Analytic variations on the airy distribution
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 +1/3, -5/3, -11/3 etc. and -7/3, -13/3 -19/3 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 byproduct of this approach is the existence of relations between moments of the Airy distribution, the asymptotic expansion of the Airy function Ai(z) at +infinity, and power symmetric functions of the zeros -alpha (k) of Ai(z).
airy function brownian excursion area parking problem linear probing hashing