The Painlevé Equations and Orthogonal Polynomials

This article is a survey of the recent studies jointly with Lies Boelen, Christophe Smet, Walter Van Assche and Lun Zhang (KULeuven, Belgium) on semi-classical continuous and discrete orthogonal polynomials and, in particular, on the connection of their recurrence coefficients to the solutions of the Painlevé equations. After recalling some basic facts about the Painlevé equations, we discuss continuous and discrete orthogonal polynomials and explain their connection.     

Defects and the Convergence of Padé Approximants

A selective survey is given of convergence results for sequences of Padé approximants. Various approaches for dealing with the convergence problems due to `defects" are discussed. Attention is drawn to the close relationship between analyticity properties of a function and the `smoothness" of its Taylor series coefficients. A new theorem on the convergence of horizontal sequences of Padé approximants to functions in the Baker–Gammel–Wills conjecture function class is presented.     

Nondiagonal Hermite–Sobolev Orthogonal Polynomials

In this work, we study algebraic and analytic properties for the polynomials { Q _n } _{n 0}, which are orthogonal with respect to the inner product where , R such that – ² > 0.     

Hermite—Padé Approximants of the Mittag-Leffler Functions

The convergence rate of type II Hermite–Padé approximants for a system of degenerate hypergeometric functions {₁F₁(1, γ; λ_jz)} _j=1 ^k is found in the case when the numbers {λ_j} _j=1 ^k are the roots of the equation λ^k = 1 or real numbers and \(\gamma\in\mathbb{C}\;\backslash\left\{0,-1,-2,...\right\}\). More general statements are obtained for approximants of this type (including nondiagonal ones) in the case of k = 2. The theorems proved in the paper complement and generalize the results obtained earlier by other authors.     

Square Matrix Padé Approximation and Convergence Acceleration of Sequences

This paper provides a new method for approximating matrix-valued functions — square Padéapproximation. Some computational methods of the approximants are given. For accelerating matrix sequences, a family of nonlinear extrapolation formulas based on the square Padé approximation is given, a convergence acceleration theorem is proved and numerical examples are presented.     

Convergence and Stability Properties of a Vector Padé Extrapolation Method

In this paper we introduce a new convergence accelerator for vector sequences — vector Padé approximation method (VPA), discuss its convergence and stability properties, and show that it is a bona fide acceleration method for some vector sequences.     

On a New Approach to the Problem of Distribution of Zeros of Hermite—Padé Polynomials for a Nikishin System

A new approach to the problem of the zero distribution of type I Hermite—Padé polynomials for a pair of functions f₁, f₂ forming a Nikishin system is discussed. Unlike the traditional vector approach, we give an answer in terms of a scalar equilibrium problem with harmonic external field which is posed on a two-sheeted Riemann surface.     

Convergence of subdiagonal Padé approximations of C 0-semigroups

Let ${(r_{n})_{n \in \mathbb{N}}}$ be the sequence of subdiagonal Padé approximations of the exponential function. We prove that for ?A the generator of a uniformly bounded C ₀-semigroup T on a Banach space X, the sequence ${(r_{n}(-t A))_{n \in \mathbb{N}}}$ converges strongly to T(t) on D(A ^α ) for ${\alpha>\frac{1}{2}}$ . Local uniform convergence in t and explicit convergence rates in n are established. For specific classes of semigroups, such as bounded analytic or exponentially γ -stable ones, stronger estimates are proved. Finally, applications to the inversion of the vector-valued Laplace transform are given.     

Convergence of simultaneous Padé approximants to two dilogarithms

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Higher order recurrences and row sequences of Hermite–Padé approximation

ABSTRACT

We obtain extensions of the Poincaré and Perron theorems for higher order recurrence relations and apply them to obtain an inverse-type theorem for row sequences of the (type II) Hermite–Padé approximation of a vector of formal power series.     

Multipoint Hermite—Padé approximations for beta functions

We construct multipoint Hermite—Padé approximations for two beta functions generating the Nikishin system with infinite discrete measures and unbounded supports. The asymptotic behavior of the approximants is studied. The result is interpreted in terms of the vector equilibrium problem in logarithmic potential theory in the presence of an external field and constraints on measure.     

Convergence of Padé approximants of partial theta functions and the Rogers-Szegö polynomials

We investigate the convergence of sequences of Padé approximants for the partial theta function $$h_q (z): = \sum\limits_{j = 0}^\infty { q^{j(j - 1)/2_{Z^j } } } , q = e^{i\theta } , \theta \in [0,2\pi ).$$ Whenθ/(2π) is irrational, this function has the unit circle as its natural boundary. We determine subrogions of ¦z¦ < 1 in which sequences of Padé approximants converge uniformly, and subrogions in which they converge in capacity, but not uniformly. In particular, we show that only a proper subsequence of the diagonal sequence {[n/n]} _n=1 ^∞ converges locally uniformly in all of ¦z¦< l; in contrast, no subsequence of any Padé row {[m/n]} _m=1 ^∞ (withn ≥ 2 fixed) can converge locally uniformly in all of ¦z¦ < 1. Further, we obtain the zero and pole distributions of sequences of Padé approximants by analyzing the zero distribution of the Rogers-Szegö polynomials $$G_n (z): = \sum\limits_{j = 0}^n {\left[ {\begin{array}{*{20}c} n \\ j \\ \end{array} } \right]} z^j , n = 0,1,2,....$$      

An Algorithm for the Computation of Hermite–Padé Approximations to the Exponential Function: Divided Differences and Hermite–Padé Forms

We present the first of two different algorithms for the explicit computation of Hermite–Padé forms (HPF) associated with the exponential function. Some roots of the algebraic equation associated with a given HPF are good approximants to the exponential in some subsets of the complex plane: they are called Hermite–Padé approximants (HPA) to this function. Our algorithm is recursive and based upon the expression of HPF as divided differences of the function texp(xt) at multiple integer nodes. Using this algorithm, we find again the results obtained by Borwein and Driver for quadratic HPF. As an example, we give an interesting family of quadratic HPA to the exponential.     

Convergence Acceleration of Vector Sequences by Vector Padé Approximation

By making use of vector Padéapproximants, a method for accelerating the convergence of vector sequences is derived. The method obtained includes the well known Henrici transformation as a special case. A main character of the method is that it can use partial vector components to accelerate convergence of the whole vector sequence. Results about the efficacy of the method are established. An algorithm and some numerical examples are given.     

Direct and Inverse Results on Row Sequences of Hermite–Padé Approximants

We give necessary and sufficient conditions for the convergence with geometric rate of the common denominators of simultaneous rational interpolants with a bounded number of poles. The conditions are expressed in terms of intrinsic properties of the system of functions used to build the approximants. Exact rates of convergence for these denominators and the simultaneous rational approximants are provided.     

Higher-Order Three-Term Recurrences and Asymptotics of Multiple Orthogonal Polynomials

The asymptotic theory is developed for polynomial sequences that are generated by the three-term higher-order recurrence

The convergence of diagonal Padé approximants and the Padé Conjecture

Questions related to the convergence problem of diagonal Padé approximants are discussed. A central place is taken by the Padé Conjecture (also known as the Baker-Gammel-Wills Conjecture). Partial results concerning this conjecture are reviewed and weaker and more special versions of the conjecture are formulated and their plausibility is investigated. Great emphasis is given to the role of spurious poles of the approximants. A conjecture by Nuttall (1970) about the number and distribution of such poles is stated and its importance for the Padé Conjecture is analyzed.