### FRACTIONAL ELLIPTIC OPERATOR OF ORDER 2/3 FROM GLAESKE-KILBAS-SAIGO FRACTIONAL INTEGRAL TRANSFORM

Rami El-Nabulsi

#### Abstract

In 2000, Andrea Raspini introduced a fractional Dirac equation of order by taking the cubic root of the Klein-Gordon equation and by proving that a 3-fold factorization of the Klein-Gordon equation leads to a fractional Dirac equation characterized by an elliptic operator of order and where the resulting γ - matrices obey an extended Clifford algebra. In this short communication, we show that an elliptic operator of order may be constructed from the generalized fractional Glaeske-Kilbas-Saigo integral which will be useful to define non-integer dimensional deformations of the canonical spectral triples in noncommutative geometry. We show that the fractional order corresponds to the highest pole of the fractional dimension spectrum of a spectral triple in the case of a four-dimensional torus as formulated by Connes-Moscovici. In this work, we give only a concise exposition of the topic whereas a complete description will be presented elsewhere.

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