众所周知, 给定微分或差分域上一组元素, 它们在常数域上线性相关当且仅当它们所对应的Wronskian行列式或者Casoratian行列式为零.文章将这个结果推广到具有微分导子和差分导子的微分差分域; 同时基于~Okugawa~的工作, 还将结果推广到特征非~0~的微分差分域.

It is well-known that given finitely many of elements in a differential filed (difference field), these elements are linearly dependent over its constant field if and only if the corresponding Wronskian determinant (Casoratian determinant) vanishes. This paper generalizes these classical results into differential-difference fields, that is, the field including both differential operators and shift  operators. Based on Okugawa's work,   the results are generalized   to differential-difference fields with positive characteristic.