Jack Polynomials and Multivariate Orthogonal Polynomials: theory, applications, and computations

We will survey the (old) combinatorial and (new) analytical definitions of the Jack polynomials as special functions of a parameter. We will define and explain the relationship between Jack and Multivariate Orthogonal Polynomials. Along the way, we will generalize univariate combinatorial quantities, like the binomial coefficients and the Pochhammer symbol.

  We will also present one or two applications of the theory to eigenvalue statistics problems. We will show how these polynomials can be computed, talk about the inherent slowness of the algorithms, and if time allows, present a few computational examples, implemented in our new Maple Library MOPs (Multivariate Orthogonal Polynomials symbolically).

No background is necessary. This is joint work with Alan Edelman and Gene Shuman.