#### Abstract

Let denote the number of bipartitions of a positive integer subject to the restriction that each part of is divisible by . In this paper, we prove some congruence properties of the function for , 11, and , for any integer , by employing Ramanujan’s theta-function identities.

#### 1. Introduction

A bipartition of a positive integer is an ordered pair of partitions such that the sum of all of the parts equals . If counts the number of bipartitions of subject to the restriction that each part of is divisible by , then the generating function of [1] is given by where The partition function is first studied by Chan [2] for the particular case by considering the function defined by Chan [2] proved that, for , Kim [3] gave a combinatorial interpretation (4). In a subsequent paper, Chan [4] showed that, for and , where is the reciprocal modulo of 8 and if is even and 0 otherwise. Inspired by the work of Ramanujan on the standard partition function , Chan [4] asked whether there are any other congruence properties of the following form: , where is prime and . Sinick [1] answered Chan’s question in negative by considering restricted bipartition function defined in (1). Liu and Wang [5] established several infinite families of congruence properties for modulo 3. For example, they proved that

Baruah and Ojah [6] also proved some congruence properties for some particular cases of by considering the generalised partition function defined by and using Ramanujan’s modular equations. Clearly, . For example, Baruah and Ojah [6] proved that Ahmed et al. [7] investigated the function for 3 and 4 and proved some congruence properties modulo 5. They also gave alternate proof of some congruence properties due to Chan [2].

In this paper, we investigate the restricted bipartition function for = 7, 11, and 5, for any integer , and prove some congruence properties modulo 2, 3, and 5 by using Ramanujan’s theta-function identities. In Section 3, we prove congruence properties modulo 2 for . For example, we prove, for , In Section 4, we deal with the function and establish the notion that if is an odd prime, , and , then In Section 5, we show that, for any integer , . We also prove congruence properties modulo 3 for Section 2 is devoted to listing some preliminary results.

#### 2. Preliminary Results

Ramanujan’s general theta function is defined by Three important special cases of are Ramanujan also defined the function as

Lemma 1. For any prime and positive integer , one has

Proof. It follows easily from the binomial theorem.

Lemma 2 (see [8, page 315]). One has

Lemma 3. One has

Proof. From (13), we have Simplifying (19) using Lemma 1 with , we arrive at the desired result.

Lemma 4 (see [9, page 286, Equation (60)]). One has

Lemma 5 (see [10, page 372]). One has

Lemma 6 (see [8, page 350, Equation (13)]). One has where

Lemma 7. One has

Proof. Employing (20) in Lemma 6 and performing simplification using Lemma 1 with , we obtain Replacing by in (28), we arrive at the desired result.

Lemma 8 (see [8, page 51, Example (v)]). One has

Lemma 9. One has

Proof. Employing (21) and (23) in Lemma 8, we obtain Simplifying (31) using Lemma 1 with , we complete the proof.

Lemma 10 (see [11, page 5, Equation (15)]). One has

Lemma 11 (see [12, Theorem ]). For any odd prime , where, for ,

Lemma 12 (see [12, Theorem ]). For any prime , one has where

Lemma 13 (see [13]). One has where and is Rogers-Ramanujan continued fraction defined by

Lemma 14 (see [8, page 345, Entry 1(iv)]). One has where and is Ramanujan’s cubic continued fraction defined by

#### 3. Congruence Identities for

Theorem 15. One has

Proof. For in (1), we have Employing (19) in (42), we obtain Employing Lemma 2 in (43), we obtain Extracting the terms involving , dividing by , and replacing by in (44), we get Employing Lemma 3 in (45), we complete the proof.

Theorem 16. One has

Proof. From Theorem 15, we obtain Employing Lemma 3 in (47), we obtain Employing Lemma 2 in (48), extracting the terms involving , dividing by , and replacing by , we obtain Employing Lemma 3 in (49) and performing simplification using Lemma 1 with , we arrive at (i).
All the terms on the right hand side of (i) are of the form . Extracting the terms involving on both sides of (i), we complete the proof of (ii).

Theorem 17. For all , one has (i),(ii),(iii)

Proof. Employing (14) in Theorem 15, we obtain Extracting those terms on each side of (50) whose power of is of the forms , , and and employing the fact that there exists no integer such that is congruent to 3, 4, and 6 modulo 7, we obtain Now, (i), (ii), and (iii) are obvious from (51).

Theorem 18. For , one has

Proof. We proceed by induction on Extracting the terms involving and replacing by in Theorem 16(i), we obtain which corresponds to the case . Assume that the result is true for , so that Employing Lemma 3 in (54), we obtain Employing Lemma 2 in (55) and extracting the terms involving , dividing by , and replacing by , we obtain Simplifying (56) using Lemmas 3 and 1 with , we obtain Extracting the terms involving and replacing by in (57), we obtain which is the case. Hence, the proof is complete.

Theorem 19. For , one has

Proof. All the terms in the right hand side of (57) are of the form , so, extracting the coefficients of on both sides of (57) and replacing by , we obtain Replacing by in (60) completes the proof.

Theorem 20. If any prime , , and , then where .

Proof. Employing Lemma 12 in (52), we obtain We consider the congruence where . The congruence (63) is equivalent to and, for , the congruence (64) has unique solution Extracting terms containing from both sides of (62) and replacing by , we obtain Extracting the coefficients of , for , on both sides of (65) and performing simplification, we arrive at the desired result.

#### 4. Congruence Identities for

Theorem 21. One has

Proof. Setting in (1), we obtain Employing (13) in (67), we obtain Employing Lemma 5 in (68), extracting the terms involving , dividing by , and replacing by , we obtain Employing Lemmas 9 and 10 in (69), we find that Extracting the terms involving and replacing by on both sides of (70) and performing simplification using Lemma 1 with , we obtain Employing Lemma 7 in (71) and using (13), we complete the proof.

Theorem 22. For any odd prime and any integer , one has

Proof. We proceed by induction on . The case corresponds to the congruence theorem (Theorem 21). Suppose that the theorem holds for , so that Employing Lemma 11 in (73), extracting the terms involving on both sides of (73), dividing by , and replacing by , we obtain Extracting the terms containing from both sides of (74) and replacing by , we arrive at which shows that the theorem is true for . Hence, the proof is complete.

Theorem 23. For any odd prime and integers and , one has

Proof. Extracting the coefficients of for on both sides of (74) and replacing by , we arrive at the desired result.

#### 5. Congruence Identities for

Theorem 24. For any positive integer , one has

Proof. Setting in (1), we obtain Using Lemma 13 in (78) and extracting the terms involving , dividing by , and replacing by , we obtain The desired result follows easily from (79).

Theorem 25. For all , one has (i),(ii),(iii)

Proof. Setting in (1), we obtain Employing Lemma 13 in (80), extracting terms involving , dividing by , and replacing by , we obtain Now, (i) follows from (81).
Simplifying (81) by using Lemma 1 with , we obtain Employing Lemma 14 in (82) and performing simplification, we obtain Extracting terms involving and on both sides of (83), we arrive at (ii) and (iii), respectively.

#### Competing Interests

The authors declare that there are no competing interests regarding the publication of this paper.

#### Acknowledgments

The first author (Nipen Saikia) is thankful to the Council of Scientific and Industrial Research of India for partially supporting the research work under Research Scheme no. 25(0241)/15/EMR-II (F. no. 25(5498)/15).