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General relation between sums of figurate numbers


Citation

Mohamat Johari, Mohamat Aidil (2013) General relation between sums of figurate numbers. Doctoral thesis, Universiti Putra Malaysia.

Abstract

In this study, we seek to find relations between the number of representations of a nonnegative integer n as a sum of figurate numbers of different types. Firstly, we give a relation between the number of representations, ck(n), of n as the sum k cubes and the number of representations, pk(n), of n as the sum of k triangular pyramidal numbers, namely under certain conditions pk(n) = c k odd (v); where c k odd denotes the number of representations as a sum of k odd cubes and the integer v is derived from n. Then we extend this problem by considering sums of s-th powers with s > 3 and the associated polytopic numbers of order s. Next, we discuss the relation between ɸ(2;k)(n), the number of representations of n as a sum of k fourth powers, and ψ(2;k)(n), the number of representations of n as a sum of k terms of the form 8γ2 + 2γ where γ is a triangular number. When 1 ≤ k ≤ 7, the relation is ɸ(2;k)(8n + k) = 2kψ (2;k) (n). We extend this result by considering the relation between the number of represen- tations of n as a sum of k 2m-th powers and the number of representations of n as a sum of k terms determined by an associated polynomial of degree m evaluated at a triangular number. Thirdly, we consider the relation between sk(n), the number of representations of n as a sum of k squares, and ek(n), the number of representations of n as a sum of k centred pentagonal numbers. When 1 ≤k ≤ 7, this relation is αkek(n) = sk (8n -3k)÷5 ; where αk = 2k + 2k-1 (k4) We extend the analysis to the number of representations induced by a partition γ of k into m parts. If corresponding number of representations of n are respectively sγ(n) and eγ(n), then βγeγ(n) = sγ(8n - 3k)÷5 where βγ = 2m + 2(m-1) (( i1/4) + (i1/2)(i2/1)+(i1/1)(i3/1) and ij denotes the number of parts of γ which are equal to j. We end this thesis with a short discussion and proposal of various open problems for further research.


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Additional Metadata

Item Type: Thesis (Doctoral)
Subject: Mathematics
Subject: Numbers, Polygonal
Call Number: IPM 2013 9
Chairman Supervisor: Professor Dato' Kamel Ariffin Bin Mohd Atan, PhD
Divisions: Institute for Mathematical Research
Depositing User: Ms. Nur Faseha Mohd Kadim
Date Deposited: 26 Mar 2019 04:14
Last Modified: 26 Mar 2019 04:14
URI: http://psasir.upm.edu.my/id/eprint/67683
Statistic Details: View Download Statistic

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