## Tohoku Mathematical Journal 2013 June SECOND SERIES VOL. 65, NO. 2

 Tohoku Math. J. 65 (2013), 199-229

Title POLYHARMONIC FUNCTIONS OF INFINITE ORDER ON ANNULAR REGIONS

Author Ognyan Kounchev and Hermann Render

(Received April 12, 2011, revised July 23, 2012)
Abstract. Polyharmonic functions $f$ of infinite order and type $\tau$ on annular regions are systematically studied. The first main result states that the Fourier-Laplace coefficients $f_{k,l}(r)$ of a polyharmonic function $f$ of infinite order and type $0$ can be extended to analytic functions on the complex plane cut along the negative semiaxis. The second main result gives a constructive procedure via Fourier-Laplace series for the analytic extension of a polyharmonic function on annular region $A(r_{0},r_{1})$ of infinite order and type less than $1/2r_{1}$ to the kernel of the harmonicity hull of the annular region. The methods of proof depend on an extensive investigation of Taylor series with respect to linear differential operators with constant coefficients.

Mathematics Subject Classification. Primary 31B30; Secondary 32A07, 42C15.

Key words and phrases. Polyharmonic function, annular region, Fourier-Laplace series, Linear differential operator with constant coefficient, Taylor series, analytical extension.