New congruences for overcubic partition pairs

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1 New congruences for overcubic partition pairs M. S. Mahadeva Naika C. Shivashankar Department of Mathematics, Bangalore University, Central College Campus, Bangalore , Karnataka, India Department of Mathematics, M. S. Ramaiah University of Applied Sciences, Peenya Campus, Bengaluru , Karnataka, India DOI 0.55/tmj Abstract In this paper, we study congruence properties of overcubic partition pairs. Let b(n) denote the number of overcubic partition pairs of n. We will establish some new Ramanujan type congruences several infinite families of congruences modulo powers of satisfied by b(n). 00 Mathematics Subject Classification. P3. 05A5, 05A7 Keywords. Overcubic partitions; Congruences; Theta function. Introduction H. C. Chan in his papers [3,, 5] studied the congruence properties of the cubic partition function a(n), which is defined by a(n)q n = (q; q) (q ; q =. (.) ) f f Here throughout this paper, for any positive k, f k is defined by f k := ( q ki ) = (q k ; q k ). (.) i= Following H. C. Chan, B. Kim [7] studied its overpartiton analog in which the overcubic partition function a(n) is given by a(n)q n = ( q; q) ( q ; q ) (q; q) (q ; q = f ) f f. (.3) Hirschhorn [6] gave an elementary proof of the results satisfied by a(n), which appeared in Kim s paper [7], Sellers [] proved several arithmetic properties satisfied by a(n) by employing elementary generating function methods. Zhao Zhong [3] established congruences modulo 5, 7 9 for the partition function b(n), defined by b(n)q n = (q; q) (q ; q ) = f f. (.) Kim [] introduced two partition statistics to explain the congruences modulo 5 7 for b(n). Since b(n) counts a pair of cubic partitions, Kim [] called b(n), the number of cubic partition Tbilisi Mathematical Journal 0() (07), pp. 7. Tbilisi Centre for Mathematical Sciences. Received by the editors: May 07. Accepted for publication: 30 September 07. Download Date // :3 PM

2 M. S. Mahadeva Naika, C. Shivashankar pairs. Recently, Kim [9] studied congruence properties of b(n) whose generating function is given by b(n)q n = ( q; q) ( q ; q ) (q; q) (q ; q ) = f f f, (.5) where b(n) denote the number of overcubic partition pairs of n. Using arithmetic properties of quadratic forms modular forms, Kim [9] derived the following two congruences b(n + 7) 0 (mod 6 ), (.6) b(9n + 3) 0 (mod 3). (.7) The congruence (.6) appears to be incorrect. It seems that it is true for modulo 5. More recently, Lin [0] studied various arithmetic properties of b(n) modulo 3 5. example, he proved for any α, n 0 For for α 0, b(3 α (3n + )) 0 (mod 3) (.) b(30 5 α ) 0 (mod 5). (.9) With this motivation, we establish some new Ramanujan like congruences infinite families of congruences modulo powers of for overcubic partition pairs b(n) by using some elementary generating function dissection formulas. Our main results can be stated as follows Theorem.. For all n 0, we have b(n + 3) 0 (mod ), (.0) b(n + 7) 0 (mod 5 ), (.) b(3n + ) 0 (mod 5 ), (.) b(3n + ) 0 (mod 5 ), (.3) b(n + 6) 0 (mod 6 ), (.) b(3n + ) 0 (mod 6 ), (.5) b(6n + 56) 0 (mod 6 ), (.) b(n + ) 0 (mod 7 ). (.7) Theorem.. For α 0, n 0 r {, 6, 6, 7}, we have b( 5 α+ n + r 5 α+ ) 0 (mod 7 ). (.) Theorem.3. For α 0, n 0 r {0, 6, 5, 7, 90, 06}, we have Theorem.. For any α 0, n 0, we have b( 7 α+ n + r 7 α+ ) 0 (mod 7 ). (.9) b( 3 α+ n α+3 ) 0 (mod ), (.0) b( 3 α+ n α+3 ) 0 (mod ). (.) Download Date // :3 PM

3 New congruences for overcubic partition pairs 9 Theorem.5. For any α 0, n 0, we have Preliminary results b( 3 α+ n α+3 ) 0 (mod 6 ), (.) b( 3 α+ n α+3 ) 0 (mod 6 ). (.3) For ab <, we define Ramanujan s general theta function f(a, b), as The special cases of f(a, b) are f(a, b) := ϕ(q) := f(q, q) = n= ψ(q) := f(q, q 3 ) = f( q) := f( q, q ) = n= a n(n+)/ b n(n )/. (.) q n = ( q; q ) (q ; q ) = f 5 f f, (.) n= q n(n+)/ = (q ; q ) (q; q ) = f f, (.3) ( ) n q n(3n )/ = (q; q) = f, (.) where the product representations arise from Jacobi s triple product identity [, p.35, Entry 9] f(a, b) = ( a; ab) ( b; ab) (ab; ab). (.5) We need the following dissection formulas to prove our main results, Lemma.. The following -dissections hold f f = f 5 f 5 f = f f f + q f f f 5f, (.6) + q f f f 0. (.7) Lemma.. is a consequence of dissection formulas of Ramanujan, collected in Berndt s book [, p. 0, Entry 5]. 3 Proof of Theorem.. To prove Theorem.. we first establish the following lemma. Download Date // :3 PM

4 0 M. S. Mahadeva Naika, C. Shivashankar Lemma 3.. We have Proof. Substituting (.7) into (.5), b(n)q n = f 5 f 0f b(n + )q n = f 0 f 36f b(n + )q n = f 6 f 3 f b(n + 3)q n = f 3 f 3f + 96q f f q f f f, (3.) + 56q f f f, (3.) + 9q f f f 30, (3.3) + 56q f 0 f f 6. (3.) b(n)q n = f f f = f f f + q f f f, (3.5) From (.7), we have b(n)q n = f f f b(n + )q n = f f f (3.6). (3.7) f = f 56 f 56f f + q f f 5f = f f f + 96q f 3 f + q f 30 f 3f + 56q 3 f 0 f f + q f f f 3 Substituting (3.) (3.9) respectively into (3.6) (3.7), we find that + 56q f f f 0 (3.) + 6q 3 f 6 f f 30. (3.9) b(n)q n = f 5 f 0f + q f 0 f 36f + 96q f f q 3 f f f + 56q f f f (3.0) b(n + )q n = f 6 f 3f + q f 3 f 3f + 9q f f f 30 Lemma 3.. follows from (3.0) (3.). This completes the proof. + 56q 3 f 0 f f 6. (3.) q.e.d. Download Date // :3 PM

5 New congruences for overcubic partition pairs Proof of Theorem.. By binomial theorem for any positive integers k m, it is easy to see that f k m In view of (3.), (3.) can be written as b(n + )q n f f f f f b(n + 6)q n 6 f f f k m (mod k ). (3.) + 56qf f ( f f f + q f f f 0 b(n + )q n f 0 f 6 f f 6 6f f + 56f f ( f 5 f 5 f ) + 56qf f (mod 9 ), (3.3) (mod 9 ) (3.) + q f f ) f 5f + 56f 9 (mod 9 ) (3.5) Congruence (.) follows from (3.5). Equating odd even powers of q from both sides of (3.5), we obtain b(n + 6)q n 6 f f 5 f 3f + 56f 9 (mod 9 ) (3.) b(n + )q n f 6 f f 3 f Congruence (.7) is immediate from (3.7). From (3.), we have b(n + 3)q n f 3 f 3f Since f m f m (mod ), (3.) becomes b(n + 3)q n f f f f 3 f 5 f f b(n + 3)q n f 3 f 5 f f + 96q f 3 f f f (mod 9 ). (3.7) (mod ). (3.) (mod ). (mod ) (3.9) Download Date // :3 PM

6 M. S. Mahadeva Naika, C. Shivashankar b(n + 7)q n 96 f 3 f f f Congruences (.0) (.) follow from (3.) (3.0). In the view of (3.), we have b(n)q n f 5 After simplification, we find that f 0f f 5 f + 96q f f 3 ( f f f b(n)q n f f f 0 + q f f f 0 b(n + )q n 0 f Substituting (.7) into (3.), we see that b(n)q n f f 0 f 6 f f ( f f f + q f f f 0 + 0q f 56 f 0 f 0 b(n)q n f 6 f f + 0q f 00 f f 3 b(n + )q n 0 f 56 0f f 0 f 0 0 f 70 f 5f 0 (mod ). (3.0) ) f (mod ). f 0f (mod ) (3.) + 96f (mod ). (3.) ) + 0q f 00 f + q f f 76 f 7 + q f ( f f f f 76 f 7 + q f f f 0 + 3q f 5 f 50f ( f f f (mod ), + q f f f 0 ) 0 (mod ) (3.3) ) 5 (mod ). Extracting the terms containing q n+ from both sides of the above equation, we obtain b(3n + )q n 3 f 5 f 50f 3 f 0 f (mod ). (3.) Download Date // :3 PM

7 New congruences for overcubic partition pairs 3 Congruence (.3) follows from (3.). Again, substituting (.6) into (3.) extracting the odd even parts of the resulting equation, we find that b(6n + )q n 3 f 5 f 5 f (mod ) (3.5) Congruence (.) follows from (3.6). Substituting (.7) into (3.), we find that b(6n + 56)q n 6 f 5 f f f (mod ). (3.6) b(n + )q n 0 f 6 f f + 96 f f f + q f 30 f 3f + 3q f 50 f 7 f 76 (mod ), b(n + )q n 0 f 6 f f + 96 f f f (mod ) (3.7) b(n + )q n f 30 f 3f f f 0 f + 3 f 50 f 7 f 76 f f 5 f 5 f + 6q f 5 f f f (mod ), b(3n + )q n f f 5 f 5 f Congruences (.) (.5) follow from (3.) (3.9). Proof of Theorem.. In view of (3.) (3.), we have (mod ) (3.) b(3n + )q n 6 f 5 f f f (mod ). (3.9) q.e.d. b(n + 6)q n 6 f f 5 f 3 f 6 f 6 f 3 (mod 7 ). (.) Download Date // :3 PM

8 M. S. Mahadeva Naika, C. Shivashankar By the definition of ψ(q), (.) becomes b(n + 6)q n 6ψ 3 (q) (mod 7 ). (.) From [, p.9, Corollary], we have Employing (.3) into (.), we obtain ψ(q) = f(q 0, q 5 ) + qf(q 5, q 0 ) + q 3 ψ(q 5 ). (.3) b(n + 6)q n 6(f(q 0, q 5 ) + qf(q 5, q 0 ) + q 3 ψ(q 5 )) 3 (mod 7 ). After simplification, extracting the terms containing q 5n+ from both sides of the resulting equation, we get b((5n + ) + 6)q n 6qψ(q 5 ) 3 (mod 7 ), b((5 n + 9) + 6)q n 6ψ(q) 3 (mod 7 ) (.) b((5 n + 5i + ) + 6)q n 0 (mod 7 ), i = 0,, 3,. (.5) Combining (.) (.), we see that Therefore, by induction on α, we have Congruence (.) follows from (.5) (.7). 5 Proof of Theorem.3. b((5 n + 9) + 6) b(n + 6) (mod 7 ). (.6) b( 5 α n α ) b(n + 6) (mod 7 ). (.7) From Entry 7(iv) on page 303 in Berndt s book [], we have the 7-dissection: Substituting (5.) into (.), we have ψ(q) = f(q, q ) + qf(q, q 35 ) + q 3 f(q 7, q ) + q 6 ψ(q 9 ). (5.) b(n + 6)q n 6(f(q, q ) + qf(q, q 35 ) + q 3 f(q 7, q ) + q 6 ψ(q 9 )) 3 (mod 7 ). Download Date // :3 PM

9 New congruences for overcubic partition pairs 5 Simplifying extracting the terms containing q 7n+ from both sides of the resulting equation, we obtain b((7n + ) + 6)q n 6q ψ(q 7 ) 3 (mod 7 ), b((7 n + ) + 6)q n 6ψ(q) 3 (mod 7 ) (5.) b((7 n + 7i + ) + 6)q n 0 (mod 7 ), i = 0,, 3,, 5, 6. (5.3) Combining (.) (5.), we see that So, by induction on α, we have Congruence (.9) follows form (5.3) (5.5). 6 Proof of Theorem.. In view of (3.) (3.7),we have b((7 n + ) + 6) b(n + 6) (mod 7 ). (5.) b( 7 α n α ) b(n + 6) (mod 7 ). (5.5) b(n + )q n ψ 7 (q) (mod ). (6.) Recently, X. Yang et al. [] have proved the following results: for α 0, ( f 3 α n + 7 ) 3α 7 q n ψ 7 (q) (mod ) f Therefore, from (6.) (6.), we see that ( 3 α+3 n + 5 ) 3α+3 7 q n q ψ 7 (q 3 ) (mod ). (6.) b( 3 α+3 n α+3 )q n q ψ 7 (q 3 ) (mod ). (6.3) Equating the terms involving q 3n q 3n+ from both sides of (6.3), we obtain This completes the proof. b( 3 α+3 (3n) α+3 ) 0 (mod ), b( 3 α+3 (3n + ) α+3 ) 0 (mod ). Download Date // :3 PM

10 M. S. Mahadeva Naika, C. Shivashankar 7 Proof of Theorem.5. Combining (3.) (3.0), we obtain b(n + 7)q n 3ψ 7 (q) (mod 6 ). (7.) Remaining part of the proof is similar to the proof of Theorem.., hence we omit the details. Internal congruences for b(n). Theorem.. For n 0, we have b(6n + ) b(3n + ) (mod 6 ), b(6n + 56) b(3n + ) (mod ). Proof. First congruence relation follows from (3.5) (3.) second one follows from (3.6) (3.9). q.e.d. Theorem.. For n 0, we have Proof. From (3.), we have b(00n + 0) 96b(0n + ) (mod ), b(00n + 30) 96b(0n + 6) (mod ), b(00n + 70) 96b(0n + ) (mod ), b(00n + 90) 96b(0n + ) (mod ). b(n + )q n = f 0 f 36f ψ (q) (mod ). (.) Substituting (.3) into (.) then extracting the terms involving q 5n+ from both sides of the resulting equation, we obtain b(0n + 0)q n (f(q, q 3 ) f(q, q ) + qf(q, q 3 )f(q, q )ψ (q 5 ) From [, p.6, (.6.7)], we have Applying (.3) into (.), we see that + q ψ (q 5 )) (mod ). (.) ψ (q) qψ (q 5 ) = f(q, q )f(q, q 3 ). (.3) b(0n + 0)q n 0q ψ (q 5 ) + 96ψ (q) (mod ), Download Date // :3 PM

11 New congruences for overcubic partition pairs 7 b(0n + 0)q n 96 b(n + )q n 0q ψ (q 5 ) (mod ). (.) Extracting the terms involving q 5n+ from both sides of the above equation, we obtain b(00n + 50) 96b(0n + 0) 0b(n + ) (mod ). Theorem follows by equating the powers containing q 5n+i for i = 0,, 3, form both sides of (.). q.e.d. Acknowledgment The authors would like thank Prof. H. M. Srivastava anonymous referees for their helpful suggestions comments to improve our original manuscript. References [] G.E. Andrews B.C. Berndt, Ramanujan s Lost Notebooks, Part I. Springer-Verlag, New York, 005. [] B.C. Berndt, Ramanujan s Notebooks, Part III. Springer-Verlag, New York, 99. [3] H.C. Chan, Ramanujan s cubic continued fraction an analog of his most beautiful identity. Int. J. Number Theory 6 no. 3(00), [] H.C. Chan, Ramanujan s cubic continued fraction Ramanujan type congruences for a certain partition function. Int. J. Number Theory, 6 no. (00), 9-3. [5] H.C. Chan, Distribution of a certain partition function modulo powers of primes. Acta Math. Sin., 7(0), [6] M.D. Hirschhorn, A note on overcubic partitions. New Zeal J. Math., (0), 9-3. [7] B. Kim, The overcubic partition function mod 3. Ramanujan Rediscovered, Ramanujan Math. Soc. Lect. Notes Ser., (00), [] B. Kim, Partition statistics for cubic partition pairs. Electron. J. Combin., (0), # P. [9] B. Kim, On partition congruences for overcubic partition pairs. Commun. Korean Math. Soc., 7(0), 77-. [0] B. L. S. Lin, Arithmetic properties of overcubic partition pairs. Electron. J. Combin. (3)(0), #P3.35. [] J. A. Sellers, Elementary proofs of congruences for the cubic overcubic partition functions. Australas. J. Combin. 60()(0), Download Date // :3 PM

12 M. S. Mahadeva Naika, C. Shivashankar [] X. Yang, S. P. Cui B. L. S. Lin, Overpartition function modulo powers of. Ramanujan J. DOI: 0.007/s [3] H. Zhao Z. Zhong, Ramanujan type congruences for a partition function. Electron. J. Combin. (0), #P5. Download Date // :3 PM