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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Hyperholomorphic functions and second order partial differential equations in $ {\bf R}\sp n$


Author: R. Z. Yeh
Journal: Trans. Amer. Math. Soc. 325 (1991), 287-318
MSC: Primary 35E20; Secondary 35A25
MathSciNet review: 1015927
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Abstract: Hyperholomorphic functions in $ {R^n}$ with $ n \geq 2$ are introduced, extending the hitherto considered hyperholomorphic functions in $ {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)!(2m + n - 2)/m!(n - 2)!$ homogeneous polynomials of degree $ m$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1991-1015927-9
PII: S 0002-9947(1991)1015927-9
Keywords: Hyperholomorphic functions, symmetric powers, Taylor formula, Leibniz formula, hyperanalyticity, real analytic solutions, polynomial bases
Article copyright: © Copyright 1991 American Mathematical Society