Elementary Polynomial Identities Involving varvec{q}-Trinomial Coefficients

Alexander Berkovich,Ali Kemal Uncu

doi:10.1007/s00026-019-00445-8

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.

Highlights

There are many areas of study he introduced that are saturated with world-class mathematicians, yet there are many more that the community is only catching up studying
We find new polynomial identities that yield Capparelli’s partition theorem in Sect
We found a similar new summation formula: Theorem 3.4

Summary

Introduction

Which implies the generalized Pascal Triangle for (1.1) with q = 1: The q-trinomial coefficients were studied in [1,2,4,5,6,7,8,9,10,14,15,16,17] It appears that the following identities are new. (q; q3)∞ , (q2; q3)∞ , and (q; q3)∞ , respectively, as L → ∞ with the aid of the q-binomial theorem This is not as easy to see that from the right-hand sides of these identities. We find new polynomial identities that yield Capparelli’s partition theorem in Sect.

Necessary Definitions and Identities

New Polynomial Identities Implying Capparelli’s Partition Theorems

Outlook