On the beta distribution, the nonlinear Fourier transform and a combinatorial problem

The paper describes some probabilistic and combinatorial aspects of the nonlinear Fourier transform associated with AKNS-ZS problems. One two main results shows that volumes a family polytopes appear in power expansion are distributed according to beta probability distribution. We establish this result by studying an Euler-type discretisation transform. This approach provides another result. discover novel discrete distribution approximating integral shape parameters. For specific choices parameters, our new has natural interpretation; numbers alternating ordered partitions integer into distinct parts essentially generating function for these can be easily expressed using constant function.