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.

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