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**Some applications of the Hermite matrix polynomials series expansions.**
*(English)*
Zbl 0929.33006

Evaluation of matrix functions using Hermite matrix polynomials was proposed to avoid computational difficulties. The results were applied to construct approximations (with a prefixed accuracy) of problems \(Y'= AY\), \(Y(0)=y_0\) (\(A\) is a matrix, \(y_0\) a vector),
\[
Y''+A^2Y= 0,\;Y(0)=P,\;Y'(0) =Q
\]
\((A\) is a matrix, \(P\) and \(Q\) are vectors), and the Sylvester matrix differential equation. The solution of the problems can be expressed in terms of \(\exp(At)\), \(\cos(At)\), \(\sin(At)\), and \(\exp(Bt)\). Some new properties of Hermite matrix polynomials were established. Hermite matrix polynomial series expansion of \(e^{At}\), \(\sin(At)\), and \(\cos(At)\) of any matrix, also with their finite series truncation, with a prefixed accuracy in a bounded domain was dealt with. Approximations were derived so that the error with respect to the exact solution is uniformly upper bounded.

Reviewer: Váslaw Burjan (Praha)

### MSC:

33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |

34A30 | Linear ordinary differential equations and systems |

41A58 | Series expansions (e.g., Taylor, Lidstone series, but not Fourier series) |

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\textit{E. Defez} and \textit{L. Jódar}, J. Comput. Appl. Math. 99, No. 1--2, 105--117 (1998; Zbl 0929.33006)

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