Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

THE NUMBER OF REPRESENTATIONS OF A POSITIVE INTEGER BY TRIANGULAR, SQUARE AND DECAGONAL NUMBERS

  • Isnaini, Uha (Mathematics & Mathematics Education National Institute of Education Nanyang Technological University) ;
  • Melham, Ray (School of Mathematical and Physical Sciences University of Technology) ;
  • Toh, Pee Choon (Mathematics & Mathematics Education National Institute of Education Nanyang Technological University)
  • Received : 2018.09.27
  • Accepted : 2019.02.07
  • Published : 2019.09.30
Download PDF
⟨ Previous Next ⟩

Abstract

Let $T_aD_b(n)$ and $T_aD^{\prime}_b(n)$ denote respectively the number of representations of a positive integer n by $a(x^2-x)/2+b(4y^2-3y)$ and $a(x^2-x)/2+b(4y^2-y)$. Similarly, let $S_aD_b(n)$ and $S_aD^{\prime}_b(n)$ denote respectively the number of representations of n by $ax^2+b(4y^2-3y)$ and $ax^2+b(4y^2-y)$. In this paper, we prove 162 formulas for these functions.

Keywords

Acknowledgement

Supported by : National Institute of Education (Singapore)

References

  1. N. D. Baruah and B. K. Sarmah, The number of representations of a number as sums of various polygonal numbers, Integers 12 (2012), Paper No. A54, 16 pp.
  2. B. C. Berndt, Ramanujan's Notebooks. Part III, Springer-Verlag, New York, 1991. https://doi.org/10.1007/978-1-4612-0965-2
  3. B. C. Berndt, Number Theory in the Spirit of Ramanujan, Student Mathematical Library, 34, American Mathematical Society, Providence, RI, 2006. https://doi.org/10.1090/stml/034
  4. H. H. Chan and P. C. Toh, Theta series associated with certain positive definite binary quadratic forms, Acta Arith. 169 (2015), no. 4, 331-356. https://doi.org/10.4064/aa169-4-3
  5. M. D. Hirschhorn, The number of representations of a number by various forms, Discrete Math. 298 (2005), no. 1-3, 205-211. https://doi.org/10.1016/j.disc.2004.08.045
  6. M. D. Hirschhorn, The number of representations of a number by various forms involving triangles, squares, pentagons and octagons, in Ramanujan rediscovered, 113-124, Ramanujan Math. Soc. Lect. Notes Ser., 14, Ramanujan Math. Soc., Mysore, 2010.
  7. M. D. Hirschhorn, The power of q, Developments in Mathematics, 49, Springer, Cham, 2017. https://doi.org/10.1007/978-3-319-57762-3
  8. G. Humby, A proof of Melham's identities, Unpublished Masters' Thesis, University of Exeter, 2017.
  9. C. G. J. Jacobi, Fundamenta Nova Theoriae Functionum Ellipticarum, Sumptibus fratrum Borntrager, 1829.
  10. R. S. Melham, Analogues of Jacobi's two-square theorem, Research Report R07-01, Department of Mathematical Sciences, University of Technology, Sydney.
  11. R. S. Melham, Analogues of Jacobi's two-square theorem: an informal account, Integers 10 (2010), A8, 83-100. https://doi.org/10.1515/INTEG.2010.008
  12. B. K. Sarmah, Proof of some conjectures of Melham using Ramanujan's $1{\psi}1$ formula, Int. J. Math. Math. Sci. 2014 (2014), Art. ID 738948, 6 pp. https://doi.org/10.1155/2014/738948
  13. C. L. Siegel, Advanced Analytic Number Theory, second edition, Tata Institute of Fundamental Research Studies in Mathematics, 9, Tata Institute of Fundamental Research, Bombay, 1980.
  14. Z.-H. Sun, The expansion of ${\Pi}^{\infty}_{k=1}(1-q^{ak})(1-q^{bk})$, Acta Arith. 134 (2008), no. 1, 11-29. https://doi.org/10.4064/aa134-1-2
  15. Z.-H. Sun, On the number of representations of n by ax(x − 1)/2 + by(y − 1)/2, J. Number Theory 129 (2009), no. 5, 971-989. https://doi.org/10.1016/j.jnt.2008.11.007
  16. Z.-H. Sun, Binary quadratic forms and sums of triangular numbers, Acta Arith. 146 (2011), no. 3, 257-297. https://doi.org/10.4064/aa146-3-5
  17. Z.-H. Sun, On the number of representations of n by $ax^2+by(y-1)/2$, $ax^2+by(3y-1)/2$ and ax(x − 1)/2 + by(3y − 1)/2, Acta Arith. 147 (2011), no. 1, 81-100. https://doi.org/10.4064/aa147-1-5
  18. Z.-H. Sun and K. S. Williams, On the number of representations of n by $ax^2+bxy+cy^2$, Acta Arith. 122 (2006), no. 2, 101-171. https://doi.org/10.4064/aa122-2-1
  19. P. C. Toh, Representations of certain binary quadratic forms as Lambert series, Acta Arith. 143 (2010), no. 3, 227-237. https://doi.org/10.4064/aa143-3-3
  20. P. C. Toh, On representations by figurate numbers: a uniform approach to the conjectures of Melham, Int. J. Number Theory 9 (2013), no. 4, 1055-1071. https://doi.org/10.1142/S1793042113500127