On the equation β(u + 2) = 6β(u + 1) -β(u) and balancing numbers
Résumé
Let A be the class of all functions(balancing functions) β : R → R that satisfy β(u+2) = 6β(u+1)-β(u) for all u ∈ R. This class is a vectoer space and satisfies the Hyers-Ulam stability in real Banach spaces. We show that these functions are strongly related to the sequence of balancing numbers and generalize it. Basic properties of B-functions are established and it is shown that lim u→∞ β(u + 1) β(u) = α, where α = 3 + 2 √ 2, for any B-functions β.
We round off by linking periodic functions to B-functions.
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