A Combinatoric Proof and Generalization of Ferguson’s Formula for k-generalized Fibonacci Numbers

Various generalizations of the Fibonacci numbers have been proposed, studied and applied over the years (see [5] for a brief list). Probably the best known are the k-generalized Fibonacci numbers F (k) n (also known as the k-fold Fibonacci, k-th order Fibonacci, k-Fibonacci or polynacci numbers), satisfying F (k) n = F (k) n−1 + F (k) n−2 + . . .+ F (k) n−k , n ≥ k , F (k) n = 0 , 0 ≤ n ≤ k − 2 , F (k) k−1 = 1 . An exhaustive bibliography of papers on the k-generalized Fibonacci numbers would cover pages, so we just give a few references. The paper of Miles [9] seems to be the oldest wellknown paper on the subject, though Knuth [6] (section 5.4.2) cites a work of Schlegel [13] dating from 1894. Numerous interesting results can be found in the pages of the Fibonacci Quarterly, see for example [2, 3, 7]. There are significant applications in computer science [6] and probability theory [11, 4, 8], the latter of which will be important for our purposes. Also much is known about “weighted” k-generalized Fibonacci numbers, with different coefficients in the recursion relation, see for example [1].