Article
Kyungpook Mathematical Journal 2023; 63(2): 155-166
Published online June 30, 2023
Copyright © Kyungpook Mathematical Journal.
Congruences for Partition Functions E O ¯ ( n ) and E O e ( n )
Riyajur Rahman and Nipen Saikia*
Department of Mathematics, Rajiv Gandhi University, Rono Hills, Doimukh 791112, Arunachal Pradesh, India
e-mail : riyajurrahman@gmail.com and nipennak@yahoo.com
Received: February 11, 2022; Revised: October 13, 2022; Accepted: January 17, 2023
Abstract
In 2018, Andrews introduced the partition functions
Keywords: Partitions of integer, congruences,
1. Introduction
A partition of a positive integer
where, for any complex number
We will use the notation, for any positive integer
Andrews [2] introduced the partition function
Andrews [2] also introduced another partition function
Andrews [2, p. 434, (1.6)] established that
Recently, Pore and Fathima [8] proved some particular congruences for
Pore and Fathima [8] also defined a new partition function
Pore and Fathima [8, Corollary 5.2] proved that
Motivated by the above work, in Section 3 of this paper, we prove infinite families of congruences modulo 2, 4, 5 and 8 for
2. Some q -series Identities
Lemma 2.1. ([3, p. 39, Entry 24(ii)])
Lemma 2.2. We have
The identity (2.2) is the 2-dissection of
from the equation (2.2) by replacing
Lemma 2.3. ([1, Lemma 2.3])
Furthermore, if
Lemma 2.4. ([4, Theorem 2.2])
where
Furthermore, if
then
Lemma 2.5. ([7])
where R(q) is the Roger-Ramanujan continued fraction defined by
Hirschhorn and Hunt [5, Lemma 2.2] proved that, if
where
Hirschhorn and Hunt [5] showed that
where
In addition to above
3. Congruences for E O ¯ ( n )
Theorem 3.1. Let
where, here and throughout the paper
Extracting the terms involving
Extracting the terms involving
Using (2.2) in (3.5), we obtain
Extracting the terms involving
Using (2.10) in (3.7), we obtain
Congruence (3.8) is the
Consider the congruence
which is equal to
For
Extracting the terms involving
which is the
Theorem 3.2. Let
Employing (2.10) in (3.14), we have
Extracting the terms involving
Employing (2.10) in (3.16), we obtain
Congruence (3.17) is the
Consider the congruence
which is equal to
For
Extracting the terms involving
which is the α + 1 case of (3.12). Thus, by the principle of mathematical induction, we arrive at (3.12). Extracting the coefficients of terms involving
Theorem 3.3. Let
Employing (2.2) in (3.23) and extracting the terms involving
Using (2.10) in (3.24), we obtain
Congruence (3.25) is the
Consider the congruence
which is equal to
For
Extracting the terms involving
which is the α + 1 case of (3.21). Thus, by the principle of mathematical induction, we arrive at (3.21). Extracting the coefficients of terms involving
Theorem 3.4. Let
Employing (2.10) in (3.32), we have
Extracting the terms involving
Congruence (3.34) is the
Extracting the term involving
Extracting the terms involving
which is the α + 1 case of (3.30). Thus, by the principle of mathematical induction, we arrive at (3.30). Extracting the coefficients of terms involving
Theorem 3.5. For any non-negative integers
Employing (2.1) and (2.8) in (3.39), we obtain
Extracting the terms involving the powers of
4. Congruences for E O e ( n )
Theorem 4.1. Let
Employing (2.10) in (4.3), we obtain
Extracting the terms involving
The remaining part of the proof is similar to proofs of the identities (3.30) and (3.31).
Theorem 4.2. Let
Extracting the terms involving
Multiplying the numerator and denominator by
Using (2.11) in (4.10), we obtain
Using (2.3) in (4.11), we obtain
Extracting the terms involving
Using (2.11) in (4.13), we obtain
The remaining part of the proof is similar to proofs of the identities (3.21) and (3.22).
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