Fischer Decomposition and Cauchy–Kovalevskaya Extension in Fractional Clifford Analysis: The Riemann–Liouville Case

Abstract

In this paper we present the basic tools of a fractional function theory in higher dimensions by means of a fractional correspondence to the Weyl relations via fractional Riemann–Liouville derivatives. A Fischer decomposition, Almansi decomposition, fractional Euler and Gamma operators, monogenic projection, and basic fractional homogeneous powers are constructed. Moreover, we establish the fractional Cauchy–Kovalevskaya extension (FCK extension) theorem for fractional monogenic functions defined on ℝd. Based on this extension principle, fractional Fueter polynomials, forming a basis of the space of fractional spherical monogenics, i.e. fractional homogeneous polynomials, are introduced. We study the connection between the FCK extension of functions of the form xPl and the classical Gegenbauer polynomials. Finally, we present an example of an FCK extension.

Clifford–Hermite Polynomials in Fractional Clifford Analysis

“…From the previous conclusions we have that the two natural operators D α and x, considered as elements of the odd part of the algebra, generate a finite-dimensional Lie superalgebra in the algebra of endomorphisms generated by the partial fractional Riemann-Liouville derivatives, the basic vector variables x j (seen as multiplication operators), and the basis of the Clifford algebra e j (for more details see [19]). Before we proceed we recall the definition of Lie superalgebra (for more details about Lie superalgebras and their connections with Clifford analysis see [2,12]).…”

“…In [19] the author showed that this simplification leads to the osp(1|2) case. Indeed, let us denote…”

“…Basic references for this mathematical field are [8,11]. Recently (see [13,18,19]) the development of an extension of Clifford analysis to the fractional case started, i.e., to the case where the first-order operator is defined via fractional derivatives -the so-called fractional Dirac operator.…”

A note on the discrete Cauchy‐Kovalevskaya extension

“…The description considered above looks tailor‐suited for operational purposes, since it combines some well‐known facts from Chebyshev polynomials with a fine integral representation involving the generalized Mittag‐Leffler function , leading to a quite unexpected link between the Cauchy problem besides the discrete Cauchy‐Kovalevskaya extension, and the Cauchy problem of heat type. Perhaps, an exploitation of this construction may be easily found for Gegenbauer polynomials, or for more general families of ultraspherical polynomials by means of Jacobi expansions (cf Vieira). For now, we will leave this question open for the interested reader.…”

“…265‐268 for the generalized theory in the context of hypercomplex variables, and to Delanghe et al, Chapter III for a wide class of examples involving spherical monogenics and other generating classes of special functions. For a fractional calculus counterpart of Kovalevskaya's result involving Gegenbauer polynomials, we refer to Vieira's work …”