Intrinsic Characterization of Representation Functions and Generalizations

Charles Helou

doi:10.1007/978-3-030-67996-5_15

Intrinsic Characterization of Representation Functions and Generalizations

Charles Helou

Penn State Brandywine

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

Abstract

Given a set A of natural numbers, i.e., nonnegative integers, we establish an intrinsic characterization of the representation function of A, which to every natural number n associates the number r_A(n) of ordered pairs (a, b) of elements a, b∈ A such that a+ b= n, thus answering an open problem stated many years ago by Mel Nathanson. We also establish similar characterizations of the characteristic function χ_A(n), which is equal to 1 or 0 according as the natural number n lies or does not lie in A, and the counting function A(n), which gives the number of elements a of A satisfying a≤ n. We then generalize to representation functions as sums of more than two elements of A.

Original language	English (US)
Title of host publication	Combinatorial and Additive Number Theory IV, CANT 2019 and 2020
Editors	Melvyn B. Nathanson
Publisher	Springer
Pages	291-305
Number of pages	15
ISBN (Print)	9783030679958
DOIs	https://doi.org/10.1007/978-3-030-67996-5_15
State	Published - 2021
Event	Workshops on Combinatorial and Additive Number Theory, CANT 2019 and 2020 - Virtual, Online Duration: Jun 1 2020 → Jun 5 2020

Publication series

Name	Springer Proceedings in Mathematics and Statistics
Volume	347
ISSN (Print)	2194-1009
ISSN (Electronic)	2194-1017

Conference

Conference	Workshops on Combinatorial and Additive Number Theory, CANT 2019 and 2020
City	Virtual, Online
Period	6/1/20 → 6/5/20

All Science Journal Classification (ASJC) codes

Mathematics(all)

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10.1007/978-3-030-67996-5_15

Cite this

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title = "Intrinsic Characterization of Representation Functions and Generalizations",

abstract = "Given a set A of natural numbers, i.e., nonnegative integers, we establish an intrinsic characterization of the representation function of A, which to every natural number n associates the number rA(n) of ordered pairs (a, b) of elements a, b∈ A such that a+ b= n, thus answering an open problem stated many years ago by Mel Nathanson. We also establish similar characterizations of the characteristic function χA(n), which is equal to 1 or 0 according as the natural number n lies or does not lie in A, and the counting function A(n), which gives the number of elements a of A satisfying a≤ n. We then generalize to representation functions as sums of more than two elements of A.",

author = "Charles Helou",

note = "Publisher Copyright: {\textcopyright} 2021, The Author(s), under exclusive license to Springer Nature Switzerland AG.; Workshops on Combinatorial and Additive Number Theory, CANT 2019 and 2020 ; Conference date: 01-06-2020 Through 05-06-2020",

year = "2021",

doi = "10.1007/978-3-030-67996-5_15",

language = "English (US)",

isbn = "9783030679958",

series = "Springer Proceedings in Mathematics and Statistics",

publisher = "Springer",

pages = "291--305",

editor = "Nathanson, {Melvyn B.}",

booktitle = "Combinatorial and Additive Number Theory IV, CANT 2019 and 2020",

}

Helou, C 2021, Intrinsic Characterization of Representation Functions and Generalizations. in MB Nathanson (ed.), Combinatorial and Additive Number Theory IV, CANT 2019 and 2020. Springer Proceedings in Mathematics and Statistics, vol. 347, Springer, pp. 291-305, Workshops on Combinatorial and Additive Number Theory, CANT 2019 and 2020, Virtual, Online, 6/1/20. https://doi.org/10.1007/978-3-030-67996-5_15

TY - GEN

T1 - Intrinsic Characterization of Representation Functions and Generalizations

AU - Helou, Charles

PY - 2021

Y1 - 2021

N2 - Given a set A of natural numbers, i.e., nonnegative integers, we establish an intrinsic characterization of the representation function of A, which to every natural number n associates the number rA(n) of ordered pairs (a, b) of elements a, b∈ A such that a+ b= n, thus answering an open problem stated many years ago by Mel Nathanson. We also establish similar characterizations of the characteristic function χA(n), which is equal to 1 or 0 according as the natural number n lies or does not lie in A, and the counting function A(n), which gives the number of elements a of A satisfying a≤ n. We then generalize to representation functions as sums of more than two elements of A.

AB - Given a set A of natural numbers, i.e., nonnegative integers, we establish an intrinsic characterization of the representation function of A, which to every natural number n associates the number rA(n) of ordered pairs (a, b) of elements a, b∈ A such that a+ b= n, thus answering an open problem stated many years ago by Mel Nathanson. We also establish similar characterizations of the characteristic function χA(n), which is equal to 1 or 0 according as the natural number n lies or does not lie in A, and the counting function A(n), which gives the number of elements a of A satisfying a≤ n. We then generalize to representation functions as sums of more than two elements of A.

UR - http://www.scopus.com/inward/record.url?scp=85115184322&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85115184322&partnerID=8YFLogxK

U2 - 10.1007/978-3-030-67996-5_15

DO - 10.1007/978-3-030-67996-5_15

M3 - Conference contribution

AN - SCOPUS:85115184322

SN - 9783030679958

T3 - Springer Proceedings in Mathematics and Statistics

SP - 291

EP - 305

BT - Combinatorial and Additive Number Theory IV, CANT 2019 and 2020

A2 - Nathanson, Melvyn B.

PB - Springer

T2 - Workshops on Combinatorial and Additive Number Theory, CANT 2019 and 2020

Y2 - 1 June 2020 through 5 June 2020

ER -

Intrinsic Characterization of Representation Functions and Generalizations

Abstract

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All Science Journal Classification (ASJC) codes

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