Quadratic forms and congruences for ℓ-regular partitions modulo 3, 5 and 7

Let b ? ( n ) be the number of ?-regular partitions of n. We show that the generating functions of b ? ( n ) with ? = 3 , 5 , 6 , 7 and 10 are congruent to the products of two items of Ramanujan's theta functions ? ( q ) , f ( - q ) and ( q ; q ) ∞ 3 modulo 3, 5 and 7. So we can express these generating functions as double summations in q. Based on the properties of binary quadratic forms, we obtain vanishing properties of the coefficients of these series. This leads to several infinite families of congruences for b ? ( n ) modulo 3, 5 and 7.

[1]  Jeremy Lovejoy,et al.  Quadratic Forms and Four Partition Functions Modulo 3 , 2011, Integers.

[2]  Basil Gordon,et al.  Divisibility of Certain Partition Functions by Powers of Primes , 1997 .

[3]  Byungchan Kim.  Overpartition pairs modulo powers of 2 , 2011, Discret. Math..

[4]  Bernard L. S. Lin,et al.  GENERALISATION OF KEITH’S CONJECTURE ON 9-REGULAR PARTITIONS AND 3-CORES , 2014, Bulletin of the Australian Mathematical Society.

[5]  Nobushige Kurokawa,et al.  Number Theory 3: Iwasawa Theory and Modular Forms , 2012 .

[6]  James A. Sellers,et al.  Arithmetic properties of partitions with even parts distinct , 2010 .

[7]  Partition identities involving gaps and weights , 1997 .

[8]  Nancy S. S. Gu,et al.  Arithmetic properties of ℓ-regular partitions , 2013, Adv. Appl. Math..

[9]  James A. Sellers,et al.  ELEMENTARY PROOFS OF PARITY RESULTS FOR 5-REGULAR PARTITIONS , 2009, Bulletin of the Australian Mathematical Society.

[10]  Ernest X. W. Xia,et al.  A PROOF OF KEITH'S CONJECTURE FOR 9-REGULAR PARTITIONS MODULO 3 , 2014 .

[11]  W. Keith.  Congruences for 9-regular partitions modulo 3 , 2013, 1306.0136.

[12]  Ken Ono,et al.  The 2-Adic Behavior of the Number of Partitions into Distinct Parts , 2000, J. Comb. Theory A.

[13]  David Furcy,et al.  Congruences for ℓ-regular partition functions modulo 3 , 2012 .

[14]  Jeremy Lovejoy.  Divisibility and Distribution of Partitions into Distinct Parts , 2001 .

[15]  Ernest X. W. Xia,et al.  Parity results for 9-regular partitions , 2014 .

[16]  David Penniston.  The pa-Regular Partition Function Modulo pj☆ , 2002 .

[17]  Bruce C. Berndt,et al.  Number theory in the spirit of Ramanujan , 2006 .

[18]  Congruences for 9-regular partitions modulo 3 , 2015 .