Two curious \(q\)-supercongruences and their extensions (Q6612607)

The first supercongruence in this paper (numerically observed by \textit{F. Rodriguez-Villegas} [Fields Inst. Commun. 38, 223--231 (2003; Zbl 1062.11038)] and proved by \textit{E. Mortenson} [J. Number Theory 99, No. 1, 139--147 (2003; Zbl 1074.11045)]) is \N\[\N\sum_{k=0}^{p-1}\,\frac{(\frac{1}{2})^2_k}{k!^2} \equiv (-1)^{(p-1)^2} (\text{mod}\,{ p^2}). \tag{a}\N\]\NThe second one is the generalization \N\[\N\sum_{k=0}^{p-1}\,\frac{\binom{d}{d-1}^d_ k}{k!^d}\equiv -\Gamma_p\left(\frac{1}{d}\right) (\text{mod}\,{ p^2}). \tag{b}\N\]\Ndue to \textit{A. Deines} et al. [Assoc. Women Math. Ser. 3, 125--159 (2016; Zbl 1376.33005)].\N\NLater on, these congruences were extended to the \(q\)-setting. Precisely, an extension of (a) was given by \textit{V. J. W. Guo} and \textit{J. Zeng} [J. Number Theory 145, 301--316 (2014; Zbl 1315.11015)], while an extension of (b) was proved by \textit{V. J. W. Guo} [Ramanujan J. 60, No. 3, 825--835 (2023; Zbl 1523.11003)].\N\NIn the paper under review, the authors prove complements to two analogous \(q\)-supercongruences which were formulated as conjectures in the V. J. W. Guo's paper [loc. cit.]. These are given in Theorems 1.1 and 1.2 in the paper, along with a third result.\N\NIn the introduction we find three theorems; two of them are reproduced here:\N\NTheorem [V. J. W. Guo, loc. cit.; Conjecture 5.2]. Let \(d\geq 4\) be an even integer and\N\(n\) an integer with \(n\equiv -1 (\text{mod}\,d)\) and \(n\geq 2d-1\). Then \N\[\N\sum_{k=0}^{n-1}\,\frac{(q^{d+1};q^d)_k^{d-1}) (q^{1-d};q^d)_k q^{dk}}{(q^d;q^d)_k^d}\]\[\equiv\N\frac{(1-q)(1-q^{d-1})(q^d;q^d)_{n-1-(n+1)/d}}{-(-1)^{(n+1)/d}(q^d;q^d)_{(n+1)/d}^{d-1}} q^{(d(d+n)(n+1)-(n+1)^2)/(2d)-1}\ (\text{mod}\ \Phi_n(q)^2). \N\]\N\NTheorem [V. J. W. Guo, loc. cit.; Conjecture 5.3]. Let \(d\geq 3\) be an odd integer and\N\(n\) a positive integer with \(n\equiv -1 (\text{mod}\,d)\). Then \N\[\N\sum_{k=0}^{n-1}\,\frac{(q^{d+1};q^d)_k^{d-2}) (q;q^d)_k^2 q^{dk}}{(q^d;q^d)_k^d} \]\[\equiv\N\frac{(1-q)^2 (q^d;q^d)_{n-1-(n+1)/d}}{(q^d;q^d)_{(n+1)/d}^{d-1}} q^{(d(d+n)(n+1)-(n+1)^2)/(2d)-2}\ (\text{mod}\ \Phi_n(q)^2). \N\]\N\NIn the remaining sections of the paper, the three theorems are proved and generalized.