Three families of \(q\)-supercongruences from a quadratic transformation of Rahman (Q6071778)

The main results of the paper are given by the following theorems: Theorem. Let \(d\) and \(r\) be positive integers such that \(d+r\) is odd, \(d>r\) and gcd\((d,r)=1\). Let \(n\) be a positive integer satisfying \(n\equiv d+r (\hbox{mod}\ 2d)\). Then there holds: \[ \sum_{k=0}^{(dn+n-r)/(2d)}\ [3dk+r]\frac{(q^r;q^{2d})_k (q^r,q^r,q^{d-r};q^d)_k}{(q^d;q^d)_k (q^{2d},q^{2d},q^{d+2r};q^{2d})_k} q^{d(k^2+k)/2} \equiv 0\ (\hbox{mod}\ \Phi_n(q)^3). \] Theorem. Let \(d\) and \(r\) be positive integers such that \(r\) is odd, \(d>r\) and gcd\((d,r)=1\). Let \(n\) be a positive integer satisfying \(n\equiv -r(\hbox{mod}\ 2d)\). Suppose that \((d,r)\not\in \{(3,1),(4,3)\}\). Then there holds: \[ \sum_{k=0}^{(dn+n-r)/d}\ [3dk+r]\frac{(q^r;q^{2d})_k (q^r,q^r,q^{d-r};q^d)_k}{(q^d;q^d)_k (q^{2d},q^{2d},q^{d+2r};q^{2d})_k} q^{d(k^2+k)/2} \equiv 0\ (\hbox{mod}\ \Phi_n(q)^2). \] Theorem. Let \(d\) and \(r\) be positive integers such that \(r\) is odd, \(d>r\) and gcd\((d,r)=1\). Let \(n\) be a positive integer satisfying \(n\equiv r(\hbox{mod}\ 2d)\). Then there holds: \begin{align*} \sum_{k=0}^{(n-r)/d}\ [3dk+r]\frac{(q^r;q^{2d})_k (q^r,q^r,q^{d-r};q^d)_k}{(q^d;q^d)_k (q^{2d},q^{2d},q^{d+2r};q^{2d})_k} q^{d(k^2+k)/2} \\ \equiv [n]\frac{(q^d;q^{2d})_{(n-r)/(2d)}}{(q^{d+2r};q^{2d})_{(n-r)/(2d)}} q^{(r-d)(n-r)/(2d)})\equiv 0\ (\hbox{mod}\ \Phi_n(q)^3). \end{align*} For completeness we give here some used notations: (1) \(q\)-shifted factorial: \((a;q)_0=1,\ (a;q)_n=(1-a)(1-aq)\cdots (1-aq^{n-1})\hbox{ for }n\geq 1\hbox{ or }n=\infty\); (2) Condensed notation: \((a_1,a_2,\ldots,a_m;q)_n=(a_1;q)_n(a_2;q)_n\cdots (a_m;q)_n\hbox{ for }n\geq 0\hbox{ or }n=\infty\); (3) \(q\)-integer: \([n]=(1-q^n)/(1-q)\); (4) \(n\)-th cyclotomic polynomial: \[ \Phi_n(q)= \ \ \prod_{\substack{1\leq k\leq n \\ \hbox{gcd} (k,n)}}\ \ (q-\zeta^k)\hbox{ with }\zeta\hbox{ being any }n\text{-th root of unity}. \] The concluding remarks contains open problems for further study in the form of 4 conjectures.