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Structural formulas for matrix-valued orthogonal polynomials related to \(2\times 2\) hypergeometric operators. (English) Zbl 1491.42041

This paper deals with matrix-valued orthogonal polynomials, in particular of size \(2 \times 2\). After recalling results of A. J. Durán [Int. Math. Res. Not. 2010, No. 5, 824–855 (2010; Zbl 1203.33006)] and A. J. Durán and F. A. Grünbaum [Int. Math. Res. Not. 2004, No. 10, 461–484 (2004; Zbl 1073.33009)] among other references on the subject matter, the authors consider the weight matrix function \[W^{(\alpha, \beta, \nu)}(t) = t^{\alpha} (1- t)^{\beta}\, \tilde{W}^{(\alpha, \beta, \nu)} (t)\,,\; \textrm{for}\; t \in (0,1),\] with \[\tilde{W}^{(\alpha, \beta, \nu)} (t) = \left(\begin{array}{cc} \frac{\nu\left(\kappa_{\nu,\beta}+2\right)}{\kappa_{\nu,-\beta}}t^2 - \left(\kappa_{\nu,\beta}+2\right)t+(\alpha +1) & (\alpha + \beta +2)t -(\alpha+1) \\ (\alpha + \beta +2)t -(\alpha+1) & -\frac{\nu\left(\kappa_{-\nu,\beta}+2\right)}{\kappa_{-\nu,-\beta}}t^2 - \left(\kappa_{-\nu,\beta}+2\right)t+(\alpha +1) \end{array} \right)\]
where \(\alpha, \beta, \nu \in \mathbb{R}\), \(\alpha, \beta > - 1\) and \(|\alpha - \beta | < |\nu| < \alpha +\beta + 2\) and \(\kappa_{\pm \nu,\pm \beta}= \alpha \pm \nu \pm \beta\).
The sequence of matrix-valued monic orthogonal polynomials associated with the weight function \(W^{(\alpha, \beta, \nu)}(t)\), notated by \(\left( P_{n}^{(\alpha, \beta, \nu)} \right)_{n \geq 0}\), was introduced in [C. Calderón et al., J. Approx. Theory 248, Article ID 105299, 17 p. (2019; Zbl 1426.42022)] and some properties are explored therein, namely the corresponding three-term recurrence relation.
In this paper, the authors provide several other characteristics of \(\left( P_{n}^{(\alpha, \beta, \nu)} \right)_{n \geq 0}\) such that a Rodrigues formula, an explicit expression for these polynomials in terms of Jacobi polynomials, a formula for the norm of monic orthogonal polynomials \(P_{n}^{(\alpha, \beta, \nu)}\,,\, n \geq 0\), and the three-term recurrence relation fulfilled by the normalised sequence.
Furthermore, the sequence of monic polynomials defined by the derivatives of order \(k\) of \(P_{n}^{(\alpha, \beta, \nu)} \), denoted by \(\left( P_{n}^{(\alpha, \beta, \nu, k)} \right)_{n \geq k}\), is written in terms of hypergeometric functions and many other interesting results follow the presentation of the weight matrix \(W^{(k)}\) that defines the orthogonality of the sequence \(\left( P_{n}^{(\alpha, \beta, \nu, k)} \right)_{n \geq k}\).
Besides, we find a Rodrigues formula for \(\left( P_{n}^{(\alpha, \beta, \nu, k)} \right)_{n \geq k}\) and a precise account of the algebra of second-order differential operators associated with the weight matrix \(W^{(\alpha, \beta, \nu)}(t)\).

MSC:

42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
47S10 Operator theory over fields other than \(\mathbb{R}\), \(\mathbb{C}\) or the quaternions; non-Archimedean operator theory
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
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