi\Sigma^*$-field extensions over a difference field that satisfies 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 that are defined over the arising sums and products of a homogeneous linear difference equation.

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$\Pi\mathbf\Sigma^*$-域扩展中线性常微分方程的有理和超几何解

我们提出了一个完整的算法，可以在 $\Pi\Sigma^*$ 域的设置下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的有理解。更一般地说，我们为一大类差分字段提供了一个灵活的框架，该框架由满足某些算法属性的差分字段上的 $\Pi\Sigma^*$ 字段扩展塔构建。因此，可以根据参数化线性差分方程的分量内出现的不定嵌套和和乘积计算所有解，并且可以找到在齐次线性差分方程的出现和和乘积上定义的所有超几何解.