Almansi Theorem and Mean Value Formula for Quaternionic Slice-regular Functions
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Advances in Applied Clifford Algebras
Almansi Theorem and Mean Value Formula for Quaternionic Slice-regular Functions Alessandro Perotti∗ Abstract. We prove an Almansi Theorem for quaternionic polynomials and extend it to quaternionic slice-regular functions. We associate to every such function f , a pair h1 , h2 of zonal harmonic functions such ¯h2 . We apply this result to get mean value formulas and that f = h1 − x Poisson formulas for slice-regular quaternionic functions. Mathematics Subject Classification. Primary 30G35, Secondary 31B30, 33C50, 33C55. Keywords. Quaternionic polynomials, Slice-regular functions, Zonal harmonics, Almansi theorem, Mean value property, Poisson formula.
1. Introduction In this note we prove a quaternionic Almansi-type Theorem for polynomials with quaternionic coefficients and, more generally, for slice-regular functions of a quaternionic variable (see Sect. 3 for definitions and references about this function theory). We show that to every slice-regular function f defined on a domain Ω of the skew field H of quaternions, it is possible to associate two uniquely defined harmonic functions h1 , h2 such that f (x) = h1 (x)−xh2 (x) on Ω. This result allows to apply well-known results of harmonic function theory on R4 to obtain a mean value property for slice-regular functions and a Poisson-type formula over three-dimensional spheres. An interesting aspect of these last results is their validity over any sphere in Ω, without requiring any symmetry with respect to the real axis. This work was supported by GNSAGA of INdAM, and by the grants “Progetto di Ricerca INdAM, Teoria delle funzioni ipercomplesse e applicazioni”, and PRIN “Real and Complex Manifolds: Topology, Geometry and holomorphic dynamics” of Ministero dell’universit` ae della ricerca. This article is part of the Topical Collection on ISAAC 12 at Aveiro, July 29–August 2, 2019, edited by Swanhild Bernstein, Uwe Kaehler, Irene Sabadini, and Franciscus Sommen. ∗ Corresponding
author.
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A. Perotti
Adv. Appl. Clifford Algebras
An Almansi-type Theorem for slice-regular functions on real Clifford algebras Rn (also called slice-monogenic functions [3]) has been proved in [14]. The quaternionic algebra can be identified with the Clifford algebra R2 . However, slice-monogenic functions are naturally defined on the paravector space Rn+1 . Therefore the results of [14] for n = 2 are related to harmonic function theory in R3 instead of the one in R4 H. The quaternionic results obtained in the present paper present some similarities with those obtained in [14] for n = 4, since slice-regular functions on R3 , as those on H, are linked to harmonic functions of four real variables (see [13]). We recall that similar Almansi-type decompositions have been established in other settings, for example in Clifford analysis [10], for Dunkl operators [15] and in the umbral calculus setting [4]. The paper is structured in the following way. In Sect. 2 we prove the Almansi decomposition for quaternionic polynomials, then in Sect. 3 we recall th
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