#A22 INTEGERS 7 (207) NEW CONGRUENCES FOR `-REGULAR OVERPARTITIONS Shane Chern Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania shanechern@psu.edu Received: 0/6/6, Accepted: 5/4/7, Published: 5/24/7 Abstract Recently, Shen (206) and Alanazi et al. (206) studied the arithmetic properties of the `-regular overpartition function A`(n), which counts the number of overpartitions of n into parts not divisible by `. In this note, we will present some new congruences modulo 5 when ` is a power of 5.. Introduction A partition of a natural number n is a nonincreasing sequence of positive integers whose sum is n. For example, 6 = 3 + 2 + is a partition of 6. Let p(n) denote the number of partitions of n. We also agree that p(0) =. It is well-known that the generating function of p(n) is given by p(n)q n =, (q; q) where we adopt the standard notation (a; q) = Y ( aq n ). For any positive integer `, a partition is called `-regular if none of its parts are divisible by `. Let b`(n) denote the number of `-regular partitions of n. We know that its generating function is b`(n)q n = (q`; q`) (q; q). On the other hand, an overpartition of n is a partition of n in which the first occurrence of each distinct part can be overlined. Let p(n) be the number of overpartitions of n. We also know that the generating function of p(n) is p(n)q n = ( q; q) = (q2 ; q 2 ) (q; q) (q; q) 2.
INTEGERS: 7 (207) 2 Many authors have studied the arithmetic properties of b`(n) and p(n). We refer the interested readers to the Introduction part of [6] and references therein for detailed description. In [5], Lovejoy introduced a function A`(n), which counts the number of overpartitions of n into parts not divisible by `. According to Shen [6], this type of partition can be named as `-regular overpartition. He also obtained the generating function of A`(n), that is, A`(n)q n = (q`; q`) 2 (q 2 ; q 2 ) (q; q) 2 (q 2`; q 2`). () Meanwhile, he presented several congruences for A 3 (n) and A 4 (n). For A 3 (n), he got A 3 (4n + ) 0 (mod 2), A 3 (4n + 3) 0 (mod 6), A 3 (9n + 3) 0 (mod 6), A 3 (9n + 6) 0 (mod 24). More recently, Alanazi et al. [] further studied the arithmetic properties of A`(n) under modulus 3 when ` is a power of 3. They also gave some congruences satisfied by A`(n) modulo 2 and 4. In this note, our main purpose is to study the arithmetic properties of A`(n) when ` is a power of 5. When ` = 5 and 25, we connect A`(n) with r 4 (n) and r 8 (n) respectively, where r k (n) denotes the number of representations of n by k squares. The method for ` = 5 also applies to other prime `. When ` = 25, we show that A 25 (25n) A 25 (625n) (mod 5). This can be regarded as an analogous result of the following congruence for overpartition function p(n) proved by Chen et al. (see [4, Theorem.5]) p(25n) p(625n) (mod 5). When ` = 5 with 4, we obtain new congruences similar to a result of Alanazi et al. (see [, Theorem 3]), which states that A 3 (27n + 8) 0 (mod 3) holds for all and 3. 2. New Congruence Results 2.. ` = 5 One readily sees from the binomial theorem that for any prime p, (q; q) p (q p ; q p ) (mod p). (2)
INTEGERS: 7 (207) 3 Setting p = 5 and applying it to (), we have Now let (q; q) A 5 (n)q n 2 4 (q 2 ; q 2 (mod 5). (3) ) '(q) := n= It is well-known that (see [2, p. 37, Eq. (22.4)]) We therefore have q n2. '( q) = (q; q)2 (q 2 ; q 2 ). Theorem. For any positive integer n, we have ( r 4 (n) if n is even A 5 (n) r 4 (n) if n is odd (mod 5). (4) We know from [3, Theorem 3.3.] that r 4 (n) = 8d (n) where d (n) = d. d n, 4-d Let p 6= 5 be an odd prime and k be a nonnegative integer. One readily verifies that 4k+3 4 p i (k + ) i 0 (mod 5). i=0 i= Note also that d (n) is multiplicative. Thus we conclude Theorem 2. Let p 6= 5 be an odd prime and k be a nonnegative integer. Let n be a nonnegative integer. We have where i 2 {, 2,..., p }. A 5 (p 4k+3 (pn + i)) 0 (mod 5), (5) Example. If we take p = 3, k = 0, and i =, then holds for all. A 5 (8n + 27) 0 (mod 5) (6)
INTEGERS: 7 (207) 4 We also note that if an odd prime p is congruent to 9 modulo 0, then + p 0 (mod 5). We therefore have A 5 (p(pn + i)) 0 (mod 5) for i 2 {, 2,..., p }. On the other hand, if we require + p + p 2 0 (mod 5), then (2p + ) 2 3 (mod 5). However, since ( 3 5) = (here ( ) denotes the Legendre symbol), such p does not exist. The above observation yields Theorem 3. Let p 9 (mod 0) be a prime and n be a nonnegative integer. We have A 5 (p(pn + i)) 0 (mod 5), (7) where i 2 {, 2,..., p }. Example 2. If we take p = 9 and i =, then holds for all. A 5 (36n + 9) 0 (mod 5) (8) We should mention that this method also applies to other primes `. In fact, if we set p = ` in (2) and apply it to (), then ( r` (n) if n is even A`(n) (mod `). (9) r` (n) if n is odd Recall that the explicit formulas of r 2 (n) and r 6 (n) are also known. From [3, Theorems 3.2. and 3.4.], we have r 2 (n) = 4 d n (d), and r 6 (n) = 6 d n (n/d)d 2 4 d n (d)d 2, where 8 >< n (mod 4), (n) = n 3 (mod 4), >: 0 otherwise. Through a similar argument, we conclude that Theorem 4. For any nonnegative integers n, k, odd prime p, and i 2 {, 2,..., p }, we have A 3 (p 2k+ (pn + i)) 0 (mod 3) where p 3 (mod 4), A 3 (p 3k+2 (pn + i)) 0 (mod 3) where p (mod 4), A 7 (p 6k+5 (pn + i)) 0 (mod 7) where p 6= 7.
INTEGERS: 7 (207) 5 2.2. ` = 25 Analogous to the congruences under modulus 3 for A 9 (n) obtained by Alanazi et al. [], we will present some arithmetic properties of A 25 (n) modulo 5. Rather than using the technique of dissection identities, we build a connection between A 25 (5n) and r 8 (n) and then apply the explicit formula of r 8 (n). It is necessary to mention that this method also applies to the results of Alanazi et al. as the following relation holds A 9 (n) ( ) n r 8 (n) (mod 3). Note that A 25 (n)q n = (q25 ; q 25 ) 2 (q 2 ; q 2 ) (q 50 ; q 50 ) (q; q) 2 0 (q 5 ; q 5 ) 2 5 @ (q 0 ; q 0 p(n)q n A (mod 5). ) Extracting powers of the form q 5n from both sides and replacing q 5 by q, we have 0 (q; q) A 25 (5n)q n 2 5 @ (q 2 ; q 2 p(5n)q n A ) '( q) 8 (mod 5). Here we use the following celebrated result due to Treneer [7] p(5n)q n '( q) 3 (mod 5). It is known that the explicit formula of r 8 (n) (see [3, Theorem 3.5.4]) is given by r 8 (n) = 6( ) n 3 (n), where 3 (n) = d n ( ) d d 3. We therefore conclude that Theorem 5. For any positive integer n, we have A 25 (5n) 3 (n) (mod 5). (0) One also readily deduces several congruences from Theorem 5.
INTEGERS: 7 (207) 6 Theorem 6. Let p 6= 5 be an odd prime and k be a nonnegative integer. Let n be a nonnegative integer. We have where i 2 {, 2,..., p }. Proof. It is easy to see that A 25 (5p 4k+3 (pn + i)) 0 (mod 5), () 3 (p4k+3 ) = 4k+3 i=0 p 3i (k + ) 4 i 0 (mod 5). i= Note also that 3 (n) is multiplicative. The theorem therefore follows. Furthermore, if p 9 (mod 0) is a prime, then + p 3 0 (mod 5). This yields Theorem 7. Let p 9 (mod 0) be a prime and n be a nonnegative integer. We have A 25 (5p(pn + i)) 0 (mod 5), (2) where i 2 {, 2,..., p }. 2.3. ` = 25 From the generating function (), we have A 25 (n)q n = (q25 ; q 25 ) 2 (q 2 ; q 2 ) (q; q) 2 (q 250 ; q 250 = '( q 25 ) p(n)q n. ) Extracting terms of the form q 25n and replacing q 25 by q, we have p(25n)q n. A 25 (25n)q n = '( q) According to [4, Eq. (5.3)], we know that p(25(5n ± )) 0 (mod 5). We also know from [2, p. 49, Corollary (i)] that '( q) = '( q 25 ) 2qM ( q 5 ) + 2q 4 M 2 ( q 5 ), where M (q) = f(q 3, q 7 ) and M 2 (q) = f(q, q 9 ). Here f(a, b) is the Ramanujan s theta function defined as f(a, b) := n= a n(n+)/2 b n(n )/2 = ( a; ab) ( b; ab) (ab; ab).
INTEGERS: 7 (207) 7 Now we extract terms involving q 5n from P A 25(25n)q n and replace q 5 by q, then p(625n)q n (mod 5). A 25 (625n)q n '( q 5 ) Finally, we use the following congruence from [4, Theorem.5] for p(n) and obtain which coincides with We therefore conclude p(25n) p(625n) (mod 5), A 25 (625n)q n '( q 5 ) A 25 (25n)q n '( q 5 ) Theorem 8. For any nonnegative integer n, we have 2.4. ` = 5 with 4 p(25n)q n (mod 5), p(25n)q n (mod 5). A 25 (25n) A 25 (625n) (mod 5). (3) We know from () that in this case A 5 (n)q n = ' q 5 p(n)q n. Note that for 4, ' q 5 is a function of q 625. Hence A 5 (625n + 25) (resp. A 5 (625n + 500)) is a linear combination of values of p(625n + 25) (resp. p(625n + 500)). Thanks to [4, Eq. (5.3)], we know that p(625n + 25) 0 (mod 5) and p(625n + 500) 0 (mod 5) hold for all. Hence Theorem 9. For any nonnegative integer n and positive integer 4, we have A 5 (625n + i) 0 (mod 5), (4) where i = 25 and 500. Note also that for 2, extracting terms of the form q 25n and replacing q 25 by q, we obtain A 5 (25n)q n = ' q 2 5 p(25n)q n.
INTEGERS: 7 (207) 8 On the other hand, we extract terms involving q 625n from P A 5 +2(n)qn and replace q 625 by q, then A 5 +2(625n)q n = ' q 5 2 p(625n)q n. Thanks again to [4, Theorem.5], which states that p(25n) p(625n) (mod 5) for all, we conclude Theorem 0. For any nonnegative integer n, we have for all 2. It follows from Theorems 9 and 0 that A 5 (25n) A 5 +2(625n) (mod 5) (5) Theorem. For any nonnegative integer n and positive integer 4, we have A 5 (5 2j (625n + i)) 0 (mod 5) (6) for all integers 0 apple j apple ( 4)/2, where i = 25 and 500. Acknowledgements The author would like to express gratitude to James A. Sellers for some interesting discussions as well as thank the reviewer for several helpful comments. References [] A. M. Alanazi, A. O. Munagi, and J. A. Sellers, An infinite family of congruences for `-regular overpartitions, Integers 6 (206), Paper No. A37, 8 pp. [2] B. C. Berndt, Ramanujan s Notebooks. Part III, Springer-Verlag, New York, 99. [3] B. C. Berndt, Number Theory in the Spirit of Ramanujan, Student Mathematical Library, 34. American Mathematical Society, Providence, RI, 2006. [4] W. Y. C. Chen, L. H. Sun, R. H. Wang, L. Zhang, Ramanujan-type congruences for overpartitions modulo 5, J. Number Theory 48 (205), 62 72. [5] J. Lovejoy, Gordon s theorem for overpartitions, J. Combin. Theory Ser. A 03 (2003), no. 2, 393 40. [6] E. Y. Y. Shen, Arithmetic properties of l-regular overpartitions, Int. J. Number Theory 2 (206), no. 3, 84 852. [7] S. Treneer, Congruences for the coe cients of weakly holomorphic modular forms, Proc. London Math. Soc. (3) 93 (2006), no. 2, 304 324.