Relation between sum of 2mth powers and polynomials of triangular numbers

• Main Authors: Mohamat Johari, Mohamat Aidil, Mohd Atan, Kamel Ariffin, Sapar, Siti Hasana
• Format: Article
• Language: English
• Published: Pushpa Publishing House 2014
• Online Access: http://psasir.upm.edu.my/id/eprint/35199/1/Relation%20between%20sum%20of%202mth%20powers%20and%20polynomials%20of%20triangular%20numbers.pdf; http://psasir.upm.edu.my/id/eprint/35199/; http://www.pphmj.com/abstract/8678.htm
• Institution: Universiti Putra Malaysia

Let Ф (m, k)(n) denote the number of representations of an integer n as a sum of k 2mth powers and Ψ (m, k)(n) denote the number of representations of an integer n as a sum of k polynomial Pm(γ), where γ is a triangular number. We show that Ф (2, k)(8n + k) = 2k Ψ(2,k) (n) for 1 ≤ k ≤ 7. A general relation between the number of representations (formula presented) and the sum of its associated polynomial of triangular numbers for any degree m ≥ 2 is given as Ф(m, k) (8n + k) = 2k Ψ (m, k) (n).