Three-fold symmetric Hahn-classical multiple orthogonal polynomials

Abstract

We characterize all the multiple orthogonal threefold symmetric polynomial sequences whose sequence of derivatives is also multiple orthogonal. Such a property is commonly called the Hahn property and it is an extension of the concept of classical polynomials to the context of multiple orthogonality. The emphasis is on the polynomials whose indices lie on the step line, also known as 2-orthogonal polynomials. We explain the relation of the asymptotic behavior of the recurrence coefficients to that of the largest zero (in absolute value) of the polynomial set. We provide a full characterization of the Hahn-classical orthogonality measures supported on a 3-star in the complex plane containing all the zeros of the polynomials. There are essentially three distinct families, one of them 2-orthogonal with respect to two confluent functions of the second kind. This paper complements earlier research of Douak and Maroni.

Multiple orthogonal polynomials associated with confluent hypergeometric functions

“…In Section 4 we explain that particular cases of the type II polynomials on the step line, characterised here, have appeared in [20] as the components of 3-fold symmetric Hahn-classical 2-orthogonal polynomials on star-like sets. A polynomial sequence {P n (x)} n∈N is said to be 3-fold symmetric if, for any n ∈ N, This property is commonly referred to as 2-symmetry, as introduced in [7, Definition 5.1].…”

“…A polynomial sequence {P n (x)} n∈N is said to be 3-fold symmetric if, for any n ∈ N, This property is commonly referred to as 2-symmetry, as introduced in [7, Definition 5.1]. We opt to follow the terminology in [20], as it gives a better picture of the intrinsic symmetry. The definition is equivalent to say that there exist three polynomial sequences P…”

On a new class of 2-orthogonal polynomials, I: the recurrence relations and some properties

“…In particular, for d = 1, the obtained results correspond to the two (2 1 ) families of symmetric classical OPS (Hermite and Gegenbauer), and for d = 2, there are exactly four (2 2 ) families of 2-symmetric classical 2-OPS. For more details we refer the reader to [11,18].…”

“…Under these specific conditions one gets a simple system that was integrally solved by the authors in [11] obtaining the four families of 2-symmetric 2-OPS. Recall that these polynomials are recently reviewed in [18]. When α n = 0, n 1, and β n = 0, n 0, the resulting polynomials belong to the Appell's class [11,14].…”

“…Besides, the polynomials Pi n , n 0, satisfy the differential-recurrence formula Mention that these polynomials are the first ones investigated in [18] where the authors gave most notably the integral representations of their corresponding linear functionals via the Airy function Ai(x) and its derivative. Obviously, we will take account of these representations in [23].…”