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Jacobi matrix differential equation, polynomial solutions, and their properties. (English) Zbl 1069.33007

One way of generalizing the hypergeometric function \[ F\left(a,b;c;z\right)=\sum_{n\geq 0}{(a)_n(b)_n\over (c)_n }{z^n\over n!} \] consists to replace the scalar parameters by matrix in \({\mathbb C}^{r\times r}\) to obtain \[ F\left(A,B;C;z\right)=\sum_{n\geq 0}{(A)_n(B)_n\left[(C)_n\right]^{-1}\over n!}z^n \] where \(A\), \(B\) and \(C\) are matrices in \({\mathbb C}^{r\times r}\) for which \(C+nI\) is invertible for every \(n\geq 0\) and \((A)_n\) designates the matrix version of the Pochhammer symbol defined by \[ (A)_n=A(A+I)\ldots(A+(n-1)I),\quad n\geq 1,\quad \text{and } \quad (A)_0=I. \] With such condition on \(C\), we have also \((C)_n=\Gamma(C+nI)\Gamma^{-1}(C)\). \(\Gamma^{-1}(C)\) is well defined since the reciprocal scalar Gamma function, \(\Gamma^{-1}(z)={1\over \Gamma(z)}\), is an entire function of the complex variable \(z\).
In this paper, the authors deal with Jacobi matrix polynomials \(P_n^{(A,B)}(x)\), \(n\geq 0\), defined by \[ P_n^{(A,B)}(x)={(-1)^n\over n!}F\left(A+B+(n+1)I,-nI;B+I;{1+x\over 2}\right)\Gamma^{-1}(B+I)\Gamma(B+(n+1)I) \] where \(A\) and \(B\) are two matrices in \({\mathbb C}^{r\times r}\) satisfying the spectral conditions \[ \text{Re} (z)>-1, \forall z\in\sigma(A),\quad\text{and}\quad \text{Re} (z)>-1, \forall z\in\sigma(B), \] \(\sigma(A)\) being the spectrum of \(A\). For the scalar case \(r=1\), taking \(A=a>-1\) and \(B=b>-1\), \(P_n^{(a,b)}\) coincides with the classical Jacobi polynomial. Some well known properties of \(P_n^{(a,b)}\) were extented in this paper to the matrix case. That turns out to be a differential equation, a Rodrigues formula, an orthogonality, and a three terms recurrence relation.

MSC:

33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
15A54 Matrices over function rings in one or more variables
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