Multiple Hilbert transforms associated with polynomials

Two-dimensional Hilbert Transforms'

Multiple orthogonal polynomials associated with Macdonald functions

We consider multiple orthogonal polynomials corresponding to two Macdonald functions (modified Bessel functions of the second kind), with emphasis on the polynomials on the diagonal of the Hermite-Padé table. We give some properties of these polynomials: differential properties, a Rodrigues type formula and explicit formulas for the third order linear recurrence relation. 1 Macdonald functions ...

Computing Hilbert Class Polynomials

We present and analyze two algorithms for computing the Hilbert class polynomial HD. The first is a p-adic lifting algorithm for inert primes p in the order of discriminant D < 0. The second is an improved Chinese remainder algorithm which uses the class group action on CM-curves over finite fields. Our run time analysis gives tighter bounds for the complexity of all known algorithms for comput...

Darboux transforms on band matrices, weights and associated polynomials

Classically, it is well known that a single weight on a real interval leads to orthogonal polynomials. In ”Generalized orthogonal polynomials, discrete KP and Riemann-Hilbert problems”, Comm. Math. Phys. 207, pp. 589-620 (1999) , we have shown that m-periodic sequences of weights lead to ”moments”, polynomials defined by matrices containing these moments and 2m+1-step relations between them, th...

And Hilbert Polynomials

We show that intersection multiplicities over regular local rings can be computed using Hilbert polynomials of modules over the bigraded rings constructed by Gabber in his proof of Serre's nonnegativity conjecture. As a consequence, we give a simpler proof of a criterion in Kurano and Roberts 7] for intersection multiplicities to be positive. 1. Introduction In 11], Serre introduced a deenition...