Relation between sum of 2mth powers and polynomials of triangular numbers
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 r...
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| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
Pushpa Publishing House
2014
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| 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 |
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| Institution: | Universiti Putra Malaysia |
| Language: | English |
| Summary: | 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). |
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