## 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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### References:

 [1] Duran, A.J., On orthogonal polynomials with respect to a positive definite matrix of measures, Canadian J. of math., 47, 88-112, (1995) · Zbl 0832.42014 [2] Duran, A.J.; Van Assche, W., Orthogonal matrix polynomials and higher order recurrence relations, Linear algebra appl., 219, 261-280, (1995) · Zbl 0827.15027 [3] Duran, A.J.; López-Rodriguez, P., Orthogonal matrix polynomials: zeros and Blumenthal’s theorem, J. approx. theory, 84, 96-118, (1996) · Zbl 0861.42016 [4] Draux, A.; Jokung-Nguena, O., Orthogonal polynomials in a non-commutative algebra. non-normal case, IMACS annals of computing and appl. maths., 9, 237-242, (1991) · Zbl 0841.33004 [5] Jódar, L.; Defez, E.; Ponsoda, E., Orthogonal matrix polynomials with respect to linear matrix moment functionals: theory and applications, J. appr. theory and its applications, 12, 1, 96-115, (1996) · Zbl 0853.42022 [6] Defez, E.; Jódar, L., Chebyshev matrix polynomials and second order matrix differential equations, Utilitas Mathematica, 61, 107-123, (2002) · Zbl 0998.15034 [7] Defez, E.; Hervás, A.; Law, A.; Villanueva-Oller, J.; Villanueva, R.J., Progressive transmission of images: PC-based computations, using orthogonal matrix polynomials, Mathl. comput. modelling, 32, 1125-1140, (2000) · Zbl 0965.68117 [8] Defez, E.; Jódar, L.; Law, A.; Ponsoda, E., Three-term recurrences and matrix orthogonal polynomials, Utilitas Mathematica, 57, 129-146, (2000) · Zbl 0962.05064 [9] James, A.T., Special functions of matrix and single argument in statistics, (), 497-520 [10] Geronimo, J.S., Scattering theory and matrix orthogonal polynomials on the real line, Circuit systems signal process, 1, 3-4, 471-494, (1982) · Zbl 0506.15010 [11] Jódar, L.; Company, R.; Navarro, E., Laguerre matrix polynomials and systems of second, order differential equations, Applied numer. math., 15, 53-63, (1994) · Zbl 0821.34010 [12] Jódar, L.; Company, R.; Ponsoda, E., Orthogonal matrix polynomials and systems of second order differential equations, Diff. equations and dynam. syst., 3, 3, 269-288, (1995) · Zbl 0892.33004 [13] Defez, E.; Jódar, L., Some applications of the Hermite matrix polynomials series expansions, J. comput. appl. math., 99, 105-117, (1998) · Zbl 0929.33006 [14] Jódar, L.; Defez, E.; Ponsoda, E., Matrix quadrature and orthogonal matrix polynomials, Congressus numerantium, 106, 141-153, (1995) · Zbl 0836.33004 [15] Sinap, A.; Van Assche, W., Polynomial interpolation and Gaussian quadrature for matrix valued functions, Linear algebra appl., 207, 71-114, (1994) · Zbl 0810.41028 [16] Defez, E.; Law, A.; Villanueva-Oller, J.; Villanueva, R.J., Matrix cubic splines for progressive 3D imaging, J. math. imag. and vision, 17, 41-53, (2002) · Zbl 1020.68099 [17] Chihara, T.S., An introduction to orthogonal polynomials, (1978), Gordon and Breach New York · Zbl 0389.33008 [18] Dunford, N.; Schwartz, J., () [19] Golub, G.; Van Loan, C.F., Matrix computations, (1995), Johns Hopkins Univ. Press New York [20] Hille, E., Lectures on ordinary differential equations, (1969), Addison-Wesley Baltimore · Zbl 0179.40301 [21] Jódar, L.; Cortes, J.C., On the hypergeometric matrix function, J. comput. appl. math, 99, 205-217, (1998) · Zbl 0933.33004 [22] Jódar, L.; Cortes, J.C., Closed form solution of the hypergeometric matrix differential equation, Mathl. comput. modelling, 32, 1017-1028, (2000) · Zbl 0985.33006 [23] Rainville, E.D., Special functions, (1960), Chelsea New York · Zbl 0050.07401
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