5. Conclusions

Laguerre, Hermite, Legendre, Tchebycheff and Gegenbauer polynomials are obtained in a simple and straightforward way using basic linear algebra concepts, such as the eigenvalue and the eigenvector of a matrix. Once the matrix of the corresponding differential operator is obtained, the eigenvalues of this matrix are found, and the elements of its eigenvectors correspond to the coefficients of each kind of polynomials. Using a larger matrix, higher order polynomials may be found; however, the general case for an nxn matrix was not obtained since it seems that in this general case, standard methods would be easier to use. The main advantage of this method lies in its easiness, since it relies on simple linear algebra concepts. This method contrasts in simplicity with standard methods based on solving the differential equation using power series, using the generating function, using the Rodrigues formula, or using a contour integral.

### Acknowledgements

V. Aboites acknowledges support and useful conversations with Prof. Ernst Wintner from TU-Wien, Dr. Matei Tene from TU-Delft and Dr. Klaus Huber from Berlin. The authors acknowledge the professional English proof reading service provided by Mr. Mario Ruiz Berganza.

### Author details

Vicente Aboites\* and Miguel Ramírez Centro de Investigaciones en Óptica, León, México

\*Address all correspondence to: aboites@cio.mx

© 2019 The Author(s). Licensee IntechOpen. This chapteris distributed underthe terms oftheCreative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
