## Orthogonal matrix polynomials and systems of second order differential equations.(English)Zbl 0892.33004

The system of second order differential equations of the form $(1-t^2) X''(t)+ tDX'(t)+ CX(t)=0,\;| t|<1$ was considered. An explicit analytic expression for the introduced orthogonal matrix polynomials as the solution of the above system was constructed without increasing the problem dimension. These solutions are of the $$X(t)= X_1(t)c_1+ X_2(t) c_2$$, $$c_i\in R^m$$, $$i=1,2$$. To establish a connection with the differential equations through the Frobenius method the scalar theory of Legendre and Gegenbauer polynomials was generalized. Expressions for such matrix polynomials were given using an appropriate generating matrix function. Orthogonality properties of the Gegenbauer matrix polynomials and an upper bound of them were discussed.
Reviewer: V.Burjan (Praha)

### MSC:

 33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) 33C55 Spherical harmonics 15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory 34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.