General relation between sums of figurate numbers

• Main Author: Mohamat Johari, Mohamat Aidil
• Format: Thesis
• Language: English
• Published: 2013
• Online Access: http://psasir.upm.edu.my/id/eprint/67683/1/IPM%202013%209%20IR.pdf; http://psasir.upm.edu.my/id/eprint/67683/
• Institution: Universiti Putra Malaysia

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.