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Characteristic, Counting, and Representation Functions Characterized

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Combinatorial and Additive Number Theory II (CANT 2015, CANT 2016)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 220))

Abstract

Given a set A of natural numbers, i.e., nonnegative integers, there are three distinctive functions attached to it, each of which completely determines A. These are the characteristic function \(\chi _{A}(n)\) which is equal to 1 or 0 according as the natural number n lies or does not lie in A, the counting function A(n) which gives the number of elements a of A satisfying \(a\le n\), and the representation function \(r_{A}(n)\) which counts the ordered pairs (a, b) of elements \(a, b\in A\) such that \(a+b=n\). We establish direct relations between these three functions. In particular, we express each one of them in terms of each other one. We also characterize the representation functions by an intrinsic recursive relation which is a necessary and sufficient condition.

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Authors and Affiliations

  1. Penn State Brandywine, 25 Yearsley Mill Rd, Media, PA, 19063, USA

    Charles Helou

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  1. Charles Helou
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Correspondence to Charles Helou .

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Editors and Affiliations

  1. Department of Mathematics, Lehman College (CUNY), Bronx, New York, USA

    Melvyn B. Nathanson

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Helou, C. (2017). Characteristic, Counting, and Representation Functions Characterized. In: Nathanson, M. (eds) Combinatorial and Additive Number Theory II. CANT CANT 2015 2016. Springer Proceedings in Mathematics & Statistics, vol 220. Springer, Cham. https://doi.org/10.1007/978-3-319-68032-3_9

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