Abstract
AbstractThis manuscript contains a small portion of the algebraic theory of orthogonal polynomials developed by Maroni and their applicability to the study and characterization of the classical families, namely Hermite, Laguerre, Jacobi, and Bessel polynomials. It is presented a cyclical proof of some of the most relevant characterizations, particularly those due to Al-Salam and Chihara, Bochner, Hahn, Maroni, and McCarthy. Two apparently new characterizations are also added. Moreover, it is proved through an equivalence relation that, up to constant factors and affine changes of variables, the four families of polynomials named above are the only families of classical orthogonal polynomials.
Funder
Centro de Matemática, Universidade de Coimbra
Publisher
Springer Science and Business Media LLC
Subject
Applied Mathematics,Mathematics (miscellaneous)
Reference23 articles.
1. Al-Salam, W.A.: Characterization theorems for orthogonal polynomials. In: Orthogonal Polynomials (Columbus, OH, 1989), 1–24, NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 294. Kluwer Academic Publishers, Dordrecht (1990)
2. Andrews, G., Askey, R., Roy, R.: Special Functions. Cambridge University Press (1999)
3. Bourbaki, N.: Algebra I: Chapters 1–3. Translated from the French. Elements of Mathematics (Berlin). Springer, Berlin. Reprint of the 1989 English translation edition (1998)
4. Bourbaki, N.: Topological vector spaces. Chapters 1–5. Translated from the French by H. G. Eggleston and S. Madan. Elements of Mathematics (Berlin). Springer (1987)
5. Castillo, K., Mbouna, D., Petronilho, J.: On the functional equation for classical orthogonal polynomials on lattices. J. Math. Anal. Appl. 515, 126390 (2022)