Andrews-Beck type congruences for overpartitions

We prove Andrews-Beck type congruences for overpartitions concerning the Drank and M2-rank. To prove congruences, we establish the generating function for weighted D-rank (respectively, M2-rank) moment of overpartitions and find a connection with the second D-rank (respectively, M2-rank) moment for overpartitions. Mathematics Subject Classifications: 11P81, 05A17


Introduction
Ramanujan's congruences for the partition function p(n) are one of remarkable results in the theory of partitions: p(5n + 4) ≡ 0 (mod 5), p(7n + 5) ≡ 0 (mod 7), p(11n + 6) ≡ 0 (mod 11), Dyson [8] defined the rank of a partition, which is defined as the largest part minus the number of parts, conjectured combinatorial explanations for the Ramanujan congruences modulo 5 and 7, and conjectured the existence of a crank function for partitions that could provide a combinatorial proof of Ramanujan's congruences modulo 11. Atkin and Swinnerton-Dyer [3] proved Dyson's conjecture on the rank. Andrews and Garvan [2] found the crank function and proved that the crank explains all Ramanujan congruences modulo 5, 7 and 11.
In the recent article [7], Chern also provided a list of over 70 Andrews-Beck type congruences involving0 N T (b, k, b) and M ω (b, k, n).
In this paper, we prove Andrews-Beck type congruence on N T (b, k, n) and N T 2(b, k, n) modulo 4 and 8 as follows.

Weighted D-rank moments of overpartitions
Using standard combinatorial arguments in partition theory as [11, Proposition 1.1], we find that where P n is the set of overpartitions of n.
Here and throughout the rest of the paper, we use the standard q-series notation, (a) n = (a; q) n := n k=1 (1 − aq k−1 ), (a 1 , . . . , a m ) n = (a 1 , . . . , a m ; q) n := (a 1 ) n · · · (a m ) n , and [a 1 , . . . , a m ] n = [a 1 , . . . , a m ; q] n = (a 1 , q/a 1 , . . . , a m , q/a m ) n , for n ∈ N 0 ∪ {∞}. We will give two proofs for Theorem 1. For the first proof of Theorem 1, we will establish the generating function for the weighted D-rank moment of overpartitions and compare it with the ordinary and symmetrized D-rank moments. Here, the ordinary and symmetrized D-rank moments are defined by where N (m, n) denotes the number of overpartitions of n with D-rank m.
It follows that Then if we differentiate it by z and evaluate it at z = 1, we get which proves the first part. If we apply [ ∂ ∂z (z ∂ ∂z )] z=1 to R(1, z, q), then we have the generating function for the second D-rank moment as follows.
Proof of Theorem 1. By Theorem 4, we have Since the last sum involves only terms where the power of q is even, the result follows.
From Theorem 1, we can have the following congruence for the second D-rank moment.
Corollary 5. For all integers n 0, In fact, we prove more detailed results on congruences of N T (b, k, n), which also deduce the congruence in Theorem 1.
3 Weighted M 2 -rank moments of overpartitions As in Section 2, we find the generating function for the weighted M 2 -rank moments of overpartitions and compare it with the ordinary and symmetrized M 2 -rank moments. We have the ordinary and symmetrized M 2 -rank moments defined by It follows that Proof of Theorem 7. If we differentiate R2(x, z, q) by x and evaluate it at x = 1, we get (−1, −q; q 2 ) n (xq) n (zq 2 , xq 2 /z; q 2 ) n ∂ ∂x log x n (xq 2 /z; q 2 ) n x=1 = n 1 (−1, −q; q 2 ) n q n (zq 2 , q 2 /z; q 2 ) n n + n m=1 q 2m z − q 2m .
Then applying [∂/∂z] z=1 gives which is the first part. Next, to compare with the M 2 -rank moments, when we apply Then the second part follows from comparing (3.1) with (3.2) and the following relation between the ordinary and symmetrized M 2 -rank moments Lastly, we have the last identity by considering the generating function for η2 2 (n) [10, Theorem 2.1].
From the generating function for weighted M 2 -rank moment of overpartitions, we can prove Theorem 2.
We prove Theorem 3 using generalized Lambert series identities with above theta function identities.
Also, we have the following congruences between D-rank and M 2 -rank moments by Theorem 3 and (1.2).