We derive a Taylor formula for
matrix-valued functions, in particular for hyperholomorphic functions. The latter
functions are matrix-valued functions that satisfy a certain type of first order
systems, for which we make no ellipticity assumption. For solutions of higher
order linear partial differential equations with constant coefficients in the
plane we show the existence of hyperconjugates, an obvious generalization of
harmonic conjugates in complex analysis. By way of hyperconjugates we
find series expansions for solutions of partial differential equations in terms
of polynomial solutions. These polynomials form a basis for real analytic
solutions at the origin. An algorithm for obtaining all such polynomials is
summarized at the end. This paper continues in the tradition of hypercomplex
analysis.
