Some Congruence Properties of a Restricted Bipartition Function

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).