### 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 ^{5}

where

Hirschhorn and Hunt [5] showed that

where _{5}

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 ^{p}

Extracting the terms involving ^{pn}^{p}

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 ^{p}

Extracting the terms involving ^{pn}^{p}

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 ^{pn+j}

**Theorem 3.3.** Let

^{2n}

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 ^{p}

Extracting the terms involving ^{pn}^{p}

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 ^{8n+k}^{8n}^{8}

Congruence (3.34) is the

Extracting the term involving ^{p}

Extracting the terms involving ^{pn}^{p}

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 ^{pn+j}

**Theorem 3.5.** For any non-negative integers

^{2n}

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 ^{4n}^{4}

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 ^{2n}^{2}

Multiplying the numerator and denominator by _{1}^{2}

Using (2.11) in (4.10), we obtain

Using (2.3) in (4.11), we obtain

Extracting the terms involving ^{2n+1}^{2}

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).

### References

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