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Some Congruence Properties of a Restricted Bipartition Function [c.sub.N](n).

1. Introduction

A bipartition of a positive integer n is an ordered pair of partitions ([lambda], [mu]) such that the sum of all of the parts equals n. If [c.sub.N] (n) counts the number of bipartitions ([lambda], [mu]) of n subject to the restriction that each part of [mu] is divisible by N, then the generating function of [c.sub.N] (n) [1] is given by

[[infinity].summation over (n=0)] [c.sub.N](n) [q.sup.n] = 1/[(q;q).sub.[infinity]][([q.sup.N]; [q.sup.N]).sub.[infinity]], (1)

where

[mathematical expression not reproducible]. (2)

The partition function [c.sub.N] (n) is first studied by Chan [2] for the particular case N = 2 by considering the function [c.sub.2] (n) defined by

[[infinity].summation over (n=0)] [c.sub.2](n) [q.sup.n] = 1/[(q;q).sub.[infinity]] [([q.sup.2];[q.sup.2]).sub.[infinity]]. (3)

Chan [2] proved that, for n [greater than or equal to] 0,

[c.sub.2] (3n + 2) [equivalent to] 0 (mod3). (4)

Kim [3] gave a combinatorial interpretation (4). In a subsequent paper, Chan [4] showed that, for k [greater than or equal to] 1 and n [greater than or equal to] 0,

[c.sub.2] ([3.sup.k] n + [s.sub.k]) [equivalent to] 0(mod[3.sup.k+[delta](k)]), (5)

where [s.sub.k] is the reciprocal modulo [3.sup.k] of 8 and [delta](k) = 1 if k is even and 0 otherwise. Inspired by the work of Ramanujan on the standard partition function p(n), Chan [4] asked whether there are any other congruence properties of the following form: [c.sub.2] (ln + k) [equivalent to] 0 (mod l), where l is prime and 0 [less than or equal to] k [less than or equal to] l. Sinick [1] answered Chan's question in negative by considering restricted bipartition function [c.sub.N] (n) defined in (1). Liu and Wang [5] established several infinite families of congruence properties for [c.sub.5] (n) modulo 3. For example, they proved that

[c.sub.5] ([3.sup.2[alpha]+1] n + 7 x [3.sup.2[alpha]] + 1/4) [equivalent to] 0 (mod3), [alpha] [greater than or equal to] 1, n [greater than or equal to] 0. (6)

Baruah and Ojah [6] also proved some congruence properties for some particular cases of [c.sub.N] (n) by considering the generalised partition function [mathematical expression not reproducible] defined by

[mathematical expression not reproducible] (7)

and using Ramanujan's modular equations. Clearly, [mathematical expression not reproducible]. For example, Baruah and Ojah [6] proved that

[mathematical expression not reproducible]. (8)

Ahmed et al. [7] investigated the function [C.sub.N] (n) for N = 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 [c.sub.N] (n) for n = 7, 11, and 5l, for any integer l [greater than or equal to] 1, 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 [c.sub.7] (n). For example, we prove, for [alpha] [greater than or equal to] 0,

[c.sub.7] ([2.sup.2[alpha]+1] n + 5 x [2.sup.2[alpha]] + 1/3) [equivalent to] 0 (mod2). (9)

In Section 4, we deal with the function [c.sub.11] (n) and establish the notion that if p is an odd prime, 1 [less than or equal to] j [less than or equal to] p - 1, and [alpha] [greater than or equal to] 0, then

[c.sub.11] (4[p.sup.2[alpha]+1] (pn + j) + [p.sup.2[alpha]+2] + 1/2) [equivalent to] 0 (mod2). (10)

In Section 5, we show that, for any integer l [greater than or equal to] 1, [c.sub.5l](5n + 4) = 0 (mod5). We also prove congruence properties modulo 3 for [c.sub.15] (n). Section 2 is devoted to listing some preliminary results.

2. Preliminary Results

Ramanujan's general theta function f(a, b) is defined by

f(a, b) = [[infinity].summation over (n=0) [a.sup.n(n+1)/2] [b.sup.n(n-1)/2], [absolute value of (ab)] < 1. (11)

Three important special cases of f(a, b) are

[mathematical expression not reproducible], (12)

[psi](q) := f(q, [q.sup.3]) = [[infinity].summation over (n=0)] [q.sup.n(n+1)/2] = [([q.sup.2];[q.sup.2]).sup.2.sub.[infinity]]/[(q;q).sub.[infinity]], (13)

[mathematical expression not reproducible]. (14)

Ramanujan also defined the function [chi](q) as

[chi](q) = [(-q;[q.sup.2]).sub.[infinity]]. (15)

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

[([q.sup.pm];[q.sub.pm]).sub.[infinity]] [equivalent to] [([q.sup.m];[q.sup.m]).sup.p.sub.[infinity]] (mod p). (16)

Proof. It follows easily from the binomial theorem.

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

[psi] (q) [psi] ([q.sup.7]) = [phi] ([q.sup.28]) [psi] ([q.sup.8]) + q[psi] ([q.sup.14]) [psi] ([q.sup.2]) + [q.sup.6][psi]([q.sup.56]) [phi] ([q.sup.4]). (17)

Lemma 3. One has

[psi](q)[psi]([q.sup.7]) [equivalent to] [(q;q).sup.3.sub.[infinity]] ([q.sup.7];[q.sup.7]) (mod 2). (18)

Proof. From (13), we have

[psi](q)[psi]([q.sup.7]) = [([q.sup.2];[q.sup.2]).sup.2.sub.[infinity]][([q.sup.14];[q.sup.14]).sup.2.sub.[infinity]]/[(q;q).sub.[infinity]][([q.s up.7];[q.sup.7]).sub.[infinity]]. (19)

Simplifying (19) using Lemma 1 with p = 2,we arrive at the desired result.

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

[phi](-q) = [(q;q).sup.2.sub.[infinity]]/[([q.sup.2];[q.sup.2]).sub.[infinity]], (20)

[psi](-q) = [(q;q).sub.[infinity]] [([q.sup.4];[q.sup.4]).sub[infinity]i]/[([q.sup.2];[q.sup.2]).sub.[infinity]], (21)

f(q) = [([q.sup.2];[q.sup.2]).sup.3.sub.[infinity]]/[(q;q).sub.[infinity]][([q.sup.4];[q.sup.4]).sub.[infinity]], (22)

[chi](q) = [([q.sup.2];[q.sup.2]).sup.2.sub.[infinity]]/[(q;q).sub.[infinity]][([q.sup.4];[q.sup.4]).sub.[infinity]]. (23)

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

[mathematical expression not reproducible]. (24)

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

f(q,[q.sup.2]) = [phi](-[q.sup.3])/[chi](-q), (25)

where

[chi](-q) = [(q;q).sub.[infinity]]/[([q.sup.2];[q.sup.2]).sub.[infinity]]. (26)

Lemma 7. One has

f([q.sup.11];[q.sup.22]) [equivalent to] [([q.sup.11];[q.sup.11]).sub.[infinity]] (mod 2). (27)

Proof. Employing (20) in Lemma 6 and performing simplification using Lemma 1 with p = 2,we obtain

f(q;[q.sup.2]) = [(q;q).sub.[infinity]] (mod 2). (28)

Replacing q by [q.sup.11] in (28), we arrive at the desired result.

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

f(q,[q.sup.5]) = [psi](-[q.sup.3]) [chi] (q). (29)

Lemma 9. One has

f(q,[q.sup.5]) [equivalent to] [([q.sup.3];[q.sup.3]).sup.3.sub.[infinity]]/[(q;q).sub.[infinity]] (mod 2). (30)

Proof. Employing (21) and (23) in Lemma 8, we obtain

[mathematical expression not reproducible]. (31)

Simplifying (31) using Lemma 1 with p = 2,we complete the proof.

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

[mathematical expression not reproducible]. (32)

Lemma 11 (see [12, Theorem 2.1]). For any odd prime p,

[mathematical expression not reproducible], (33)

where, for 0 [less than or equal to] k [less than or equal to] (p - 3)/2,

[k.sup.2] + k/2 [not equivalent to] [p.sup.2] - 1/8 (mod p). (34)

Lemma 12 (see [12, Theorem 2.2]). For any prime p [greater than or equal to] 5, one has

[mathematical expression not reproducible], (35)

where

[mathematical expression not reproducible]. (36)

Lemma 13 (see [13]). One has

[mathematical expression not reproducible], (37)

where F(q) := [q.sup.-1/5] R(q) and R(q) is Rogers-Ramanujan continued fraction defined by

[mathematical expression not reproducible]. (38)

Lemma 14 (see [8, page 345, Entry 1(iv)]). One has

[(q;q).sub.sup.3.[infinity]] = [([q.sup.9];[q.sup.9]).sup.3.sub.[infinity]] (4[q.sup.3][W.sup.2] ([q.sup.3]) - 3q + [W.sup.-1] ([q.sup.3])), (39)

where W(q) = [q.sup.-1/3] G(q) and G(q) is Ramanujans cubic continued fraction defined by

[mathematical expression not reproducible]. (40)

3. Congruence Identities for [c.sub.7] (n)

Theorem 15. One has

[[infinity].summation over (n=0)] [c.sub.7] (2n+1)[q.sup.n] [equivalent to] [(q;q).sub.[infinity]] [([q.sup.7];[q.sup.7]).sub.[infinity]] (mod 2). (41)

Proof. For N = 7 in (1), we have

[[infinity].summation over (n=0)] [c.sub.7](n)[q.sup.n] = 1/[(q;q).sub.[infinity]][([q.sup.7];[q.sup.7]).sub.[infinity]]. (42)

Employing (19) in (42), we obtain

[[infinity].summmation over (n=0)] [c.sub.7](n)[q.sup.n] = [psi](q)[psi]([q.sup.7])/[([q.sup.2];[q.sup.2]).sup.2.sub.[infinity]][([q.sub.14];[q.sub.14]).sup.2.sub.[infinity]]. (43)

Employing Lemma 2 in (43), we obtain

[mathematical expression not reproducible]. (44)

Extracting the terms involving [q.sup.2n+1], dividing by q, and replacing [q.sup.2] by q in (44), we get

[mathematical expression not reproducible]. (45)

Employing Lemma 3 in (45), we complete the proof.

Theorem 16. One has

[mathematical expression not reproducible]. (46)

Proof. From Theorem 15, we obtain

[mathematical expression not reproducible]. (47)

Employing Lemma 3 in (47), we obtain

[mathematical expression not reproducible]. (48)

Employing Lemma 2 in (48), extracting the terms involving [q.sup.2n+1], dividing by q, and replacing [q.sup.2] by q, we obtain

[mathematical expression not reproducible]. (49)

Employing Lemma 3 in (49) and performing simplification using Lemma 1 with p = 2, we arrive at (i).

All the terms on the right hand side of (i) are of the form [q.sup.2n]. Extracting the terms involving [q.sup.2n+1] on both sides of (i), we complete the proof of (ii).

Theorem 17. For all n [greater than or equal to] 0, one has

(i) [c.sub.7] (14n+7) [equivalent to] 0 (mod2), (ii) [c.sub.7] (14n+9) [equivalent to] 0 (mod2), (iii) [c.sub.7] (14n+ 13) [equivalent to] 0 (mod2).

Proof. Employing (14) in Theorem 15, we obtain

[mathematical expression not reproducible]. (50)

Extracting those terms on each side of (50) whose power of q is of the forms 7n + 3, 7n + 4, and 7n + 6 and employing the fact that there exists no integer n such that n(3n+ 1)/2 is congruent to 3, 4, and 6 modulo 7, we obtain

[mathematical expression not reproducible]. (51)

Now, (i), (ii), and (iii) are obvious from (51).

Theorem 18. For [alpha] [greater than or equal to] 1, one has

[[infinity].summation over (n=0)] [c.sub.7]([2.sup.2[alpha]+1] n + [2.sup.2[alpha]+1] + 1/3)[q.sup.n] [equivalent to] [(q;q).sub.[infinity]][([q.sup.7];[q.sup.7]).sub.[infinity]] (mod 2). (52)

Proof. We proceed by induction on [alpha]. Extracting the terms involving [q.sup.2n] and replacing [q.sup.2] by q in Theorem 16(i), we obtain

[[infinity].summation over (n=0)] [c.sub.7] (8n + 3) [q.sup.n] [equivalent to] [(q; q).sub.[infinity]] [([q.sup.7];[q.sup.7]).sub.[infinity]] (mod 2), (53)

which corresponds to the case [alpha] = 1. Assume that the result is true for [alpha] = k [greater than or equal to] 1, so that

[[infinity].summation over (n=0)] [c.sub.7]([2.sup.2k+1] n + [2.sup.2k+1] + 1/3)[q.sup.n] [equivalent to] [(q;q).sub.[infinity]][([q.sup.7];[q.sup.7]).sub.[infinity]](mod 2). (54)

Employing Lemma 3 in (54), we obtain

[[infinity].summation over (n=0)][c.sub.7]([2.sup.2k+1] n + [2.sup.2k+1] + 1/3)[q.sup.n] = [equivalent to] [psi](q)[psi]([q.sup.7])/[(q;q).sup.2.sub.[infinity]][([q.sup.7];[q.sub.7]).sup.2.sub.[infinity]] (mod 2). (55)

Employing Lemma 2 in (55) and extracting the terms involving [q.sub.2n+1], dividing by q, and replacing [q.sup.2] by q, we obtain

[[infinity].summation over (n=0)] [c.sub.7] ([2.sup.2k+1] (2n + 1) + [2.sup.2k+1] + 1/3)[q.sup.n] [equivalent to] [psi](q)[psi]([q.sup.7])/[(q;q).sub.[infinity]][([q.sup.7];[q.sup.7]).sub.[infinity]](mod 2). (56)

Simplifying (56) using Lemmas 3 and 1 with p = 2,we obtain

[[infinity].summation over (n=0)] [c.sub.7] ([2.sup.2(k+1)+1] n + [2.sup.2(k+1)+1] + 1/3)[q.sup.n] [equivalent to] [psi](q)[psi]([q.sup.7])/[(q;q).sub.[infinity]][([q.sup.14];[q.sup.14]).sub.[infinity]](mod 2). (57)

Extracting the terms involving [q.sup.2n] and replacing [q.sup.2] by q in (57), we obtain

[[infinity].summation over (n=0)] [c.sub.7]([2.sup.2(k+1)+1] n + [2.sup.2(k+1)+1] + 1/3)[q.sup.n] [equivalent to] [(q;q).sub.[infinity]][([q.sup.7];[q.sup.7]).sub.[infinity]] (mod 2), (58)

which is the [alpha] = k+1 case. Hence, the proof is complete.

Theorem 19. For [alpha] [greater than or equal to] 0, one has

[c.sub.7] ([2.sup.2[alpha]+1] n + 5 x [2.sup.2[alpha]] + 1/3) [equivalent to] 0 (mod 2). (59)

Proof. All the terms in the right hand side of (57) are of the form [q.sup.2n], so, extracting the coefficients of [q.sup.2n+1] on both sides of (57) and replacing k by [alpha], we obtain

[c.sub.7]([2.sup.2([alpha]+1)+1] n + 5 x [2.sup.2([alpha]+1)] + 1/3) [equivalent to] 0 (mod 2). (60)

Replacing [alpha] + 1 by [alpha] in (60) completes the proof.

Theorem 20. If any prime p [greater than or equal to] 5, (-7/p) = -1, and [alpha] [greater than or equal to] 0, then

[c.sub.7] ([2.sup.2[alpha]+1] [p.sup.2] n + [2.sup.2[alpha]+1] p(3j + p) + 1/3) [equivalent to] 0 (mod 2), (61)

where 1 [less than or equal to] j [less than or equal to] p - 1.

Proof. Employing Lemma 12 in (52), we obtain

[mathematical expression not reproducible]. (62)

We consider the congruence

3[k.sup.2] + k/2 + 7 x 3[m.sup.2]/2 [equivalent to] 8[p.sup.2] - 8/24 (mod p), (63)

where -(p - 1)/2 [less than or equal to] k, m [less than or equal to] (p - 1)/2. The congruence (63) is equivalent to

[(6k + 1).sup.2] + 7 [(6m + 1).sup.2] [equivalent to] 0 (mod p) (64)

and, for (-7/p) = -1, the congruence (64) has unique solution k = m = ([+ or -] p - 1)/6. Extracting terms containing [mathematical expression not reproducible] from both sides of (62) and replacing [q.sup.p] by q, we obtain

[[infinity].summation over (n=0)] [c.sub.7] ([2.sup.2[alpha]+1] pn + [2.sup.2[alpha]+1] [p.sup.2] + 1/3)[q.sup.n] [equivalent to] [([q.sup.p];[q.sup.p]).sub.[infinity]]([q.sup.7p];[q.sup.7p])(mod 2). (65)

Extracting the coefficients of [q.sup.pn+j], for 1 [less than or equal to] j [less than or equal to] p - 1,on both sides of (65) and performing simplification, we arrive at the desired result.

4. Congruence Identities for [c.sub.11] (n)

Theorem 21. One has

[[infinity].summation over (n=0) [c.sub.11] (4n + 1) [q.sup.n] [equivalent to] [([q.sup.2];[q.sup.2]).sup.2.sub.[infinity]]/[(q;q).sub.[infinity]] = [psi](q) (mod 2). (66)

Proof. Setting N = 11 in (1), we obtain

[[infinity].summation over (n=0)] [c.sub.11](n) [q.sup.n] = 1/[(q;q).sub.[infinity]] [([q.sup.11];[q.sup.11]).sub.[infinity]]. (67)

Employing (13) in (67), we obtain

[[infinity].summation over (n=0)] [c.sub.11] (n) [q.sup.n] = [psi](q) [psi]([q.sup.11])/[([q.sup.2];[q.sup.2]).sup.2.sub.[infinity]] [([q.sup.22];[q.sup.22]).sup.2.sub.[infinity]]. (68)

Employing Lemma 5 in (68), extracting the terms involving [q.sup.2n+1], dividing by q, and replacing [q.sup.2] by q, we obtain

[mathematical expression not reproducible]. (69)

Employing Lemmas 9 and 10 in (69), we find that

[mathematical expression not reproducible]. (70)

Extracting the terms involving [q.sup.2n] and replacing [q.sup.2] by q on both sides of (70) and performing simplification using Lemma 1 with p = 2,we obtain

[mathematical expression not reproducible]. (71)

Employing Lemma 7 in (71) and using (13), we complete the proof.

Theorem 22. For any odd prime p and any integer [alpha] [greater than or equal to] 0, one has

[[infinity].summation over (n=0)] [c.sub.11] (4[p.sup.2[alpha]] n + [p.sup.2[alpha]] + 1/2) [q.sup.n] [equivalent to] [psi](q) (mod 2). (72)

Proof. We proceed by induction on [alpha]. The case [alpha] = 0 corresponds to the congruence theorem (Theorem 21). Suppose that the theorem holds for [alpha] = k [greater than or equal to] 0, so that

[[infinity].summation over (n=0)] [c.sub.11] (4[p.sup.2k] n + [p.sup.2k] + 1/2) [q.sup.n] [equivalent to] [psi](q) (mod2). (73)

Employing Lemma 11 in (73), extracting the terms involving [mathematical expression not reproducible] on both sides of (73), dividing by [mathematical expression not reproducible], and replacing [q.sup.p] by q, we obtain

[[infinity].summation over (n=0)] [c.sub.11] (4[p.sup.2k+1] n + [p.sup.2(k+1)] + 1/2) [q.sup.n] [equivalent to] [psi]([q.sup.p]) (mod2). (74)

Extracting the terms containing [q.sup.pn] from both sides of (74) and replacing [q.sup.p] by q, we arrive at

[[infinity].summation over (n=0)] [c.sub.11] (4[p.sup.2k] n + [p.sup.2k] + 1/2) [q.sup.n] [equivalent to] [psi](q) (mod2). (75),

which shows that the theorem is true for [alpha] = k + 1. Hence, the proof is complete.

Theorem 23. For any odd prime p and integers [alpha] [greater than or equal to] 0 and 1 [less than or equal to] j [less than or equal to] p - 1, one has

[c.sub.11] (4[p.sup.2[alpha]+1] (pn + j) + [p.sup.2[alpha]+2] + 1/2) [equivalent to] 0 (mod2). (76)

Proof. Extracting the coefficients of [q.sup.pn+j] for 1 [less than or equal to] j [less than or equal to] p - 1 on both sides of (74) and replacing k by [alpha], we arrive at the desired result.

5. Congruence Identities for [c.sub.5l] (n)

Theorem 24. For any positive integer l, one has

[c.sub.5l] (5n + 4) [equivalent to] 0 (mod5). (77)

International Journal of Analysis Proof. Setting N = 5l in (1), we obtain

[[infinity].summmation over (n=0)] [c.sub.5l](n)[q.sup.n] = 1/[(q;q).sub.[infinity]][([q.sup.5l];[q.sup.5l]).sub.[infinity]]. (78)

Using Lemma 13 in (78) and extracting the terms involving [q.sup.5n+4], dividing by [q.sup.4], and replacing [q.sup.5] by q, we obtain

[mathematical expression not reproducible]. (79)

The desired result follows easily from (79).

Theorem 25. For all n [greater than or equal to] 0, one has

(i) [c.sub.15] (5n + 4) [equivalent to] 0 (mod5), (ii) [c.sub.15] (15n+9) [equivalent to] 0 (mod3), (iii) [c.sub.15] (15n+ 14) = 0 (mod3).

Proof. Setting N = 15 in (1), we obtain

[[infinity].summation over (n=0)] [c.sub.15](n) [q.sup.n] = 1/[(q;q).sub.[infinity]][([q.sup.15];[q.sup.15]).sub.[infinity]]. (80)

Employing Lemma 13 in (80), extracting terms involving [q.sup.5n+4], dividing by [q.sup.4], and replacing q by q, we obtain

[mathematical expression not reproducible]. (81)

Now, (i) follows from (81).

Simplifying (81) by using Lemma 1 with p = 3,we obtain

[mathematical expression not reproducible]. (82)

Employing Lemma 14 in (82) and performing simplification, we obtain

[mathematical expression not reproducible]. (83)

Extracting terms involving [q.sup.3n+1] and [q.sup.3n+2] on both sides of (83), we arrive at (ii) and (iii), respectively.

http://dx.doi.org/10.1155/2016/9037692

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).

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[9] N. D. Baruah, J. Bora, and N. Saikia, "Some newproofs of modular relations for the Gollnitz-Gordon functions," Ramanujan Journal, vol. 15, no. 2, pp. 281-301, 2008.

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Nipen Saikia and Chayanika Boruah

Department of Mathematics, Rajiv Gandhi University, Rono Hills, Doimukh, Arunachal Pradesh 791112, India

Correspondence should be addressed to Nipen Saikia; nipennak@yahoo.com

Received 12 April 2016; Accepted 10 July 2016

Academic Editor: Ahmed Zayed
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