Analytic Variations on the Airy Distribution

Abstract

The Airy distribution (of the ``area'' type) occurs as a limit distribution of cumulative parameters in a number of combinatorial structures, like path length in trees, area below walks, displacement in parking sequences, and it is also related to basic graph and polyomino enumeration. We obtain curious explicit evaluations for certain moments of the Airy distribution, including moments of orders -1 , -3 , -5 , etc., as well as +\frac13 , -\frac53 , -\frac 11 3 , etc. and -\frac73 , -\frac 13 3 , -\frac 19 3 , etc. Our proofs are based on integral transforms of the Laplace and Mellin type and they rely essentially on ``non-probabilistic'' arguments like analytic continuation. A by-product of this approach is the existence of relations between moments of the Airy distribution, the asymptotic expansion of the Airy function \Ai(z) at +∈fty , and power symmetric functions of the zeros -αk of \Ai(z) .