Combinatorial proofs for identities related to generalizations of the mock theta functions $$\omega (q)$$ω(q) and $$\nu (q)$$ν(q)

Abstract

The two partition functions p ω (n) and p ν (n) were introduced by Andrews, Dixit and Yee, which are related to the third order mock theta functions ω(q) and ν(q), respectively. Recently, Andrews and Yee analytically studied two identities that connect the refinements of p ω (n) and p ν (n) with the generalized bivariate mock theta functions ω(z; q) and ν(z; q), respectively. However, they stated these identities cried out for bijective proofs. In this paper, we first define the generalized trivariate mock theta functions ω(y, z; q) and ν(y, z; q). Then by utilizing odd Ferrers graph, we obtain certain identities concerning to ω(y, z; q) and ν(y, z; q), which extend some early results of Andrews that are related to ω(z; q) and ν(z; q). In virtue of the combinatorial interpretations that arise from the identities involving ω(y, z; q) and ν(y, z; q), we finally present bijective proofs for the two identities of Andrews-Yee.

Generalizations of the Andrews-Yee identities associated with the mock theta functions $ω(q)$ and $ν(q)$

“…In this paper, we have partially answered the question posed by Li and Yang in their paper [18] and obtained generalizations of the Andrews-Yee identities for ω 1 (α, z; q) and ν(α, z; q), defined in (1.17) and (1.18), respectively. While ν(α, z; q) is essentially equivalent to their trivariate generalization of ν(q), the function ω 1 (α, z; q) is different from the trivariate generalization of ω(q) that they consider in their paper.…”

“…Hence, to realize our goal of generalizing (1.10) and (1.11), it is natural to consider ω 1 (α, z; q) instead of ω 0 (α, z; q). Also, the other trivariate generalization of ν(q) that we study, namely, ν(α, z; q), is the same as that considered by Li and Yang [18] and also by Choi [13], in view of (1.14), since ν(α, z; q) = ν 0 (−z, −α/z; q).…”

“…Thus obtaining a generalization of the identity (1.10) of Andrews and Yee for ω 0 (y, z; q) might be very crucial in that it may lead to partition-theoretic interpretations of mock theta functions of odd order, which are special cases of g 3 (x, q), as explicated in [16,Section 5]. However, at the end of their paper [18], Li and Yang surmise that finding generalizations of (1.10) and (1.11) for ω 0 (y, z; q) and ν 0 (y, z; q) remains mysterious. While we have not been able to fully resolve this mystery, we have partially resolved it.…”

“…Andrews and Yee also expressed the difficulty in obtaining bijective proofs of their results. Recently, F. Z. K. Li and J. Y. X. Yang [18] overcame this difficulty and provided bijective proofs of (1.10) and (1.11). In the same paper, Li and Yang also gave three-variable generalizations of ω(z; q) and ν(z; q), namely, ω 0 (y, z; q) := ∞ n=0 y n z n q 2n 2 +2n (yq; q 2 ) n+1 (zq; q 2 ) n+1 , (1.12) ν 0 (y, z; q) := ∞ n=0 y n z n q n 2 +n (yq; q 2 ) n+1 .…”

Combinatorics of some fifth and sixth order mock theta functions

“…They also proved that another third order mock theta function ν(q) is related to these partitions with distinct parts. In [12], Andrews and Yee have given two variable generalizations of the results of [10] and Li and Yang [20] have further studied these generalized results combinatorially.…”