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

A series representation of the discrete fractional Laplace operator of arbitrary order