Generalizations of Capparelli's identity

Abstract

Using jagged overpartitions, we give three generalizations of a weighted word version of Capparelli's identity due to Andrews, Alladi and Gordon, and present several corollaries.

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

“…Lepowsky and Milne [17] also showed that the Rogers-Ramanujan identities arise in character formulas for the level 2 standard modules for A (2) 2 , which was later proven using Z-algebras by Capparelli [5]. Capparelli additionally used Z-algebras to construct the level 3 standard modules for A (2) 2 , which relied on the discovery of two conjectural partition identities (which were first proven by Andrews [2], and later by Capparelli [6]; an analytic sum-side for Capparelli's identities was recently given by [7]). These identities were a significant development in the theory of vertex operator algebras, as they were the first notable examples of sum-product identities that had not previously appeared, but were instead discovered using vertex-operator-theoretic techniques.…”

Staircases to Analytic Sum-Sides for Many New Integer Partition Identities of Rogers-Ramanujan Type

“…x i +j+k+2l q 3(i +j +k +2l )(i +j +k +2l −1) 2 +3i +2j+ 3j (j −1) 2 +4k+ 3k (k −1) 2 +3l q 3 ; q 3 i q 3 ; q 3 j q 3 ; q 3 k q 6 ; q 6 l , (7.3) which is exactly [DL,Equation (2.6)] with r → k, s → j, t → i, → l, q → q 3 , a → q −2 , b → q −4 . We may instead arrive at a different analytic sum-side by first simplifying the expression (7.1).…”

A unifying combinatorial approach to refined little Göllnitz and Capparelli's companion identities

“…Then, we have s o 1 (π) t o 2 (π) u e 1 (π) v e 2 (π) q |π| , (3. 16) i.e., the polynomial S µ (s, t, u, v, q, q). Similarly we consider the following four further specializations, corresponding to taking s = v = 0, t = u = 0, t = v = 0 and s = u = 0 respectively, and get the following theorem, which is a bounded version of (3.5) ∼ (3.8), where the unspecified sums are over all quadruples (i, j, k, l) ∈ N 4 such that i+j +k+l = N. Theorem 3.13.…”

“…The proof of the other two formulae can be given similarly and thus omitted. + π = (20,17,16,11,10,9,6,5,4,2) π 1 = (12,11,10,9,8,7,6,5,4,2) π 2 = (8, 6, 6, 2, 2, 2) Figure 4. π = π 1 + π 2 π 1 = (12, 11, 10, 9, 8, 7, 6, 5, 4, 2) π 6,4 = (12, 10, 8, 7, 6, 5, 4, 3, 2, 1) π = (4, 4, 4, 4) + Figure 5.…”

“…Finally in 1995, both identities were proven by Alladi, Andrews and Gordon [1]. For recent study on Capparelli's identities, see for example [11,15,16,22]. In Capparelli's original identities there are the infinite product sides or the modular sides and the "gap condition" sides.…”