Andrews–Gordon type series for Capparelli's and Göllnitz–Gordon identities

Abstract

We construct an evidently positive multiple series as a generating function for partitions satisfying the multiplicity condition in Schur's partition theorem. Refinements of the series when parts in the said partitions are classified according to their parities or values mod 3 are also considered. Direct combinatorial interpretations of the series are provided.

Elementary Polynomial Identities Involving $$\varvec{q}$$-Trinomial Coefficients

“…We note that (4.6) first appeared in Kurşungöz [13]. In fact, this is equivalent to the Cappareli's second partition theorem.…”

“…The identity (4.3) was independently proposed by Kanade-Russell [12] and Kurşungöz [13]. They showed that (4.3) is equivalent to the following partition theorem.…”

Polynomial identities implying Capparelli's partition theorems

“…We follow Kurşungöz's ideas [16,17] and start with the partition (written in ascending order) (4.1) π = ( 2, 4 , 8, 10 , . .…”

“…We now describe rules of motion for the parts of π in the style of Kurşungöz [16,17], which would convert π into another Capparelli partition with the smallest part ≥ 2 and with n pairs and m singletons, just as in (4.1). These rules of motion are bijective and, in principal, can be used in reverse to transform any Capparelli partition into a unique minimal configuration.…”

“…Independently, Kurşungöz [16,17] discovered the same generating functions' representations with some slight difference in the representation of (1.7): Provided that the Capparelli Partition Theorem is valid, these imply m,n≥0 q Q(m,n) (q; q) m (q 3 ; q 3 ) n = (−q 2 , −q 4 ; q 6 ) ∞ (−q 3 ; q 3 ) ∞ , (1.9) m,n≥0 q Q(m,n)+m+3n (q; q) m (q 3 ; q 3 ) n + m,n≥0 q Q(m,n)+3m+6n+1 (q; q) m (q 3 ; q 3 ) n = (−q, −q 5 ; q 6 ) ∞ (−q 3 ; q 3 ) ∞ , (1.10) m,n≥0 q Q(m,n)+m (q; q) m (q 3 ; q 3 ) n + m,n≥0 q Q(m,n)+4m+6n+1 (q; q) m (q 3 ; q 3 ) n = (−q, −q 5 ; q 6 ) ∞ (−q 3 ; q 3 ) ∞ . (1.11) We would like to remark that Sills [20] discovered different series representations of the product in (1.9).…”