  This is a detailed version of the paper math.FA/0212273. The main motivation
for this work was to find an explicit formula for a "Szego-regularized"
determinant of a zeroth order pseudodifferential operator (PsDO) on a Zoll
manifold. The idea of the Szego-regularization was suggested by V. Guillemin
and K. Okikiolu. They have computed the second term in a Szego type expansion
on a Zoll manifold of an arbitrary dimension. In the present work we compute
the third asymptotic term in any dimension. In the case of dimension 2, our
formula gives the above mentioned expression for the Szego-redularized
determinant of a zeroth order PsDO. The proof uses a new combinatorial
identity, which generalizes a formula due to G.A.Hunt and F.J.Dyson. This
identity is related to the distribution of the maximum of a random walk with
i.i.d. steps on the real line. The proof of this combinatorial identity
together with historical remarks and a discussion of probabilistic and
algebraic connections has been published separately.

Functional Analysis;Combinatorics;Probability