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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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