Congruences modulo 8 for \((2,\, k)\) -regular overpartitions for odd \(k > 1\)

Arabian Journal of Mathematics, Dec 2017

In this paper, we study various arithmetic properties of the function \(\overline{p}_{2,\,\, k}(n)\), which denotes the number of \((2,\,\, k)\)-regular overpartitions of n with odd \(k > 1\). We prove several infinite families of congruences modulo 8 for \(\overline{p}_{2,\,\, k}(n)\). For example, we find that for all non-negative integers \(\beta , n\) and \(k\equiv 1\pmod {8}\), \(\overline{p}_{2,\,\, k}(2^{1+\beta }(16n+14))\equiv ~0\pmod {8}\).

A PDF file should load here. If you do not see its contents the file may be temporarily unavailable at the journal website or you do not have a PDF plug-in installed and enabled in your browser.

Alternatively, you can download the file locally and open with any standalone PDF reader:

https://link.springer.com/content/pdf/10.1007%2Fs40065-017-0195-z.pdf

Congruences modulo 8 for \((2,\, k)\) -regular overpartitions for odd \(k > 1\)

Arabian Journal of Mathematics Congruences modulo 8 for (2, k )-regular overpartitions for odd k > 1 Chandrashekar Adiga 0 1 2 M. S. Mahadeva Naika 0 1 2 D. Ranganatha 0 1 2 C. Shivashankar 0 1 2 Mathematics Subject Classification 0 1 2 0 D. Ranganatha ( 1 M. S. Mahadeva Naika 2 C. Adiga Department of Studies in Mathematics, University of Mysore , Manasagangotri, Mysore, Karnataka 570006 , India In this paper, we study various arithmetic properties of the function p2, k (n), which denotes the number of (2, k)-regular overpartitions of n with odd k > 1. We prove several infinite families of congruences modulo 8 for p2, k (n). For example, we find that for all non-negative integers β, n and k ≡ 1 (mod 8), 5, 4 + 1, 2 + 2 + 1, 2 + 1 + 1 + 1, 1 + 1 + 1 + 1 + 1. - 05A15 · 05A17 · 11P83 1 Introduction Let N denote the set of natural numbers and N0 = N ∪ {0}. A partition of an integer n ∈ N is a non-increasing sequence of positive integers that sum to n. For a positive integer > 1, a partition is called -regular if none of the parts is divisible by . For example, the 3-regular partitions of 5 are An overpartition of n ∈ N0 is a partition of n in which the first occurrence of a number may be overlined. For example, the overpartitions of 3 are 3, 3, 2 + 1, 2 + 1, 2 + 1, 2 + 1, 1 + 1 + 1, 1 + 1 + 1. The generating function for p(n), the number of overpartitions of n with p(0) = 1 is given by [ 5 ] ∞ n=0 p(n)qn = n=1 An extensive study of overpartitions can be found in the work of Corteel and Lovejoy [ 5 ]. Let po(n) denote the number of overpartitions of n into odd parts. The generating function for po(n) is given by Many mathematicians have extensively studied the arithmetic properties of po(n) and they have also established several Ramanujan-type congruences satisfied by po(n) (for example, one can see [ 4,7 ]). Let A (n) denote the number of -regular overpartitions of n. The generating function for A (n) is given by where and for m ∈ N ∞ n=0 A (n)qn = f2 f 2 , This function was introduced and investigated by Lovejoy [ 9 ]. Later, Shen [ 13 ] discovered several Ramanujanlike congruences for A3(n) and A4(n). Since then, a number of congruence properties for various -regular overpartition functions have been proved. (For example, one can see [ 2,3,10,12 ].) Very recently, the arithmetic properties of -regular overpartition pairs have been studied in [11]. Definition 1.1 A partition of n is said to be a (2, k)-regular overpartition of n, if it is both 2 and k-regular overpartition of n. Motivated by the above works, in this paper we prove infinite families of congruences modulo 8 for p2, k (n) which enumerate the number of (2, k)-regular overpartitions of n, for infinitely many values of odd k > 1. For example, we prove the following theorems: Theorem 1.2 If n, α ∈ N0 and k ≡ 1 (mod 4), then p2, k (22+αn) ≡ p2, k (22n) (mod 8), p2, k (21+α(4n + 2)) ≡ 7 p2, k (4n + 2) (mod 8). Theorem 1.3 Let p ≡ 5, 7 (mod 8) and k ≡ 5 (mod 8) such that k p = 1. Then for all n, β ∈ N0, we have It is easy to see that the generating function for p2, k (n) is = (q; q2)∞(−qk ; q2k )∞ (1.1) (1.2) (1.3) (1.4) For example, setting k = 3 in (1.4), we obtain ∞ n=0 p2, 3(n)qn The (2, 3)-regular overpartitions of the integer 6 are We note that work of Lin [ 8 ] on overpartition pairs into odd parts gives numerous congruences for p2,3(n) modulo 3. 2 Set of preliminary results In order to prove the main congruences of this paper, we collect some dissection formulas in this section. For |ab| < 1, Ramanujan’s general theta function f (a, b) is defined by [ 1 ] The following lemma is a consequence of Entry 25 of (v) and (vi) in [1, pp. 35–36]. Lemma 2.1 The following 2-dissection formulas are true: f 5 Ranganatha [ 12 ] established the p-dissection formula for f12 which can be stated as follows: 2 Lemma 2.2 [12, Theorem 3.2] If p ≥ 5 is a prime and ± p − 1 6 := p−61 , if p ≡ 1 (mod 6), − p−1 , if p ≡ −1 (mod 6), 6 From [ 6 ], we recall the following p-dissection formula for ff212 : Lemma 2.3 [6, Theorem 2.1] For any odd prime p, we have Furthermore, m22+m ≡ p28−1 (mod p) for 0 ≤ m ≤ p−23 . f (a, b) = ∞ n=−∞ an(n+1)/2bn(n−1)/2. For notational convenience, in this section, we assume that all congruences are modulo 8, k ∈ N is odd and p is a prime, unless stated otherwise. We first establish the following generating function for p2, k (2n) modulo 8. Lemma 3.1 For any odd k > 1 ∈ N and n ∈ N0, we have Proof Substituting (2.1) and (2.2) in (1.4), we find that (3.1) (3.2) (3.3) (3.4) (3.5) (3.6) (3.7) (3.8) (3.9) ∞ n=0 p2, k (n)q n = = − 4q k+1 f4k f4 f126 f126k + 4q k f43 f83k + 4q f49 + 6q k −21 f8k f4k f8 Extracting the terms involving even powers of q in the above identity where k > 1 ∈ N is odd, we obtain In view of (2.2), we have for any s ∈ N Using (3.4), we can rewrite (3.3) as follows: 1 f 2 f 2 = 1 s It is easy to check that, for any m, Employing (3.6)–(3.8) into (3.5), we obtain (3.1). Lemma 3.2 For any odd k > 1 ∈ N and n ∈ N0, we have p2, k (2n)q n ≡ f 5 f 5 f 3 f 3 4 4k 8 8k Proof From (3.2) with odd k > 1 , we have By (3.4), we find that ∞ n=0 p2, k(2n + 1)qn = 2 In view of (3.6), (3.7), (3.10) and (3.11), we obtain the required congruence. Theorem 3.3 If n,α ∈ N0 and k ≡ 1 (mod 4), then Proof Extracting the odd and even powers of q in (3.1), where k ≡ 1 (mod 4), we obtain p2, k(22+αn) ≡ p2, k(22n) (mod 8), p2, k(21+α(4n + 2)) ≡ 7p2, k(4n + 2) (mod 8). ∞ n=0 ∞ n=0 Applying (2.1), we deduce that Substituting (3.16) into (3.14) and then using (3.6)–(3.8) in the resulting congruence, we can rewrite (3.14) as Congruences (3.12) and (3.13) follow from the above two congruences and by induction on α ≥ 0. Lemma 3.4 For any k ≡ 1 (mod 8) and n ∈ N0, we have ∞ n=0 p2, k(8n + 6)qn ≡ 4q 3k4−3 f156k f25k f322k f42k + 4q k−41 f43 f23k + 4q k−21 f43 f23k + 4 f156 f25 (mod 8). (3.17) f322 f42 (3.10) (3.11) (3.12) (3.13) (3.14) (3.15) (3.16) Proof Substituting (3.4) into (3.15), we see that p2, k (4n + 2)qn ≡ 2q k −21 f4 f43k f85 f85k and for 1 ≤ j ≤ p − 1, Proof If k ≡ 1, then terms appearing on the right side of (3.17) are powers of q2 and thus equating the odd Congruence (3.22) follows from the above congruence and (3.13). Next, we turn to prove (3.23). From (3.17), we have , Invoking (3.20) and (3.21) into (3.19), we obtain the required congruence. prime. The Legendre symbol is defined by In order to state the congruence properties for p2, k (n), we need the following definition: Let p ≥ 3 be a a p if a is a quadratic residue modulo p and p a, := −1 if a is a quadratic nonresidue modulo p and p a, Theorem 3.5 Let p ≡ 5, 7 (mod 8) and k ≡ 1 (mod 8) such that k p = 1. Then for all n and β ∈ N0, we Extracting the odd powers of q in (3.18) for k ≡ 1 (mod 8), we find that where k ≡ 1 and Now, consider the following congruence relations: ∞ n=0 a(n)q n p−1 where − 2 ≤ m1, r1 ≤ p−21 , 0 ≤ m2, r2 ≤ p−23 . Because −2 p = −1, the first congruence holds if and only if m1 = r1 = ± p6−1 and the second congruence holds if and only if m2 = r2 = p−21 , since −p2k = −1. p2−1 Substituting Lemmas 2.2 and 2.3 into (3.25), extracting the terms involving q pn+3 8 in the resulting identity, canceling q3 p28−1 on both sides and then replacing q p by q, we deduce ∞ n=0 a pn + 3 · The terms appearing on the right side are powers of q p and thus for 1 ≤ j ≤ p − 1, we have and From (3.26) and by induction on β ∈ N0, we find that for all n ∈ N0 2 a p n + 3 · Finally, replacing n by p2β+2n + p2β+1 j + 3 · p2β+2−1 in (3.24) and then using (3.29), we obtain (3.23). This 8 Theorem 3.6 Let p ≡ 5, 7 (mod 8) and k ≡ 5 (mod 8) such that k p = 1. Then for all n, β ∈ N0, we have Proof Extracting the terms involving odd powers of q in (3.18) where k ≡ 5, we see that Now, consider the following congruence relations: 8 · (3m12 + m1) + (3r12 + r1) ≡ 3 · Substituting Lemmas 2.2 and 2.3 into (3.31), extracting the terms involving qpn+3· p24−1, we obtain ∞ n=0 b pn + 3 · p2 − 1 qn = p2qp3k8−3 f156kp f25kp 3 f322kp f42kp + qpk−41 f43p f2kp + qpk−85 fp3 fk3p 4 where 1 ≤ j ≤ p − 1. From (3.33) and by induction on β, we find that for n ∈ N0, By (3.35) and (3.34), we find that which implies that and we have and ≡ 0 (mod 2). Theorem 3.7 Let p ≥ 3andk ≡ 3 (mod 8)suchthat −k p = −1.Thenforα,β,n ∈ N0 and1 ≤ j ≤ p−1, p2, k(22+αn) ≡ p2, k(22n) (mod 8) p2, k 22+α 2p2β+2n + 2p2β+1 j + p2β+2 ≡ 3p2,k 4p2β+2n + 4p2β+1 j + 2p2β+2 (mod 8). (3.32) (3.33) (3.34) (3.35) (3.36) (3.37) (3.38) Proof Extracting the even and odd powers of q in (3.1), where k ≡ 3 (mod 4), we find that p2, k (8n) ≡ p2, k (4n), ∞ n=0 p2, k (8n + 4)q ≡ 3 n p2, k (4n + 2)q n Extracting the even and odd powers of q in (3.41) where k ≡ 3 and then using (3.39) and (3.40), we obtain which implies that ∞ n=0 ∞ n=0 (3.39) (3.40) (3.41) (3.42) (3.43) (3.44) (3.45) (3.46) Congruence (3.37) follows from (3.42) and by induction on α ∈ N0. ∞ n=0 c(n)q n Now, consider the congruence relation where 0 ≤ m, r ≤ p−23 and p ≥ 3. The above congruence relation holds if and only if m = r = p−21 , since p2−1 = −1. Applying Lemma 2.3 into (3.44) and then extracting the terms of the form q pn+ 2 , we obtain ∞ n=0 c pn + where 1 ≤ j ≤ p − 1. From the above two identities and by induction, we find that From (3.43), (3.44) and (3.46), we have p2, k 8 p2β+2n + 8 p2β+1 j + 4 p2β+2 ≡ 3 p2, k 4 p2β+2n + 4 p2β+1 j + 2 p2β+2 . Proof In view of (3.4), (3.6) and (3.7), we deduce that Combining (3.40) and (3.49) and then extracting the odd and even powers of q where k obtain ≡ 3 (mod 4), we n n p2, k (8n + 2)q , Substituting (3.52) in (3.50) and then extracting the terms involving odd powers of q where k ≡ 3, we obtain Lemma 3.9 For all n ∈ N0 and k ≡ 7 (mod 8), we have Lemma 3.8 If k ≡ 3 (mod 8), then for all n ∈ N0, we have and and ∞ (mod 4). (3.49) (3.47) (3.48) (3.50) (3.51) (3.52) (3.53) (3.54) (3.55) p2, k (16n + 6)q n k+1 ≡ 4 f8 f1 + 4q 8 fk3 f43 we obtain (3.54). Congruence (3.55) follows from (3.52) and (3.50). Proof Substituting (3.53) in (3.51) and then extracting the terms involving the even powers of q where k ≡ 7, 1 ≤ j ≤ p − 1, we have and Proof From (3.47), we have where k ≡ 3 and where 1 ≤ j ≤ p − 1. By (3.60) and (3.61), we have The congruence (3.56) follows from (3.58) and (3.62). The proof of the congruence (3.57) is analogous to the proof of (3.56), except that in place of (3.47), (3.48) is used. Theorem 3.11 Let p ≡ 5,7 (mod 8) and k ≡ 7 (mod 8) such that kp = 1. Then for all n,β ∈ N0 and For p ≡ 1,7 and 0 ≤ m,r ≤ p−23, the following congruences hold true if and only if m = r = p−21, since −2k p = −1. Substituting Lemma 2.3 into (3.59) and then extracting the terms involving qpn+5· p28−1, we obtain ∞ n=0 p2 − 1 qn = qpk−83 f23p fk3p + qpk−21 fp3 f43kp, d pn + 5 · 8 and so we have ∞ n=0 p2, k(16n + 10) ≡ 4d(n), Proof Rewrite (3.54) in the form where k ≡ 7 and Now, consider the following congruence relations: 2 ≡ 3 · p−23 . The first congruence relation holds true if and only if = 1, the second congruence relation holds true if and only if m2 = r2 = p−21 . Substituting Lemmas 2.2, 2.3 into (3.66) and then extracting the terms involving q pn+3· p28−1 , we obtain which implies that ∞ n=0 f pn + 3 · p2 − 1 qn = p2 f8 p f p + q p k +81 fk3p f43p, 8 where 1 ≤ j ≤ p − 1. From the above two identities, we deduce that Finally, replacing n by p2β+2n + p2β+1 j + 3 · p2β+82−1 in (3.65) and then using the above identity, we arrive at (3.63). The proof of the congruence (3.64) is similar to the proof of (3.63), except that in place of (3.54), (3.55) is used. Theorem 3.12 Let p ≡ 5, 7 (mod 8) and k ≡ 1 (mod 8) such that have k p = 1. Then for all n, β ∈ N0, we p2, k (8n + 7) ≡ 0 (mod 8) p2, k 8 p2β+2n + p2β+1 + 3 p2β+2 Proof Extracting the terms involving the odd powers of q in (3.9) where k ≡ 1 (mod 4), we obtain p2, k (4n + 3)qn ≡ 4q k −21 f23 f43k + 4 f29 + 4q 3k4−3 f29k + 4q k −41 f43 f23k . (3.65) (3.66) (3.67) (3.68) (3.69) From (3.25) and (3.71), we have which implies that ∞ n=0 p2, k (8n + 3)q n ≡ 4 a(n)qn, ∞ n=0 p2, k (8n + 3) ≡ 4a(n). Now, the congruence (3.68) follows from the above congruence and (3.29). If we extract the even and odd powers of q in (3.69) where k ≡ 5, we obtain the following lemma: Lemma 3.13 If k ≡ 5 (mod 8) and n ∈ N0, then and p2, k (8n + 3)q n For k ≡ 1, equating the coefficients of odd powers of q, we arrive at (3.67). Again, from (3.69), we have Employing above congruence in (3.70), we obtain p2, k (8n + 3)q ≡ n Lemma 3.14 If k ≡ 1 (mod 8) and for n ∈ N0, then Proof Let k ≡ 1 (mod 4). By (3.9), we see that p2, k (8n + 5)q ≡ n 4q k −21 f13 f43k + 4q(k−1)/8 fk3 f43 In view of (3.52), (3.53) and (3.75), we have ∞ n=0 p2, k (4n + 1)q n ≡ 2 Extracting the terms involving q2n+1 in (3.76) where k ≡ 1, we obtain the required congruence. and and ≡ ∞ n=0 p2, k (16n + 6)qn, p2, k (8n + 3) ≡ p2, k (16n + 6). p2, k (16n + 10) ≡ p2, k (8n + 5). (3.77) (3.78) (3.79) (3.80) (3.81) (3.82) (3.83) (3.84) (3.85) n=0 which yields Changing n to 2n + 1 in (3.79), we obtain Finally, from (3.63), (3.84) and (3.64), (3.85), we obtain (3.82) and (3.83), respectively. It follows from (3.81) with k ≡ 3 and (3.48), In view of (3.57) and the above congruence, we obtain (3.78). Theorem 3.16 Let p ≡ 5, 7 (mod 8) and k ≡ 7 (mod 8) such that k p n, β ∈ N0, we have p2, k (16n + 14) ≡ p2, k (8n + 7). = 1. Then for 1 ≤ j ≤ p − 1 and p2, k 8 p2β+2n + 8 p2β+1 j + 3 p2β+2 ≡ 0 (mod 8) p2, k 8 p2β+2n + 8 p2β+1 j + 5 p2β+2 Proof Let k ≡ 7. From (3.6), (3.54) and (3.81), we see that ∞ n=0 We can easily prove the following theorem with the help of Lemmas 3.13, 2.2 and 2.3: ≡ 0 In view of Lemmas 2.3 and 3.14, we can prove the following congruence: = 1. Then for n, β ∈ N0 and 1 ≤ j ≤ p − 1, we have ≡ 0 and Acknowledgements The authors would like to thank the anonymous referees for their valuable suggestions. Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http:// creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. 1. Adiga , C. ; Berndt , B.C. ; Bhargava , S. ; Watson, G.N. : Chapter 16 of Ramanujan's second notebook: theta functions and q-series . Mem. Am. Math. Soc. 315 , 1 - 91 ( 1985 ) 2. Adiga , C. ; Ranganatha , D. : Congruences modulo powers of 2 for -regular overpartitions . J. Ramanujan Math. Soc . 32 , 147 - 163 ( 2017 ) 3. Alanazi , A.M. ; Munagi , A.O. ; Sellers , J.A. : An infinite family of congruences for -regular overpartitions . Integers 16 , #A37 ( 2016 ) 4. Chen , S.C. : On the number of overpartitions into odd parts . Discrete Math . 325 , 32 - 37 ( 2014 ) 5. Corteel , S. ; Lovejoy , J. : Overpartitions. Trans. Am. Math. Soc . 356 , 1623 - 1635 ( 2004 ) 6. Cui , S.-P.; Gu , N.S.S.: Arithmetic properties of -regular partition . Adv. Appl . Math. 51 , 507 - 523 ( 2013 ) 7. Hirschhorn , M.D.; Sellers , J.A. : Arithmetic properties of overpartitions into odd parts . Ann. Comb. 10 , 353 - 367 ( 2006 ) 8. Lin , B.L.S.: Arithmetic properties of overpartitions pairs into odd parts . Electron. J. Comb . 19 ( 2 ), P17 ( 2012 ) 9. Lovejoy , J.: Gordon's theorem for overpartitions . J. Comb. Theory Ser. A 103 , 393 - 401 ( 2003 ) 10. Naika , M.S.M. ; Gireesh , D.S. : Congruences for Andrews's singular overpartitions . J. Number Theory 165 , 109 - 130 ( 2016 ) 11. Naika , M.S.M. ; Shivashankar , C. : Arithmetic propertities of -regular overpartition pairs . Turk. J. Math . 41 ( 3 ), 756 - 774 ( 2017 ) 12. Ranganatha , D. : On some new congruences for -regular overpartitions . Palest. J. Math. 7 ( 1 ), 345 - 362 ( 2018 ) 13. Shen , E.Y.Y.: Arithmetic properties of -regular overpartitions . Int. J. Number Theory 12 ( 3 ), 841 - 852 ( 2016 ) 14. Xia , E.X.W. ; Yao , O.X.M.: Analogues of Ramanujan's partition identities . Ramanujan J . 31 , 373 - 396 ( 2013 )


This is a preview of a remote PDF: https://link.springer.com/content/pdf/10.1007%2Fs40065-017-0195-z.pdf

Chandrashekar Adiga, M. S. Mahadeva Naika, D. Ranganatha, C. Shivashankar. Congruences modulo 8 for \((2,\, k)\) -regular overpartitions for odd \(k > 1\), Arabian Journal of Mathematics, 2017, 1-15, DOI: 10.1007/s40065-017-0195-z