On generalized Fibonacci k-sequences and Fibonacci k-numbers

Abstract

In this paper analogs of Fibonacci sequences and Fibonacci numbers as well as Fibonacci functions (the case n = 2) for cases n = 3, 4. … are introduced. It is shown that these analogs are related to each other in a regular manner and that if (Formula presented) for a Fibonacci k-sequence, then (Formula presented) Many identities of types similar to those which hold for the case n = 2 (i.e., the Fibonacci case) are also established, indicating the existence of a larger theory of which the Fibonacci case is an integrated part.