2019/11/15 by Mongkhon Tuntapthai, Tuntapthai, Mongkhon
Computer Science · Mathematics · Physics and Astronomy · #Algebraic structures and combinatorial models #Coding theory and cryptography #Combinatorics (math.CO) #FOS: Mathematics #Nonlinear Waves and Solitons
paper · pdf · doi:10.48550/arxiv.1911.06449
openalex publication_date 2019/11/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The q-integer is the polynomial [n]q = 1 + q + q2 + … + qn-1. For every sequences of polynomials \mathcal S = \sm(q)\m=1^∞, \mathcal T = \tm(q)\m=1^∞, \mathcal U = \um(q)\m=1^∞ and \mathcal V = \vm(q)\m=1^∞, define an addition rule for three q-integers by ⊕\mathcal S,\mathcal T,\mathcal U,\mathcal V ([m]q, [n]q, [k]q) = sm (q) [m]q + tm (q) [n]q + um(q) [k]q + vm (q) [n]q [k]q . This is called the first kind of quadratic addition rule for three q-integers, if ⊕\mathcal S,\mathcal T,\mathcal U,\mathcal V ([m]q, [n]q, [k]q) = [m+n+k]q for all positive integers m, n, k. In this paper the first kind of quadratic addition rules for three q-integers are determined when sm(q)≡ 1. Moreover, the solution of the functional equation for a sequence of polynomials \fn(q)\n=1^∞ given by fm+n+k (q) = fm (q) + qm fn (q) + qm fk (q) + qm (q-1) fn (q) fk (q) for all positive integers m, n, k, are computed.