In 1978 M. Lassalle obtained
an analogue of the Laurent series for holomorphic functions on the complexification
of a compact symmetric space and proved a Plancherel type formula for such
functions. In 2002 J. Faraut established such a formula, which he calls Gutzmer’s
formula, for all noncompact Riemannian symmetric spaces. This was immediately put
into use by B. Krotz, G. Olafsson and R. Stanton to characterise the image of the
heat kernel transform. In this article we prove an analogue of Gutzmer’s formula for
the Heisenberg motion group and use it to characterise Poisson integrals associated
to the sublaplacian. We also use the Gutzmer’s formula to study twisted Bergman
spaces.
