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We are interested here in the application of ∞-generalized Fibonacci sequences (∞-GFS for short), to study some properties of Bernoulli and Genocchi numbers and their related classical numbers and polynomials. That is, properties of this class of sequences, allows us to derive new recursive relations for generating Bernoulli and Genocchi numbers, and their related polynomials.

In this paper we investigate the generalized Pell numbers of order r ≥ 2 through the properties of their related fundamental system of generalized Pell numbers. That is, the generalized Pell number of order r ≥ 2; are expressed as a linear combination of a fundamental system of generalized Pell numbers. The properties of this fundamental system are examined and results can be established for generalized Pell numbers of order r ≥ 2. Some identities and combinatorial results are established. Moreover, the analytic study of the fundamental system of generalized Pell is provided. Furthermore, the generalized Pell Cassini identity type is provided.

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Este artigo apresenta fórmulas explícitas para a solução de recorrências lineares homogêneas de ordem 2 com coeficientes constantes determinada através de funções geradoras ordinárias. Além disso, aplicações do método de resolução em cada caso estudado são exibidos e as relações entre as fórmulas de Binet e as expressões obtidas via funções geradoras são discutidos.