On a class of polynomials connected to Bell polynomials

In this paper, we study a class of sequences of polynomials linked to the sequence of Bell polynomials. Some sequences of this class have applications on the theory of hyperbolic differential equations and other sequences generalize Laguerre polynomials and associated Lah polynomials. We discuss, for these polynomials, their explicit expressions, relations to the successive derivatives of a given function, real zeros and recurrence relations. Some known results are significantly simplified.Comment: 13 page

Derivadas formales con respecto a gramáticas generativas

Los conceptos de función formal y derivada formal a partir de gramáticas independientes del contexto, presentados por William Chen en 1993, son los fundamentos de un cálculo gramatical, en el cual tienen sentido ciertas operaciones básicas. Desde su concepción, este cálculo ha sido empleado por diversos autores, principalmente Shi-Mei Ma y Dominique Dumont, para la representación de series de potencias, permitiendo generar familias de números especiales y obtener identidades para ciertas familias de polinomios. Recientemente, se ha estudiado la conexión entre gramáticas independientes del contexto y análisis combinatorio, dando como resultado un amplio campo de investigación en el cual se enmarca este trabajo. En particular, se estudia la construcción de gramáticas que generen familias de polinomios y números, con propiedades especiales, con el objetivo de estudiar las propiedades de dichos objetos combinatorios mediante técnicas gramáticales. Adicionalmente se propone una generalización de este cálculo gramatical al considerar gramáticas matriciales en lugar de gramáticas independientes del contexto.Abstract: The concepts of formal functions and formal derivative based on context-free grammars, introduced by William Chen in 1993, are the foundations for a grammatical calculus in which certain basic operations make sense. This calculus has been used by several authors, including Shi-Mei Ma and Dominique Dumont, for the representation of formal power series. Thus, allowing the generation of families of special numbers and proving identities for some families of polynomials. Recently, the connection between context-free grammars and combinatorial analysis has been giving rise to a broad research field on which the present document is framed. In particular, we approach the problem of constructing grammars that generate families of polynomials or numbers, having special given properties, so that we can obtain properties for those combinatorial objects by grammatical techniques. In addition, a generalization of this grammatical calculus is proposed herein by considering matrix grammars instead of context-free grammars.Doctorad