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Quadratic addition rules for three $q$-integers.
| Content Provider | Semantic Scholar |
|---|---|
| Author | Tuntapthai, Mongkhon |
| Copyright Year | 2019 |
| Abstract | The $q$-integer is the polynomial $[n]_q = 1 + q + q^2 + \dots + q^{n-1}$. For every sequences of polynomials $\mathcal S = \{s_m(q)\}_{m=1}^\infty$, $\mathcal T = \{t_m(q)\}_{m=1}^\infty$, $\mathcal U = \{u_m(q)\}_{m=1}^\infty$ and $\mathcal V = \{v_m(q)\}_{m=1}^\infty$, define an addition rule for three $q$-integers by $\oplus_{\mathcal S,\mathcal T,\mathcal U,\mathcal V} ([m]_q, [n]_q, [k]_q) = s_m (q) [m]_q + t_m (q) [n]_q + u_m(q) [k]_q + v_m (q) [n]_q [k]_q .$ This is called the first kind of quadratic addition rule for three $q$-integers, if $\oplus_{\mathcal S,\mathcal T,\mathcal U,\mathcal V} ([m]_q, [n]_q, [k]_q) = \left[m+n+k\right]_q$ for all positive integers $m$, $n$, $k$. In this paper the first kind of quadratic addition rules for three $q$-integers are determined when $s_m(q)\equiv 1$. Moreover, the solution of the functional equation for a sequence of polynomials $\{f_n(q)\}_{n=1}^\infty$ given by $f_{m+n+k} (q) = f_m (q) + q^m f_n (q) + q^m f_k (q) + q^m (q-1) f_n (q) f_k (q)$ for all positive integers $m$, $n$, $k$, are computed. |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | https://arxiv.org/pdf/1911.06449v1.pdf |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
