Hyperholomorphic functions and second order partial differential equations in 𝑅ⁿ

Abstract

Hyperholomorphic functions in R n {R^n} with n ≥ 2 n \geq 2 are introduced, extending the hitherto considered hyperholomorphic functions in R 2 {R^2} . A Taylor formula is derived, and with it a unique representation theorem is proved for hyperholomorphic functions that are real analytic at the origin. Hyperanalyticity is seen to be generally a consequence of hyperholomorphy and real analyticity combined. Results for hyperholomorphic functions are applied to gradients of solutions of second order homogeneous partial differential equations with constant coefficients. Polynomial solutions of such a second order equation are obtained by a matrix algorithm. These polynomials are modified and combined to form polynomial bases for real analytic solutions. It is calculated that in such a basis there are ( m + n − 3 ) ! ( 2 m + n − 2 ) / m ! ( n − 2 ) ! (m + n - 3)!(2m + n - 2)/m!(n - 2)! homogeneous polynomials of degree m m .