Nikishin systems on star-like sets: Ratio asymptotics of the associated multiple orthogonal polynomials

Abstract

In this paper we continue the investigations initiated in [14] on ratio asymptotics of multiple orthogonal polynomials and functions of the second kind associated with Nikishin systems on star-like sets. We describe in detail the limiting functions found in [14], expressing them in terms of certain conformal mappings defined on a compact Riemann surface of genus zero. We also express the limiting values of the recurrence coefficients, which are shown to be strictly positive, in terms of certain values of the conformal mappings. As a consequence, the limits depend exclusively on the geometry of the problem; that is, on the location of the interval supports of the orthogonality measures generating the Nikishin system.

Structure of the Nuttall partition for some class of four-sheeted Riemann surfaces

“…However, for an arbitrary multivalued function (in particular, for an algebraic functions of order 4), the question of asymptotic properties of the Hermite-Padé polynomials for the family [1, f, f 2 ] remains open in the general case. 6 The corresponding results in this direction have been obtained to date only in very special cases; see, for example, [21], [15], [24]). One of the main reasons is as follows.…”

A multi-orthogonal polynomials' approach to bulk queueing theory

“…This kind of families of polynomials {K n } n∈Z+ has been studied in several publications such as [3,4,21,22,23]. We also introduce the vectors of polynomials p r = (p 1,r , .…”

“…From [3,Theorem 8] we can deduce that this system conforms a Nikishin system of measures (see [25]). The Nikishin systems on star-like sets are well studied in several publications such as [21,22,23]. Given a complex measure µ, its Cauchy transform is the function…”

Viskovatov algorithm for Hermite-Padé polynomials

“…Remark 5. Hermite-Padé polynomials are known to satisfy many different recurrence relations, which are widely useful in theoretical studies of Hermite-Padé polynomials (see, for example, [12], [13], [3] and the references given therein). However, as a rule, such results are obtained for a system of Laurent series defined at the point at infinity and are, in general, more involved than the three-term relations (3.12).…”