  The signature of a sample path is a formal series of iterated integrals along
the path. The expected signature of a stochastic process gives a summary of the
process that is especially useful for studying stochastic differential
equations driven by the process. Lyons-Ni derived a partial differential
equation for the expected signature of Brownian motion, starting at a point z
in a bounded domain, until it hits to boundary of the domain. We focus on the
domain of planar unit disc centred at 0. Motivated by recently found explicit
formulae for some terms in the expected signature of this process in terms of
Bessel functions, we derive a tensor series representation for this expected
signature, coming from from studying Lyons-Ni's PDE. Although the
representation is rather involved, it simplifies significantly to give a
formula for the polynomial leading order term in each tensor component of the
expected signature.
