Zeckendorf family identities generalized

This is a detailed version of my text [2]. It contains the proof outlined in [2] in much more detail and was written for the purpose of persuading myself that my proofs are correct. This note has never been proofread by myself or anyone else. If you find any mistakes or typos, please inform me at ∆Γ@gmail.com where ∆ =darij and Γ =grinberg Thank you! Definitions. 1) A subset S of Z is called holey if it satisfies (s + 1 / ∈ S for every s ∈ S). be the Fibonacci sequence (defined by f 1 = f 2 = 1 and the recurrence relation (f n = f n−1 + f n−2 for all n ∈ N satisfying n ≥ 3)). Theorem 1 (generalized Zeckendorf family identities). Let T be a finite set, and a t be an integer for every t ∈ T. Then, there exists one and only one finite holey subset S of Z such that t∈T f n+at = s∈S f n+s for every n ∈ Z which satisfies n > max ({−a t | t ∈ T } ∪ {−s | s ∈ S}). Remarks. 1) Theorem 1 generalizes the so-called Zeckendorf family identities (which correspond to the case when all a t are = 0), which were discussed in [1]. 1 2) The condition n > max ({−a t | t ∈ T } ∪ {−s | s ∈ S}) in Theorem 1 is just a technical condition made in order to ensure that the Fibonacci numbers f n+at for all t ∈ T and f n+s for all s ∈ S are well-defined. (If we would define the Fibonacci numbers f n for integers n ≤ 0 by extending the recurrence relation f n = f n−1 + f n−2 " to the left " , then we could drop this condition.) The following is my proof of Theorem 1. It does not even try to be combinatorial-it is pretty much the opposite. While I won't use any results from analysis, the proof below has a distinctively analytic flavor.