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Almansi theorem and mean value formula for quaternionic slice-regular functions. (English) Zbl 07261286
Summary: 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 $$h_1$$, $$h_2$$ of zonal harmonic functions such that $$f=h_1-\bar{x} h_2$$. We apply this result to get mean value formulas and Poisson formulas for slice-regular quaternionic functions.
##### MSC:
 30G35 Functions of hypercomplex variables and generalized variables 31B30 Biharmonic and polyharmonic equations and functions in higher dimensions 33C50 Orthogonal polynomials and functions in several variables expressible in terms of special functions in one variable 33C55 Spherical harmonics
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##### References:
 [1] Almansi, E., Sull’integrazione dell’equazione differenziale $$\Delta^{2n}=0$$, Annali di Mat. (3), 2, 1-51 (1899) · JFM 30.0331.03 [2] Axler, S.; Bourdon, P.; Ramey, W., Harmonic function theory, volume 137 of graduate texts in mathematics (2001), New York: Springer-Verlag, New York · Zbl 0959.31001 [3] Colombo, F.; Sabadini, I.; Struppa, DC, Slice monogenic functions, Israel J. Math., 171, 385-403 (2009) · Zbl 1172.30024 [4] Faustino, N.; Ren, G., (Discrete) Almansi type decompositions: an umbral calculus framework based on $${osp}(1|2)$$ symmetries, Math. Methods Appl. Sci., 34, 16, 1961-1979 (2011) · Zbl 1244.30069 [5] Gentili, G.; Stoppato, C.; Struppa, DC, Regular functions of a quaternionic variable. Springer monographs in mathematics (2013), Berlin: Springer, Berlin · Zbl 1269.30001 [6] Gentili, G.; Struppa, DC, A new theory of regular functions of a quaternionic variable, Adv. Math., 216, 1, 279-301 (2007) · Zbl 1124.30015 [7] Ghiloni, R.; Perotti, A., Slice regular functions on real alternative algebras, Adv. Math., 226, 2, 1662-1691 (2011) · Zbl 1217.30044 [8] Ghiloni, R.; Perotti, A.; Stoppato, C., The algebra of slice functions, Trans. Amer. Math. Soc., 369, 7, 4725-4762 (2017) · Zbl 1404.30055 [9] Ghiloni, R., Perotti, A., Stoppato, C.: Division algebras of slice functions. Proc. R. Soc. Edinburgh 150(4), 2055-2082 (2020). 10.1017/prm.2019.13 · Zbl 07220516 [10] Malonek, HR; Ren, G., Almansi-type theorems in Clifford analysis, Math. Methods Appl. Sci., 25, 16-18, 1541-1552 (2002) · Zbl 1058.30050 [11] Mitrea, D., Distributions, partial differential equations, and harmonic analysis (2018), Cham: Universitext Springer, Cham · Zbl 1425.46001 [12] Perotti, A.: A four dimensional Bernstein theorem. (2019) arXiv:1903.03068, submitted [13] Perotti, A.: Slice regularity and harmonicity on Clifford algebras. In: Topics in Clifford Analysis - Special Volume in Honor of Wolfgang Sprößig. Trends Math. Springer, Basel (2019) 10.1007/978-3-030-23854-4 [14] Perotti, A.: Almansi-type theorems for slice-regular functions on Clifford algebras. Complex Variables and Elliptic Equations, (2020) 10.1080/17476933.2020.1755967 [15] Ren, G., Almansi decomposition for Dunkl operators, Sci. China Ser. A, 48, Suppl, 333-342 (2005) · Zbl 1131.43010
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