Type I Hermite--Padé polynomials for a set of functions $f_0, f_1, ..., f_s$ at infinity, $Q_{n,0}$, $Q_{n,1}$, ..., $Q_{n,s}$, is defined by the asymptotic condition $$ R_n(z):=\bigl(Q_{n,0}f_0+Q_{n,1}f_1+Q_{n,2}f_2+...+Q_{n,s}f_s\bigr)(z) =\mathcal O (\frac1{z^{s n+s}}), \quad z\to\infty, $$ with the degree of all $Q_{n,k}\leq n$. We describe an approach for finding the asymptotic zero distribution of these polynomials as $n\to \infty$ under the assumption that all $f_j$'s are semiclassical, i.e. their logarithmic derivatives are rational functions. In this situation $R_n$ and $Q_{n,k}f_k$ satisfy the same differential equation with polynomials coefficients.
We discuss in more detail the case when $f_k$'s are powers of the same function $f$ ($f_k=f^k$); for illustration, the simplest non trivial situation of $s=2$ and $f$ having two branch points is analyzed in depth. Under these conditions, the ratio or comparative asymptotics of these polynomials is also discussed.
From methodological considerations and in order to make the situation clearer, we start our exposition with the better known case of Padé approximants (when $s=1$).
In the present paper we prove a Stieltjes type theorem on the convergence of a sequence of rational functions associated with a mixed type Hermite-Padé approximation problem of a Nikishin system of functions and analyze the ratio asymptotic of the corresponding Hermite-Padé polynomials.
We study the logarihtnmic asymptotic of multiple orthogonal polynomials arising in a mixed type Hermite-Padé approximation problem associated with the rational perturbation of a Nikishin system of functions. The formulas obtained allow to give exact estimates of the rate of convergence of the corresponding Hermite-Padé approximants.
We propose an algorithm for producing Hermite-Padé polynomials of type I for an arbitrary tuple of $m+1$ formal power series $[f_0,\dots,f_m]$, $m\geq1$, about $z=0$ ($f_j\in{\mathbb C}[[z]]$) under the assumption that the series have a certain (`general position') nondegeneracy property. This algorithm is a straightforward extension of the classical Viskovatov algorithm for construction of Padé polynomials (for $m=1$ our algorithm coincides with the Viskovatov algorithm). The algorithm proposed here is based on a recurrence relation and has the feature that all the Hermite-Padé polynomials co…
Motivated by the Novikov equation and its peakon problem, we propose a new mixed type Hermite--Padé approximation whose unique solution is a sequence of polynomials constructed with the help of Pfaffians. These polynomials belong to the family of recently proposed partial-skew-orthogonal polynomials. The relevance of partial-skew-orthogonal polynomials is especially visible in the approximation problem germane to the Novikov peakon problem so that we obtain explicit inverse formulae in terms of Pfaffians by reformulating the inverse spectral problem for the Novikov multipeakons. Furthermore, w…
On a general method for deriving a fourth-order differential equation satisfied by Laguerre-Hahn orthogonal polynomials with new results for the class 0 analogous to Hermite
University of Monastir（莫纳斯提尔大学）；Sorbonne Université（索邦大学）；Universidade do Porto（波尔图大学）
We consider sequences of biorthogonal polynomials with respect to a Cauchy type convolution kernel and give the weak and ratio asymptotic of the corresponding sequences of biorthogonal polynomials. The construction is intimately related with a mixed type Hermite-Padé approximation problem whose asymptotic properties is also revealed.
We give the strong asymptotic of Cauchy biorthogonal polynomials under the assumption that the defining measures are supported on non intersecting intervals of the real line and satisfy Szegő's condition. The biorthogonal polynomials are connected with certain mixed type Hermite-Padé polynomials, which verify full orthogonality relations with respect to some varying measures. Thus, the strong asymptotic of orthogonal polynomials with respect to varying measures plays a key role in the study.
Polynomials $Q_n(z)$, $n=0,1,\ldots,$ that are multi-orthogonal with respect to a Nikishin system of $p\geq 1$ compactly supported measures over the star-like set of $p+1$ rays $S_+:=\{z\in \mathbb{C}: z^{p+1}\geq 0 \}$ are investigated. We prove that the Nikishin system is normal, that the polynomials satisfy a three-term recurrence relation of order $p+1$ of the form $z Q_{n}(z)=Q_{n+1}(z)+a_{n}\,Q_{n-p}(z)$ with $a_n>0$ for all $n\geq p$, and that the nonzero roots of $Q_n$ are all simple and located in $S_+$. Under the assumption of regularity (in the sense of Stahl and Totik) of the measu…
We find and discuss asymptotic formulas for orthonormal polynomials $P_{n}(z)$ with recurrence coefficients $a_{n}, b_{n}$. Our main goal is to consider the case where off-diagonal elements $a_{n}\to\infty$ as $n\to\infty$. Formulas obtained are essentially different for relatively small and large diagonal elements $b_{n}$. Our analysis is intimately linked with spectral theory of Jacobi operators $J$ with coefficients $a_{n}, b_{n}$ and a study of the corresponding second order difference equations. We introduce the Jost solutions $f_{n}(z)$, $n\geq -1$, of such equations by a condition for $…
