Asymptotics of Type I Hermite–Padé Polynomials for Semiclassical Functions

Abstract

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 conditionwith the degree of all Q n,k ≤ n. We describe an approach for finding the asymptotic zero distribution of these polynomials as n → ∞ 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).

Equivalence of a Scalar and a Vector Equilibrium Problem for a Pair of Functions Forming a Nikishin System

“…In [22] an example of a multivalued analytic function f is given such that the pair f, f 2 forms a Nikishin system (note that in [22] the concept of a Nikishin system is a little bit more general than that given by E. M. Nikishin himself). There exist classes of multivalued analytic functions f such that the pair of functions f , f 2 can be naturally looked upon as a complex Nikishin system (see [15], [10], [22]). In connection with the new approach of [21] to the problem of efficient continuation of a given germ of a multivalued analytic function, this fact seems to be one of the main impetus for the study of equilibrium problems pertaining to complex Nikishin systems.…”

“…For further advances in this vector approach, see [1], [14], [15], and [2]. At the same time, the vector approach faces certain difficulties in the solution of problems involving complex Nikishin systems of the form f, f 2 , where, for example, f is a multivalued function of Laguerre class (see [10]). In [19] (see also [20]) a new approach to the solution of the problem on the limit zeros distribution of the Hermite-Padé polynomials of type I was proposed.…”

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

“…The corresponding result was announced in [50]; the author intends to give the proof of this result in a separate paper. We note the papers [40], [46] and [24], in which the equilibrium problem for a mixed Green-logarithmic potential was employed for the study of the limit distribution of the zeros of Hermite-Padé polynomials for a tuple [1, f 1 , f 2 ], where a pair of functions f 1 , f 2 forms a generalized (complex) Nikishin system (see also [12], [47], [32], [41]). The method of investigation proposed in the present paper is different from that of [40], [46] and [24].…”

Zero distribution for Angelesco Hermite–Padé polynomials

“…The conjecture is supported by the results from [44] related to function f from the class f k ∈ L of the form f k (z) = m k i=1 (z − a i,k ) α i,k with m k i=1 α i,k = 0 (see also [33] and [68]). For function of this class the weighted Hermite-Padé polynomials q n,k (z)f k (z) and the remainder satisfy the same differential equation with polynomial coefficient of order s + 1 (polynomials -coefficient depend on n but their degrees are bounded by constants not depending on n).…”