Proofs and reductions of various conjectured partition identities of Kanade and Russell

Abstract

We prove seven of the Rogers-Ramanujan type identities modulo 12 that were conjectured by Kanade and Russell. Included among these seven are the two original modulo 12 identities, in which the products have asymmetric congruence conditions, as well as the three symmetric identities related to the principally specialized characters of certain level 2 modules of A (2) 9 . We also give reductions of four other conjectures in terms of single-sum basic hypergeometric series.Note that h 8,N = h 9,N and so the functional equation is the same.As with H 5 , we cannot apply Proposition 2.2 directly. We setand note that H 8 (xq 2 ) = H 9 (x). Equation (1.20) then gives a q-hypergeometric series representation for H 8 (q 2 ). To find a q-hypergeometric series representation for J 8 (1), we use the recurrence J 8 (x) = 1 + xq 3 J 8 xq 2 + xq 2 (1 + xq) J 8 xq 4 + x 2 q 6 1 − xq 4 J 8 xq 6 .This follows by setting x → xq 2 in (4.7), multiplying by xq 3 , and then subtracting the resulting equation from (4.7). We apply Proposition 2.2 with a = b = 1 to find that J 8 (1) = q; q 4 ∞ n≥0

Proofs of some partition identities conjectured by Kanade and Russell

“…For completeness, we show how to prove the conjectures (1.3a)-(1.3e) in our approach. Bringmann et al [5] also obtain these results by reducing triple summations to single summations, which are either the same as ours or easily seen to be equivalent. The main virtue of our approach is the more streamlined way to obtain such reductions as explained in Sect.…”

“…The identities (1.3a)-(1.3e) were proved by Bringmann et al [5]. In the present work, we give a more streamlined proof of those results, and prove the remaining conjectures (1.3f)-(1.3i) for the first time.…”

“…For the first two cases, Bringmann et al [5] found equivalent reductions but for the last two cases, their reductions are more complicated. Replacing q by q 1/2 , it follows that the conjectures (1.3f)-(1.3i) are equivalent to the following summations.…”

“…In [5], Bringmann et al showed that the nine conjectures (1.3) can be reduced to sums of single series, which they could evaluate in the first five cases. In this section, we will obtain more general reductions using the following integral representation of the function (1.2).…”

Linked Partition Ideals, Directed Graphs and $q$-Multi-Summations

“…There are many identities of the same flavor, including the Andrews-Gordon identity [2,12], the Göllnitz-Gordon identities [11,13], the Capparelli identities [8] and so forth. In 2014, Kanade and Russell [16] further proposed six challenging conjectures on Rogers-Ramanujan type identities, the latter two of which were proved in 2018 by Bringmann, Jennings-Shaffer and Mahlburg [7].…”