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

TL;DR

This paper introduces a new series representation for the discrete fractional Laplace operator of any positive real order, extending understanding beyond integer powers and ensuring consistency with existing theories.

Contribution

It develops a novel series representation for the discrete fractional Laplace operator of arbitrary positive real order, including convergence and consistency proofs.

Findings

Series representation converges to known integer power cases

Representation is consistent with existing theoretical results

Extends fractional Laplace operator analysis to arbitrary positive real orders

Abstract

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.