Some new relations between T(a1,a2,a3,a4,a5;n) and N(a1,a2,a3,a4,a5;n)

Vandna and Mandeep Kaur
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 29, 2023, Number 2, Pages 216–225
DOI: 10.7546/nntdm.2023.29.2.216-225
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Authors and affiliations

Vandna
Department of Mathematics, Lovely Professional University
Phagwara-144411, Punjab, India

Mandeep Kaur
Department of Mathematics, Abhayapuri College
Abhayapuri, Assam-783384, India

Abstract

Let N(a_1,a_2,a_3,a_4,a_5;n) and T(a_1,a_2,a_3,a_4,a_5;n) count the representations of n as a_1x_1^2+a_2x_2^2+a_3x_3^2+a_4x_4^2+a_5x_5^2 and a_1X_1(X_1+1)/2+a_2X_2(X_2+1)/2+a_3X_3(X_3+1)/2+a_4X_4(X_4+1)/2+a_5X_5(X_5+1)/2, respectively, where a_1,a_2,a_3,a_4,a_5 are positive integers, x_1,x_2,x_3,x_4,x_5 are integers and n,X_1,X_2,X_3,X_4,X_5 are nonnegative integers. In this paper, we establish some new relations between N(a_1,a_2,a_3,a_4,a_5;n) and T(a_1,a_2,a_3,a_4,a_5;n). Also, we prove that T(a_1,a_2,a_3,a_4,a_5;n) is a linear combination of N(a_1,a_2,a_3,a_4,a_5;m) and N(a_1,a_2,a_3,a_4,a_5;m/4), where m=8n+a_1+a_2+a_3+a_4+a_5, for various values of a_1,a_2,a_3, a_4,a_5.

Keywords

  • Sum of squares
  • Sum of triangular numbers
  • Theta function identities

2020 Mathematics Subject Classification

  • 11D85
  • 11E25

References

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Manuscript history

  • Received: 18 July 2022
  • Revised: 27 March 2023
  • Accepted: 18 April 2023
  • Online First: 25 April 2023

Copyright information

Ⓒ 2023 by the Authors.
This is an Open Access paper distributed under the terms and conditions of the Creative Commons Attribution 4.0 International License (CC BY 4.0).

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Cite this paper

Vandna, & Kaur, M. (2023). Some new relations between T(a1,a2,a3,a4,a5;n) and N(a1,a2,a3,a4,a5;n). Notes on Number Theory and Discrete Mathematics, 29(2), 216-225, DOI: 10.7546/nntdm.2023.29.2.216-225.

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