Some Congruence Properties of a Restricted Bipartition Function c N ( n )

where s k is the reciprocal modulo 3 of 8 and δ(k) = 1 if k is even and 0 otherwise. Inspired by the work of Ramanujan on the standard partition function p(n), Chan [4] asked whether there are any other congruence properties of the following form: c 2 (ln + k) ≡ 0 (modl), where l is prime and 0 ≤ k ≤ l. Sinick [1] answered Chan’s question in negative by considering restricted bipartition function c N (n) defined in (1). Liu and Wang [5] established several infinite families of congruence properties for c 5 (n)modulo 3. For example, they proved that


Introduction
A bipartition of a positive integer  is an ordered pair of partitions (, ) such that the sum of all of the parts equals .If   () counts the number of bipartitions (, ) of  subject to the restriction that each part of  is divisible by , then the generating function of   () [1] is given by where The partition function   () is first studied by Chan [2] for the particular case  = 2 by considering the function  2 () defined by Chan [2] proved that, for  ≥ 0,  2 (3 + 2) ≡ 0 (mod3) .
All the terms on the right hand side of (i) are of the form  2 .Extracting the terms involving  2+1 on both sides of (i), we complete the proof of (ii).
Theorem 18.For  ≥ 1, one has Proof.We proceed by induction on .Extracting the terms involving  2 and replacing  2 by  in Theorem 16(i), we obtain which corresponds to the case  = 1.Assume that the result is true for  =  ≥ 1, so that Employing Lemma 3 in (54), we obtain Employing Lemma 2 in (55) and extracting the terms involving  2+1 , dividing by , and replacing  2 by , we obtain Simplifying (56) using Lemmas 3 and 1 with  = 2, we obtain Extracting the terms involving  2 and replacing  2 by  in (57), we obtain which is the  =  + 1 case.Hence, the proof is complete.
Theorem 19.For  ≥ 0, one has Proof.All the terms in the right hand side of (57) are of the form  2 , so, extracting the coefficients of  2+1 on both sides of (57) and replacing  by , we obtain Replacing  + 1 by  in (60) completes the proof.
Theorem 22.For any odd prime  and any integer  ≥ 0, one has Proof.We proceed by induction on .The case  = 0 corresponds to the congruence theorem (Theorem 21).Suppose that the theorem holds for  =  ≥ 0, so that Employing Lemma 11 in (73), extracting the terms involving  +( 2 −1)/8 on both sides of (73), dividing by  ( 2 −1)/8 , and replacing   by , we obtain Extracting the terms containing   from both sides of (74) and replacing   by , we arrive at which shows that the theorem is true for  =  + 1.Hence, the proof is complete.
Theorem 23.For any odd prime  and integers  ≥ 0 and 1 ≤  ≤  − 1, one has Proof.Extracting the coefficients of  + for 1 ≤  ≤  − 1 on both sides of (74) and replacing  by , we arrive at the desired result. .

Congruence
The desired result follows easily from (79).