The classical Herglotz theorem states that a sequence (ak) ∞ k=−∞ of complex numbers is the sequence of the Fourier-Stieltjes coefficients of a non-decreasing function g defined on the interval 〈0, 2π〉 if and only if the sequence (ak)k=−∞ is positive definite. Recall that ak = ∫ 2π 0 e dg(t) is said to be the kth FourierStieltjes coefficient of a function g (with bounded variation defined on th...

For each irreducible module of the symmetric group on N objects there is a set of parametrized nonsymmetric Jack polynomials in N variables taking values in the module. These polynomials are simultaneous eigenfunctions of a commutative set of operators, self-adjoint with respect to certain Hermitian forms. These polynomials were studied by the author and J.-G. Luque using a Yang–Baxter graph te...

Polynomial solutions to the generalized Lamé equation, the Stieltjes polynomials, and the associated Van Vleck polynomials have been studied since the 1830’s, beginning with Lamé in his studies of the Laplace equation on an ellipsoid, and in an ever widening variety of applications since. In this paper we show how the zeros of Stieltjes polynomials are distributed and present two new interlacin...

We study a class of kinetic-type differential equations ∂ϕt/∂t+ϕt=Q̂ϕt, where Q̂ is an inhomogeneous smoothing transform and, for every t≥0, ϕt the Fourier–Stieltjes probability measure. show that under mild assumptions on above equation possesses unique solution and represent this as characteristic function certain stochastic process associated with continuous time branching random walk pertaini...