On the beta distribution, the nonlinear Fourier transform and a combinatorial problem - Saksida, Pavle (Ars mathematica contemporanea, 2024)
The paper describes some probabilistic and combinatorial aspects of the nonlinear Fourier transform associated with the AKNS-ZS problems. We show that the volumes of a family of polytopes that appear in a power expansion of the nonlinear Fourier transform are distributed according to the beta probability distribution. We establish this result by studying an Euler-type discretization of the nonlinear Fourier transform. This approach leads to the combinatorial problem of finding the number of alternating ordered partitions of an integer into a fixed number of distinct parts. We find the explicit formula for these numbers and show that they are essentially distributed according to a novel discretization of the beta distribution for a suitable choice of the shape parameters. We also find the generating functions of the numbers of alternating sums. These functions are expressed in terms of the our discrete nonlinear Fourier transform.
Keywords: beta distribution; discretization; diskretizacija; distribucija beta; nelinearna Fourierova transformacija; nonlinear Fourier transform