Multiple orthogonal polynomials as special functions

type I multiple orthogonal polynomials
(An,1,...,An,r) is a vector ofr polynomials, withAn,j of
degreenj −1, for which
x k
x |n|−1
r
An,j dµj(x) = 0, k = 0,1,...,|n|−2
j=1
r
An,j dµj(x) = 1.
j=1
|n| linear conditions for|n| unknowns.
Solution exists and unique: n is a normal index for type I (⇔for
type II).
Notation:
Qn(x) =
r
j=1
An,j(x)wj(x), wj(x) = dµj(x)
dµ(x) .
Multiple orthogonal
polynomials
definition
type II multiple
orthogonal polynomials
type I multiple
orthogonal polynomials
Determinantal point
processes
Recurrence relations
Various examples
5 / 27