On Certain Wronskians of Multiple Orthogonal Polynomials

Abstract

Abstract. We consider determinants of Wronskian type whose entries are multiple orthogonal polynomials associated with a path connecting two multi-indices. By assuming that the weight functions form an algebraic Chebyshev (AT) system, we show that the polynomials represented by the Wronskians keep a constant sign in some cases, while in some other cases oscillatory behavior appears, which generalizes classical results for orthogonal polynomials due to Karlin and Szegő. There are two applications of our results. The first application arises from the observation that the m-th moment of the average characteristic polynomials for multiple orthogonal polynomial ensembles can be expressed as a Wronskian of the type II multiple orthogonal polynomials. Hence, it is straightforward to obtain the distinct behavior of the moments for odd and even m in a special multiple orthogonal ensemble -the AT ensemble. As the second application, we derive some Turán type inequalities for multiple Hermite and multiple Laguerre polynomials (of two kinds). Finally, we study numerically the geometric configuration of zeros for the Wronskians of these multiple orthogonal polynomials. We observe that the zeros have regular configurations in the complex plane, which might be of independent interest.

Oscillation Theorems for the Wronskian of an Arbitrary Sequence of Eigenfunctions of Schrödinger’s Equation

“…Their complex roots form very regular patterns in the complex plane, [18], which can be interpreted approximately in terms of the Ferrer's diagram of the partition that defines the sequence, [21]. Zhang and Filipuk have recently studied Wronskian determinants of multiple orthogonal polynomials, [22].…”

“…From standard Sturm Liouville theory we know that these zeros are simple and x 1 < y k < x j ∀k = 1, ..., i. From (22) we see that these are the only zeros of w ′ in (a, b), and from (21)- (23) it is also clear that…”

Casorati type determinants of some $\mathfrak {q}$-classical orthogonal polynomials

“…One can mention Turán inequality for Legendre polynomials [15] and its generalizations, especially that of Karlin and Szegő on Hankel determinants whose entries are ultraspherical, Laguerre, Hermite, Charlier, Meixner, Krawtchouk, and other families of orthogonal polynomials [10]. Karlin and Szegő's strategy was precisely to express these Hankel determinants in terms of the Wronskian of certain orthogonal polynomials of another class (see, also, [1], [4], [5], [6], [7], [8], [9]). …”