Congruences for (2, 5)-regular Bipartitions into Distinct Parts

M. Prasad1, S. R. Nayaka1 and K. V. Prasad2


1Department of Mathematics, PES College of Engineering, Mandya, Karnataka, India.
2Department of Mathematics, VSK University, Ballary, Karnataka, India.

Abstract: Let $B_{2, 5}(n)$ denote the number of $(2, 5)$-regular bipartitions of a positive integer $n$ into distinct parts . In this paper, we establish several infinite families of congruences modulo powers of $2$ for $B_{2, 5}(n)$. For example, \begin{equation*} \sum_{n=0}^{\infty}B_{2, 5}\left(2^{2\alpha+1}\cdot5^{2\beta}n+\frac{2^{2\alpha+2}\cdot5^{2\beta}-1}{3}\right)q^n \equiv 2f_1f_{5}^3 \pmod{2^2}, \end{equation*} for $\alpha, \beta \geq 0$.
Keywords: Partition identities, Theta--functions, Partition congruences, Regular partition.


Cite this article as: M. Prasad, S. R. Nayaka and K. V. Prasad, Congruences for (2, 5)-regular Bipartitions into Distinct Parts, Int. J. Math. And Appl., vol. 9, no. 2, 2021, pp. 17-25.

References
  1. B. C. Berndt, Ramanujan's Notebooks, Part III, Springer-Verlag, New York, (1991).
  2. N. Calkin, N. Drake, K. James, S. Law, P. Lee, D. Penniston and J. Radder, Divisibility properties of the $5$-regular and $13$-regular partition functions, Integers, 8(2008), \#A60.
  3. S. P. Cui and N. S. S. Gu, Arithmetic properties of $\ell$-regular partitions, Adv. Appl. Math., 51(2013), 507-523.
  4. M. D. Hirschhorn, Ramanujan's "most beautiful identity", Amer. Math. Monthly, 118(2011), 839-845.
  5. M. D. Hirschhorn and J. A. Sellers, Elementary proofs of parity results for $5$-regular partitions, Bull. Aust. Math. Soc., 81(2010), 58-63.
  6. M. S. Mahadeva Naika and B. Hemanthkumar, Arithmetic propertities of $5$-regular bipartitions, Int. J. Number Theory, 13(2)(2017), 939-956.
  7. M. S. Mahadeva Naika, B. Hemanthkumar and H. S. Sumanth Bharadwaj, Congruences modulo $2$ for certain partition functions, Bull. Aust. Math. Soc., 93(3)(2016), 400-409.
  8. M. S. Mahadeva Naika, B. Hemanthkumar and H. S. Sumanth Bharadwaj, Color partition identities arising from Ramanujan's theta functions, Acta Math. Vietnam., 44(4)(2016), 633-660.
  9. M. Prasad and K. V. Prasad, On $(\ell, m)$-regular partitions with distinct parts, Ramanujan J., 46(1)(2018), 19-27.
  10. M. Prasad and K. V. Prasad, On $5$-regular bipartitions into distinct parts, Integers, 20(A94)(2020), 1-11.
  11. S. Ramanujan, Collected papers, Cambridge university press, 1927; reprinted by Chelsea, New York, 1962; reprinted by the American mathematical society, RI, (2000).
  12. G. N. Watson, Theorems stated by Ramanujan (VII): Theorems on continued fractions, J. London Math. Soc., 4(1929), 39-48.

Back