Article
Kyungpook Mathematical Journal 2021; 61(1): 49-59
Published online March 31, 2021
Copyright © Kyungpook Mathematical Journal.
Some Congruences for Andrews' Partition Function E O ¯ ( n )
Utpal Pore and Syeda Noor Fathima*
Department of Mathematics, Ramanujan School of Mathematical Sciences, Pondicherry University, Puducherry - 605 014, India
e-mail : utpal.mathju@gmail.com and dr.fathima.sn@gmail.com
Received: March 14, 2019; Revised: October 7, 2020; Accepted: October 8, 2020
Abstract
Recently, Andrews introduced partition functions
Keywords: partitions, partitions with even parts below odd parts, congruences.
1. Introduction
A partition of a positive integer
where throughout this paper, for any complex numbers
Almost a century back Ramanujan established the following identity [7],
which in fact implies Ramanujan's congruences for
Recently, Andrews [2] introduced the partition function
Andrews [2], also defined the partition function
In Section 3 of this paper, we prove some congruences modulo 2 and 4 for the partition function
and we prove some interesting congruences modulo 10 and 20. In the Section 5, we consider
where the function
2. Preliminaries
We require the following definitions and lemmas to prove the main results in the next three sections. For
Using Jacobi's triple product identity [1, Theorem 2.8], (2.1) takes the shape
The special cases of
Lemma 2.1.
(Hirschhorn [6, p. 14, Eqn. 1.9.4]) We have the following 2-dissection of
Lemma 2.2.
(Hirschhorn [5] or Hirschhorn [6, p. 36, Eqn. 3.6.4]) we have
Lemma 2.3.
(Hirschhorn [6, p. 105, Eqn. 10.7.6])
From the Binomial Theorem, for any positive integer, k,
3. Congruences Modulo 2 and 4 for
E
O
¯
(
n
)
In this section we prove some congruences modulo 2 and 4 satisfied by
We require the following generating functions to prove congruences for
Theorem 3.1.
We have,
since there are no terms on the right in which the power of
thus by using (2.6), we obtain
It follows that
and
which is our (3.1) and (3.2). We have
It follows that
and
which is our (3.3) and (3.5). We have
It follows that
and
We have the following congruences.
Corollary 3.2.
For all
Remark 3.3.
The congruences (3.11)-(3.13) were obtained earlier by Andrews et al. [4]. Andrews et al. [3] introduced a partition function
where
He proved the congruences using the properties of mock theta function, whereas we use the q-series identities.
4. Congruences Modulo 5, 10 and 20 for
E
O
¯
(
n
)
In this section we prove some congruences modulo 5, 10 and 20 for
Theorem 4.1.
For all
From (2.7), we have
where
where
There are no terms on the right in which the power of
from which we deduce (4.1).
In the next theorem, we derive two congruences modulo 10 from the generating Łinebreak functions (3.2) and (3.5).
Theorem 4.2.
For all
Replacing
where
Substituting (4.3) and (4.9) in (4.8), we obtain
There are no terms on the right in which the power of
from which we deduce (4.6). Using (2.9) in (3.5), we have
From (2.7), we have
where
There are no terms on the right in which the power of
from which we deduce (4.7).
In the next theorem, we derive a congruences modulo 20 from the generating Łinebreak function (3.6).
Theorem 4.3.
For all
From (2.8), we have
where
There are no terms on the right in which the power of
from which we deduce (4.14).
5. Congruences for
E
O
e
(
n
)
In this section we prove some congruences modulo 2 for
Theorem 5.1.
since there are no terms on the right in which the power of
by using (2.6), we obtain
It follows that
and
We have the following congruences.
Corollary 5.2.
For all
6. Conclusion
Andrews [2, Problem 4], proposed to further investigate the properties of
Conjecture 6.1.
Andrews [2, Problem 4], proposed to further investigate the properties of
For all
Acknowledgements.
The authors would like to thank Professor Michael Hirschhorn for his valuable comments and helpful suggestions.
References
- G. E. Andrews, The theory of partitions, Cambridge Univ. Press, Cambridge, 1998.
- G. E. Andrews. Integer partitions with even parts below odd parts and the mock theta functions, Ann. Comb. 22(2018), 433-445.
- G. E. Andrews, A. Dixit, and A. J. Yee. A. Dixit and A. J. Yee, Partitions associated with the Ramanu-jan/Watson mock theta functions ω(q), ν(q) and ϕ(q), Res. Number Theory 1 (2015). Paper No. 19, 25 pp.
- G. E. Andrews, D. Passary, J. Sellers, and A. J. Yee. Congruences related to the Ramanujan/Watson mock theta functions ω(q), ν(q), Ramanujan J. 43(2017), 347-357.
- M. D. Hirschhorn and Another short proof of Ramanujan's mod 5 partition congruence. Another short proof of Ramanujan’s mod 5 partition congruence, and more, Amer. Math. Monthly 106(1999), 580-583.
- M. D. Hirschhorn, The power of q, Developments in Mathematics 49, Springer, 2017.
- Some properties of p(n), the number of partitions of n, Proc. Camb. Philos. Soc. 19(1919), 207-210.