1. Introduction
This commentary is intended to correct the errors that happened in [1]. Recall that a partition of a positive integer is a nonincreasing sequence of positive integers whose sum is . A bipartition of a positive integer is a pair of partitions such that the sum of all the parts is . Let be the number of bipartitions of . This function has some beautiful arithmetic properties (see, e.g., [2]). Recently, there is more and more research on the bipartitions with certain restrictions on each partition, for example, [3–7]. In this short review, we consider the number of bipartitions with the restriction that each part of is divisible by . Let denote the number of bipartitions of such that part of is divisible by . The generating function for is given by [8] where
We will mainly focus on the case when , where is any positive integer. In [1], the discussion on congruence identities for was based on an incorrect lemma. We will point out all the incorrect points in that paper in the next section.
2. Comments on the Results of Saikia and Boruah
In [1], Saikia and Boruah considered the congruence identities for by employing the following lemma.
Lemma 13 of [1] where is the Rogers-Ramanujan continued fraction and
However, the lemma of this version is wrong, The correct version should be as follows.
Lemma 1. Let be defined as above:
There are several proofs of this famous identity (see, e.g., [9, 10]).
In [1], all the discussion in Section was based on Lemma of [1] which has been shown above.
Since Lemma of [1] is not correct, the discussion in Section is not correct.
First, we can say that the conclusion shown in Theorem of [1], which says is correct. However, in the course of the proof, there is one mistake. The equation which is equation () of [1], is not correct. It should be
The first result of Theorem of [1] is also correct, but the proof has a problem. In the course of the proof, equation () of [1] should be rather than
However, the second conclusion and the third conclusion of Theorem of [1] are not correct. The proof of Theorem (ii, iii) of [1] relies on the claim that However, as we have pointed out above, we do not have the factor in . This is the reason why the second conclusion and the third conclusion of Theorem of [1] cannot hold.
Actually, we do have some 3-dissection formulas to find how and modulo-3 look like. Let us show them now.
Thanks to [11], we have If we let , and in Entry 31 of [12, p. 48], we can obtain where According to these two formulas, we get that When extracting terms involving , dividing by and replacing by , we obtain When extracting terms involving , dividing by and replacing by , we obtain Up to now, we have obtained the forms of and modulo-3. However, we cannot get any congruence properties of or by the forms of and modulo-3 above.
Conflicts of Interest
The author declares no conflicts of interest.