Arithmetic properties for Andrews’ (48,6)- and (48,18)-singular overpartitions

Eric H. Liu; Wenjing Du

doi:10.1515/math-2019-0026

Article Open Access

Arithmetic properties for Andrews’ (48,6)- and (48,18)-singular overpartitions

Eric H. Liu and Wenjing Du

Published/Copyright: May 11, 2019

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From the journal Open Mathematics Volume 17 Issue 1

Abstract

Singular overpartition functions were defined by Andrews. Let C_k,i(n) denote the number of (k, i)-singular overpartitions of n, which counts the number of overpartitions of n in which no part is divisible by k and only parts ±i (mod k) may be overlined. A number of congruences modulo 3, 9 and congruences modulo powers of 2 for C_k,i(n) were discovered by Ahmed and Baruah, Andrews, Chen, Hirschhorn and Sellers, Naika and Gireesh, Shen and Yao for some pairs (k, i). In this paper, we prove some congruences modulo powers of 2 for C_{48, 6}(n) and C_{48, 18}(n).

Keywords: congruence; singular overpartition; theta function

MSC 2010: 11P83; 05A17

1 Introduction

In a recent work, Andrews [1] defined combinatorial objects that he called singular overpartitions. Moreover, Andrews proved that these singular overpartitions, which depend on two parameters k and i, can be enumerated by the function C_k,i(n) which denotes the number of overpartitions of n in which no part is divisible by k and only parts ≡ ±i (mod k) may be overlined. Andrews also established the generating function for C_k,i(n). For k ≥ 3 and 1 ≤ i ≤ ⌊k2⌋, the generating function for C_k,i(n) is given by

∑n=0∞C¯k,i(n)qn=(−qi;qk)∞(−qk−i;qk)∞(qk;qk)∞(q;q)∞, (1.1)

where

(a;q)∞=∏n=0∞(1−aqn).

Furthermore, by elementary generating function manipulations, Andrews [1] proved that for all n ≥ 0,

C¯3,1(9n+3)≡C¯3,1(9n+6)≡0(mod3).

Since then, a number of congruences modulo 2, 3, 4, 8, 9, 16, 32 and 64 for C_k,i(n) have been discovered for some pairs (k, i), see for example, Ahmed and Baruah [2], Chen [3] Chen, Hirschhorn and Sellers [4], Naika and Gireesh [5], Shen [6] and Yao [7].

In this paper, we prove some new congruences modulo powers of 2 for (48,6)- and (48,18)-singular overpartitions.

2 Congruences modulo 2, 8 and 16 for C_{48, 6}(n)

In this section, we prove some congruences modulo 2, 8 and 16 for C_{48, 6}(n). It should be noted that congruences for C_{48, 6}(n) and C_{48, 18}(n) were not discovered before.

Theorem 2.1

For n ≥ 0,

C¯48,6(16n+8)≡0(mod8), (2.1)

C¯48,6(16n+12)≡0(mod8), (2.2)

and

C¯48,6(32n+28)≡0(mod16). (2.3)

Theorem 2.2

For n, k ≥ 0,

C¯48,6(22k+3n+31×22k−103)≡0(mod2), (2.4)

C¯48,6(22k+4n+17×22k+1−103)≡0(mod2), (2.5)

C¯48,6(22k+4n+23×22k+1−103)≡0(mod2), (2.6)

C¯48,6(22k+4n+49×22k−103)≡0(mod2). (2.7)

Theorem 2.3

For n, k ≥ 0,

C¯48,6(4k+2n+25×4k−103)≡1(mod2),if2n+1=m(3m−1)2forsomeintegerm,0(mod2),otherwise.

Proofs of Theorems 2.1–2.3

First, we introduce some notations. Throughout this paper, for any positive integer k, f_k is defined by

fk:=(qk;qk)∞=∏n=1∞(1−qkn),

and S(q) and T(q) are the Göllnitz-Gordon functions defined by

S(q)=1(q;q8)∞(q4;q8)∞(q7;q8)∞

and

T(q)=1(q3;q8)∞(q4;q8)∞(q5;q8)∞.

By (1.1), we have

∑n=0∞C¯48,6(n)qn=f122T(−q6)f1f6 (2.8)

and

∑n=0∞C¯48,18(n)qn=f122S(−q6)f1f6. (2.9)

It follows from (3.15) and (3.16) in [8] that

T(−q2)=12qf23f1f42−f1f4 (2.10)

and

S(−q2)=12f23f1f42+f1f4. (2.11)

Substituting (2.10) into (2.8) yields

2∑n=0∞C¯48,6(n)qn+3=1f1f3⋅f62−f3f1⋅f12f6. (2.12)

It follows from Lemma 2.6 in [9] that

f3f1=f4f6f16f242f22f8f12f48+qf6f82f48f22f16f24. (2.13)

Xia and Yao [10] proved the following 2-dissection formula for 1f1f3:

1f1f3=f82f125f22f4f64f242+qf45f242f24f62f82f12. (2.14)

By the binomial theorem, for any positive integer k and any prime p,

f1pk≡fppk−1(modpk). (2.15)

Substituting (2.14) and (2.13) into (2.12) and employing (2.15), we have

2∑n=0∞C¯48,6(2n)qn+1=1f14⋅f25f122f42f6−1f12⋅f42f6f24f8f12 (2.16)

and

2∑n=0∞C¯48,6(2n+1)qn+2=f42f65f12f2f32f122−f2f8f122f12f4f24≡f32f12⋅f42f63f2f122−1f12⋅f2f8f122f4f24(mod4). (2.17)

From Lemma 2.1 in [11],

1f12=f85f25f162+2qf42f162f25f8 (2.18)

and

1f14=f414f214f84+4qf42f84f210. (2.19)

Substituting (2.18) and (2.19) into (2.16), we obtain

∑n=0∞C¯48,6(4n)qn=2⋅1f14⋅1f1f3⋅f44f62−1f14⋅f3f1⋅f24f82f12f42f6 (2.20)

and

2∑n=0∞C¯48,6(4n+2)qn+1=f212f62f19f3f44−f22f3f44f12f15f6f82. (2.21)

Substituting (2.19), (2.14) and (2.13) into (2.20) and employing (2.15), we deduce that

∑n=0∞C¯48,6(8n)qn≡2f217f65f116f32f42f122−f213f8f122f112f43f24+4qf48f6f24f18f8f12≡2⋅1f32⋅f25f6−1f14⋅f29f8f122f43f24+4qf44f63(mod8) (2.22)

and

∑n=0∞C¯48,6(8n+4)qn=2f223f122f118f46f6+8f25f46f65f112f32f122−f212f6f24f112f8f12−4f2f45f8f122f18f24≡2⋅1f12⋅f215f122f46f6+8f46f2−1f143⋅f212f6f24f8f12−4f45f8f122f23f24(mod16). (2.23)

Substituting (2.18) and (2.19) into (2.22) and utilizing (2.15), we see that

∑n=0∞C¯48,6(16n+8)qn≡4f24f33−4f45f62f1f2f12−4qf15f62f242f34f12≡4f1F(q)(mod8), (2.24)

where

F(q)=f33f1⋅f24−f22f43f62f12−qf24f123f4. (2.25)

Hirschhorn, Garvan and Borwein [12] proved that

f33f1=f43f62f22f12+qf123f4. (2.26)

Substituting (2.26) into (2.25), we find that

F(q)=0. (2.27)

Congruence (2.1) follows from (2.24) and (2.27).

Substituting (2.18) and (2.19) into (2.23) and using (2.15) yields

∑n=0∞C¯48,6(16n+12)qn≡4f110f62f82f24f3f4+4f230f3f12f126f45f6≡4f22f63f82f12f4f12f12f3f6+f3(mod16). (2.28)

Define

∑n=0∞b(n)qn=f4f1f2. (2.29)

Replacing q by −q in (2.29) and employing (2.15) and (2.30), and using the fact that

(−q;−q)∞=f23f1f4, (2.30)

we have

∑n=0∞(−1)nb(n)qn=f1f42f24≡f1(mod4). (2.31)

Hence,

f4f1f2+f1≡∑n=0∞(1+(−1)n)b(n)qn=2∑n=0∞b(2n)q2n(mod4) (2.32)

and

f4f1f2−f1≡∑n=0∞(1−(−1)n)b(n)qn=2∑n=0∞b(2n+1)q2n+1(mod4). (2.33)

In view of (2.15), (2.28) and (2.32),

∑n=0∞C¯48,6(16n+12)qn≡8f22f63f82f12f4f12∑n=0∞b(2n)q6n≡8f6f82f2∑n=0∞b(2n)q6n(mod16). (2.34)

Congruences (2.2) and (2.3) follow from (2.34).

From (3.29) in [10],

f32f12=f44f6f122f25f8f24+2qf4f62f8f24f24f12. (2.35)

Substituting (2.18) and (2.35) into (2.17) and employing (2.15) yields

2∑n=0∞C¯48,6(4n+1)qn+1≡f26f34f16f4f12−f46f62f14f2f12f82(mod4) (2.36)

and

∑n=0∞C¯48,6(4n+3)qn+1≡f23f35f4f12f15f63−f2f62f82f14f12≡f33f1⋅f42f2−f82f2(mod2). (2.37)

Substituting (2.26) into (2.37) and using (2.15), we deduce that

∑n=0∞C¯48,6(4n+3)qn+1≡f45f62f23f12−f82f2+qf4f123f2≡qf4f123f2(mod2),

which implies that for n ≥ 0,

C¯48,6(8n+7)≡0(mod2). (2.38)

By (2.15) and (2.36),

2∑n=0∞C¯48,6(4n+1)qn+1≡1f12⋅f4f62f12−f42f62f23f12(mod4). (2.39)

Substituting (2.18) into (2.39) and employing (2.15), we have

2∑n=0∞C¯48,6(8n+5)qn+1≡f2f32f45f15f6f82−f22f32f13f6≡f32f6f4f1f2−f1(mod4). (2.40)

Thanks to (2.33) and (2.40),

∑n=0∞C¯48,6(8n+5)qn≡f32f6∑n=0∞b(2n+1)q2n≡∑n=0∞b(2n+1)q2n(mod2)(by(2.15)).

Therefore, for n ≥ 0,

C¯48,6(16n+13)≡0(mod2) (2.41)

and

C¯48,6(16n+5)≡b(2n+1)(mod2). (2.42)

It is well-known that

f1=∑n=−∞∞(−1)nqn(3n−1)2. (2.43)

Combining (2.43) and (2.31) yields

b(n)≡1(mod2),ifn=m(3m−1)2forsomeintegerm,0(mod2),otherwise. (2.44)

Thanks to (2.15) and (2.21),

2∑n=0∞C¯48,6(4n+2)qn+1≡f33f1−f3f12f1f6(mod4). (2.45)

Based on (2.15), (2.12) and (2.45), we see that for n ≥ 0,

C¯48,6(4n+10)≡C48,6(n)(mod2). (2.46)

By (2.46) and mathematical induction, for n, k ≥ 0,

C¯48,64kn+10(4k−1)3≡C¯48,6(n)(mod2). (2.47)

Replacing n by 8n + 7, 16n + 8, 16n + 12, 16n + 13 in (2.47) and using (2.38), (2.1), (2.2), (2.41) respectively, we obtain the congruences stated in Theorem 2.2.

It follows from (2.42) and (2.44) that

C¯48,6(16n+5)≡1(mod2),if2n+1=m(3m−1)2forsomeintegerm,0(mod2),otherwise. (2.48)

Theorem 2.3 follows form (2.47) and (2.48). This completes the proof.

3 Congruences modulo 2, 8 and 16 for C_{48, 18}(n)

We prove several congruences modulo 2, 8 and 16 for C_{48, 18}(n) in this section.

Theorem 3.1

For n ≥ 0,

C¯48,18(16n+11)≡0(mod8), (3.1)

C¯48,18(16n+15)≡0(mod8), (3.2)

C¯48,18(32n+15)≡0(mod16). (3.3)

Theorem 3.2

For n, k ≥ 0,

C¯48,18(22k+3n+7×22k−13)≡0(mod2), (3.4)

C¯48,18(22k+4n+25×22k−13)≡0(mod2), (3.5)

C¯48,18(22k+4n+17×22k+1−13)≡0(mod2), (3.6)

C¯48,18(22k+4n+23×22k+1−13)≡0(mod2). (3.7)

Theorem 3.3

For n, k ≥ 0,

C¯48,18(4k+2n+4k−13)≡1(mod2),if2n=m(3m−1)2forsomeintegerm,0(mod2),otherwise. (3.8)

Proofs of Theorems 3.1–3.3

In view of (2.9) and (2.11),

2∑n=0∞C¯48,18(n)qn=1f1f3⋅f62+f3f1⋅f12f6. (3.9)

Substituting (2.14) and (2.13) into (3.9) yields

2∑n=0∞C¯48,18(n)qn=f82f125f22f4f62f242+f4f16f242f22f8f48+qf45f242f24f82f12+qf82f12f48f22f16f24.

Therefore,

2∑n=0∞C¯48,18(2n)qn=f42f65f12f2f32f122+f2f8f122f12f4f24 (3.10)

and

2∑n=0∞C¯48,18(2n+1)qn=1f14⋅f25f122f42f6+1f12⋅f42f6f24f8f12. (3.11)

Substituting (2.18) and (2.19) into (3.11) yields

2∑n=0∞C¯48,18(4n+1)qn=f22f3f44f12f15f6f82+f212f62f19f3f44 (3.12)

and

∑n=0∞C¯48,18(4n+3)qn=1f14⋅f3f1⋅f24f82f12f42f6+2⋅1f1f3⋅1f14⋅f44f62. (3.13)

Substituting (2.19), (2.14) and (2.13) into (3.13) and employing (2.15), we have

∑n=0∞C¯48,18(8n+3)qn=f213f8f122f112f43f24+2f217f65f116f32f42f122+4qf48f6f24f18f8f12+8qf211f42f122f114f6≡1f14⋅f29f8f122f43f24+2⋅1f32⋅f25f6+4qf48f6f24f82f12(mod8) (3.14)

and

∑n=0∞C¯48,18(8n+7)qn=f212f6f24f112f8f12+4f2f45f8f122f18f24+2f223f122f118f46f6+8f25f46f65f112f32f122≡(1f14)3⋅f212f6f24f8f12+4f2f43f8f122f24+2⋅1f12⋅f215f122f46f6+8f46f64f2f122(mod16). (3.15)

Substituting (2.18) and (2.19) into (3.14) yields

∑n=0∞C¯48,18(16n+11)qn≡4f45f62f1f2f12+4f28f3f12f42f6+4qf15f62f242f34f12≡4f1F(q)(mod8),(by(2.15)) (3.16)

where F(q) is defined by (2.25). Congruence (3.1) follows from (2.27) and (3.16).

Substituting (2.18) and (2.19) into (3.15) yields

∑n=0∞C¯48,18(16n+15)qn≡4f110f62f82f24f3f4−4f230f3f12f126f45f6≡4f22f82f12f12f4f6f12f3f6−f3(by(2.15))≡8f22f82f12f12f4f6∑n=0∞b(2n+1)q6n+3(mod16)(by(2.33))≡8f82f12f2f6∑n=0∞b(2n+1)q6n+3(mod16)(by(2.15)). (3.17)

Congruences (3.2) and (3.3) follow from (3.17).

Based on (2.15) and (3.10),

2∑n=0∞C¯48,18(2n)qn≡f32f12⋅f42f2f6+1f12⋅f2f8f122f4f24(mod4). (3.18)

Substituting (2.18) and (2.35) into (3.18) and employing (2.15), we deduce

2∑n=0∞C¯48,18(4n)qn≡f26f62f16f4f12+f46f62f14f2f82f12≡1f12⋅f4f62f12+f2f62f12(mod4) (3.19)

and

∑n=0∞C¯48,18(4n+2)qn≡f23f3f4f12f15f6+f2f62f82f14≡f23f33f1+f82f2(mod2). (3.20)

Substituting (2.26) into (3.20) and employing (2.15), we deduce that

∑n=0∞C¯48,18(4n+2)qn≡f2f43f62f12+f82f2+qf23f123f4≡qf2f123(mod2). (3.21)

It follows from (3.21) that for n ≥ 0,

C¯48,18(8n+2)≡0(mod2). (3.22)

Substituting (2.18) into (3.19) and utilizing (2.15) and (2.32), we see that, modulo 4,

2∑n=0∞C¯48,18(8n)qn≡f2f32f45f15f6f82+f1f32f6≡f32f4f1f2f6+f1f32f6=f32f6f4f1f2+f1=2f32f6∑n=0∞b(2n)q2n≡2∑n=0∞b(2n)q2n,

which implies that for n ≥ 0,

C¯48,18(16n+8)≡0(mod2) (3.23)

and

C¯48,18(16n)≡b(2n)(mod2). (3.24)

By (2.15) and (3.12),

2∑n=0∞C¯48,18(4n+1)qn≡f3f12f1f6+f62f1f3(mod4). (3.25)

Thanks to (3.9) and (3.25), we see that for n ≥ 0,

C48,18(4n+1)≡C48,18(n)(mod2). (3.26)

By (3.26) and mathematical induction, we deduce that for n, k ≥ 0,

C48,18(4kn+4k−13)≡C48,18(n)(mod2). (3.27)

Replacing n by 8n + 2, 16n + 8, 16n + 11, 16n + 15 in (3.27) and using (3.22), (3.23), (3.1) and (3.2) respectively, we arrive at the congruences stated in Theorem 3.2. It follows from (2.44) and (3.24) that

C48,18(16n)≡1(mod2),if2n=m(3m−1)2forsomeintegerm,0(mod2),otherwise. (3.28)

Theorem 3.3 follows from (3.27) and (3.28). This completes the proof.

Acknowledgments

This work was supported by the National Science Foundation of China (grant no. 11701362).

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Received: 2018-11-03

Accepted: 2019-02-20

Published Online: 2019-05-11

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

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https://doi.org/10.1515/math-2019-0026

Keywords for this article

congruence; singular overpartition; theta function

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Arithmetic properties for Andrews’ (48,6)- and (48,18)-singular overpartitions

Article

Abstract

1 Introduction

2 Congruences modulo 2, 8 and 16 for C48, 6(n)

Theorem 2.1

Theorem 2.2

Theorem 2.3

Proofs of Theorems 2.1–2.3

3 Congruences modulo 2, 8 and 16 for C48, 18(n)

Theorem 3.1

Theorem 3.2

Theorem 3.3

Proofs of Theorems 3.1–3.3

Acknowledgments

References

Articles in the same Issue

Articles in the same Issue

Articles in the same Issue

2 Congruences modulo 2, 8 and 16 for C_{48, 6}(n)

3 Congruences modulo 2, 8 and 16 for C_{48, 18}(n)