Gaussian Fibonacci Numbers

The purpose of this note is to present a natural manner of extension of the Fibonacci numbers into the complex plane. The extension is analogous to the analytic continuation of solutions of differential equations. Although, in general, it does not guarantee permanence of form, in case of the Fibonacci numbers even that requirement is satisfied. The resulting complex Fibonacci numbers are, in fact, Gaussian integers. The applicability of this generalization will be demonstrated by the derivation of two interesting identities for the classical Fibonacci numbers. The notion of monodiffricity was introduced by Rufus P. Isaacs [ 1, 2] in 1941; for references to the more recent literature the reader is directed to two papers by the present author [3,4]. The domain of definition of monodiffric functions is the set of Gaussian integers; a complex-valued function /is said to be monodiffric at z = x + y/\ (1) 4 [f(z + i)-f(z)] = f(z+1)-f(z). i As Isaacs already observed, if / is defined on the set of integers, then the requirement of monodiffricity determines /uniquely at the Gaussian integers of the upper half-plane. We term this extension monodiffric continuation. Kurowsky [5] showed that the functional values of /may be calculated by use of the formula (2) f(x + yi) = £ ( * ) / * A f c f l W, k=o ' where the operator A is defined by the relations A°f(x) = f(x), A*f{x) = f(x + 1)- fix) and A k ffx) = A k ' 1 {A 1 Hx)) for k> 2. When applied to the Fibonacci numbers A k behaves especially nicely; one may easily prove that & k Fn = Fn_k. Therefore, via Eq. (2), one may define the Gaussian Fibonacci numbers, Fn+mi, for/? an integer, m a nonnegative integer by (3) The first few values of Fn+mi y f