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.

We show that the alternative discrete Painlevé I equation (alt-dP I ) has a unique solution which remains positive for all n ≥ 0. Furthermore, we identify this positive solution in terms of a special solution of the second Painlevé equation (P II ) involving the Airy function Ai(t). The special-function solutions of P II involving only the Airy function Ai(t) therefore have the property that they remain positive for all n ≥ 0 and all t ≥ 0, which is a new characterization of these special solutions of P II and alt-dP I .

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A new set of multiple orthogonal polynomials of both type I and type II with respect to two weight functions involving Gauss' hypergeometric function on the interval (0,1) is studied. This type of polynomials has direct applications in the investigation of singular values of products of Ginibre random matrices and are connected with branched continued fractions and total-positivity problems in combinatorics. The pair of orthogonality measures is shown to be a Nikishin system and to satisfy a matrix Pearson-type differential equation. The focus is on the polynomials whose indices lie on the step-line, for which it is shown that the differentiation gives a shift in the parameters, therefore satisfying Hahn's property. We obtain Rodrigues-type formulas for type I polynomials and functions, while a more detailed characterization is given for the type II polynomials (aka 2-orthogonal polynomials) that include an explicit expression as a terminating hypergeometric series, a third-order differential equation, and a third-order recurrence relation. The asymptotic behavior of their recurrence coefficients mimics those of Jacobi-Piñeiro polynomials, based on which their asymptotic zero distribution and a Mehler-Heine asymptotic formula near the origin are given. Particular choices of the parameters and confluence relations give some known

The classical polynomials (Hermite, Laguerre, Bessel and Jacobi) are the only orthogonal polynomial sequences (OPS) whose elements are eigenfunctions of the Bochner secondorder differential operator F (Bochner, 1929 [3]). In Loureiro, Maroni and da Rocha (2006) [18] these polynomials were described as eigenfunctions of an even order differential operator F k with polynomial coefficients defined by a recursive relation. Here, an explicit expression of F k for any positive integer k is given. The main aim of this work is to explicitly establish sums relating any power of F with F k , k 1, in other words, to bring a pair of inverse relations between these two operators. This goal is accomplished with the introduction of a new sequence of numbers: the so-called A-modified Stirling numbers, which could be also called as Bessel or Jacobi-Stirling numbers, depending on the context and the values of the complex parameter A.