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

Abstract

We use the q-binomial theorem to prove three new polynomial identities involving q-trinomial coefficients. We then use summation formulas for the q-trinomial coefficients to convert our identities into another set of three polynomial identities, which imply Capparelli's partition theorems when the degree of the polynomial tends to infinity. This way we also obtain an interesting new result for the sum of the Capparelli's products. We finish this paper by proposing an infinite hierarchy of polynomial identities.

Polynomial identities implying Capparelli's partition theorems

“…List of the related sizes and partitions, where 1 and 2 modulo 3 parts are colored differently. n π n π n π n π 16 (11, 5) 28 (17,8,3) 34 (17,12,5) 40 (18,14,8) 19 (14,5) (17, 11) (18,11,5) 43 (17,12,9,5) 22 (14,8) 31 (15,11,5) (17,11,5,1) 40 (17,11,7,5) 49 (18,15,11,5) 28 (14,9,5) (17, 11, 6) (17,12,8,3) 52 (19,17,11,5) One example of this corollary is presented in Table 1.…”