Although fractional powers of non-negative operators have received much attention in recent years, there is still little known about their behavior if real-valued exponents are greater than one. In this article, we define and study the discrete fractional Laplace operator of arbitrary real-valued positive order. A series representation of the discrete fractional Laplace operator for positive non-integer powers is developed. Its convergence to a series representation of a known case of positive integer powers is proven as the power tends to the integer value. Furthermore, we show that the new representation for arbitrary real-valued positive powers of the discrete Laplace operator is consistent with existing theoretical results.

尽管近年来非负运算符的分数幂受到了很多关注，但如果实值指数大于 1，它们的行为仍然知之甚少。在本文中，我们定义并研究了任意实值正阶的离散分数拉普拉斯算子。开发了用于正非整数幂的离散分数拉普拉斯算子的级数表示。当幂趋向于整数值时，证明了它收敛于已知正整数幂情况的级数表示。此外，我们表明离散拉普拉斯算子的任意实值正幂的新表示与现有理论结果一致。