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Register a new userThe symmetrization problem for multiple orthogonal polynomials
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We analyze the effect of symmetrization in the theory of multiple orthogonal polynomials. For a symmetric sequence of type II multiple orthogonal polynomials satisfying a high-term recurrence relation, we fully characterize the Weyl function associated to the corresponding block Jacobi matrix as well as the Stieltjes matrix function. Next, from an arbitrary sequence of type II multiple orthogonal polynomials with respect to a set of d linear functionals, we obtain a total of d+1 sequences of type II multiple orthogonal polynomials, which can be used to construct a new sequence of symmetric type II multiple orthogonal polynomials. Finally, we prove a Favard-type result for certain sequences of matrix multiple orthogonal polynomials satisfying a matrix four-term recurrence relation with matrix coefficients.
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We consider a set of measures on the real line and the corresponding system of multiple orthogonal polynomials (MOPs) of the first and second type. Under some very mild assumptions, which are satisfied by Angelesco systems, we define self-adjoint Jacobi matrices on certain rooted trees. We express their Greens functions and the matrix elements in terms of MOPs. This provides a generalization of the well-known connection between the theory of polynomials orthogonal on the real line and Jacobi matrices on $mathbb{Z}_+$ to higher dimension. We illustrate importance of this connection by proving ratio asymptotics for MOPs using methods of operator theory.
Let $p_n(x)$, $n=0,1,dots$, be the orthogonal polynomials with respect to a given density $dmu(x)$. Furthermore, let $d u(x)$ be a density which arises from $dmu(x)$ by multiplication by a rational function in $x$. We prove a formula that expresses the Hankel determinants of moments of $d u(x)$ in terms of a determinant involving the orthogonal polynomials $p_n(x)$ and associated functions $q_n(x)=int p_n(u) ,dmu(u)/(x-u)$. Uvarovs formula for the orthogonal polynomials with respect to $d u(x)$ is a corollary of our theorem. Our result generalises a Hankel determinant formula for the case where the rational function is a polynomial that existed somehow hidden in the folklore of the theory of orthogonal polynomials but has been stated explicitly only relatively recently (see [arXiv:2101.04225]). Our theorem can be interpreted in a two-fold way: analytically or in the sense of formal series. We apply our theorem to derive several curious Hankel determinant evaluations.
In this contribution we deal with sequences of monic polynomials orthogonal with respect to the Freud Sobolev-type inner product begin{equation*} leftlangle p,qrightrangle _{s}=int_{mathbb{R}}p(x)q(x)e^{-x^{4}}dx+M_{0}p(0)q(0)+M_{1}p^{prime }(0)q^{prime }(0), end{equation*}% where $p,q$ are polynomials, $M_{0}$ and $M_{1}$ are nonnegative real numbers. Connection formulas between these polynomials and Freud polynomials are deduced and, as a consequence, a five term recurrence relation for such polynomials is obtained. The location of their zeros as well as their asymptotic behavior is studied. Finally, an electrostatic interpretation of them in terms of a logarithmic interaction in the presence of an external field is given.
For the weight function $W_mu(x) = (1-|x|^2)^mu$, $mu > -1$, $lambda > 0$ and $b_mu$ a normalizing constant, a family of mutually orthogonal polynomials on the unit ball with respect to the inner product $$ la f,g ra = {b_mu [int_{BB^d} f(x) g(x) W_mu(x) dx + lambda int_{BB^d} abla f(x) cdot abla g(x) W_mu(x) dx]} $$ are constructed in terms of spherical harmonics and a sequence of Sobolev orthog onal polynomials of one variable. The latter ones, hence, the orthogonal polynomials with respect to $la cdot,cdotra$, can be generated through a recursive formula.
A new recurrence relation for exceptional orthogonal polynomials is proposed, which holds for type 1, 2 and 3. As concrete examples, the recurrence relations are given for Xj-Hermite, Laguerre and Jacobi polynomials in j = 1,2 case.
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