Multivariate orthogonal polynomials on non uniform lattices

Résumé

In this PhD Thesis we investigate multivariate orthogonal polynomials on non uniform lattices. We have made a review of univariate classical orthogonal polynomials as well as their main properties. Then, we have introduced the notion of bivariate and multivariate orthogonal polynomials as well as a class of bivariate orthogonal polynomials satisfying some properties as those of the classical univariate orthogonal polynomials. In particular, We have analyzed properties related with moments, partial divided-difference equations as well as the bivariate symmetric case on a linear lattice. We have investigated some new properties and families of bivariate orthogonal polynomials and their application in the field of quasi- geostrophic shallow water. My contribution to the material in this thesis is contained in the following original papers: P1) I. Area, M. Foupouagnigni, E. Godoy, Y. Guemo Tefo. On moments of hypergeometric bivariate weight functions. Bulletin des Sciences Math ́ematiques 141 (2017) 766-784. P2) D. D. Tcheutia, Y. Guemo Tefo, M. Foupouagnigni, E. Godoy, I. Area. Linear Partial Divided- Difference Equation Satisfied by Multivariate Orthogonal Polynomials on Quadratic Lattices. Math. Model. Nat. Phenom., 12(3 (2017) 22–50. P3) D. D. Tcheutia, M. Foupouagnigni, Y. Guemo Tefo, I. Area. Divided-difference equation and three-term recurrence relations of some systems of bivariate q-orthogonal polynomials. Journal of Difference Equations and Applications 23(12) (2017) 2004–2036. P4) Y. Guemo Tefo, I. Area, M. Foupouagnigni. Bivariate Symmetric Discrete Orthogonal Polyno- mials. In “Advances in Real and Complex Analysis with Applications” (Michael Ruzhansky, Yeol Je Cho, Praveen Agarwal, Iv ́an Area, Editors) 87–106. ISSN: 978-981-10-4336-9. P5) I. Area, Y. Guemo Tefo. Monic bivariate polynomials on quadratic and q-quadratic lattices. Mediterranean Journal of Mathematics (accepted). P6) Y. Guemo Tefo, R. Akta ̧s, I. Area, E. Gu ̈ldo ̆gan-Lekesiz. On a symmetric generalization of bivariate Sturn-Liouville problems. Bulletin of the Iranian Mathematical Society (2021). https://doi.org/10.1007/s41980-021-00605-8. done in collaboration with my advisors (Professor Iv ́an Area and Professor Mama Foupouagnigni) and co-authored with Drs. R. Akta ̧s, E. Godoy, E. Gu ̈ldo ̆gan-Lekesiz, and D. D. Tcheutia. This thesis discusses some recent knowledge and investigation on orthogonal polynomials. The following are the main topics: (1) Divided-difference equations. (2) Moments. (3) Symmetric orthogonal polynomials. (4) Monic solutions. (5) Application to a problem related with water.