Further Applications of Cubic q-Binomial Transformations

2025/09/30 by Berkovich, Alexander, Dhar, Aritram
#05A15 #05A17 #05A30 #11P81 #11P84 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2510.00130

Abstract

Consider G(N,M;α,β,K,q) = ∑_j∈ℤ(-1)^jq^{(1)/(2)Kj((α+β)j+α-β)}[\beginmatrixM+N
N-Kj\endmatrix]_q. In this paper, we prove the non-negativity of coefficients of some cases of G(N,M;α,β,K,q). For instance, for non-negative integers n and t, we prove that G(n,n;(4)/(3)+(3(3^t-1))/(2),(5)/(3)+(3(3^t-1))/(2),3^t+1,q) and G(n-(3^t-1)/(2),n+(3^t+1)/(2);(8)/(3)+2(3^t-1),(4)/(3)-(3^t-1),3^t+1,q)
are polynomials in q with non-negative coefficients. Using cubic positivity preserving transformations of Berkovich and Warnaar and some known formulae arising from Rogers-Szegö polynomials, we establish new identities such as ∑_{0≤ 3j≤ n}\dfrac(q³;q³)_n-j-1(1-q²ⁿ)q^3j2(q;q)_n-3j(q⁶;q⁶)_j = ∑_j=-∞^∞(-1)^jq^6j22n\brack n-3j_q.

Citations

• An asymptotic approach to Borwein-type sign pattern theorems
• Bressoud's identities for even moduli. New companions and related positivity results
• Some New Positive Observations
• The generalized Borwein conjecture. I. The Burge transform

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• Factors of sums and alternating sums of products of q-binomial coefficients and powers of q-integers
• Nontrivial Solutions to a Cubic Identity and the Factorization of n²+n+1
• Two new families of q-positive integers
• Elementary Polynomial Identities Involving q-Trinomial Coefficients