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In terms of the $q$-Saalsch{u}tz identity and the Chinese remainder theorem for coprime polynomials, we establish some $q$-supercongruences modulo the third power of a cyclotomic polynomial. In particular, we give a $q$-analogue of a formula due to Long and Ramakrishna [Adv. Math. 290 (2016), 773--808].
Inspired by the recent work on $q$-congruences and the quadratic summation formula of Rahman, we provide some new $q$-supercongruences. By taking $qto 1$ in one of our results, we obtain a new Ramanujan-type supercongruence, which has the same right-
Let $E_n$ be the $n$-th Euler number and $(a)_n=a(a+1)cdots (a+n-1)$ the rising factorial. Let $p>3$ be a prime. In 2012, Sun proved the that $$ sum^{(p-1)/2}_{k=0}(-1)^k(4k+1)frac{(frac{1}{2})_k^3}{k!^3} equiv p(-1)^{(p-1)/2}+p^3E_{p-3} pmod{p^4}, $
We introduce a notion of $q$-deformed rational numbers and $q$-deformed continued fractions. A $q$-deformed rational is encoded by a triangulation of a polygon and can be computed recursively. The recursive formula is analogous to the $q$-deformed Pa
The Assmus-Mattson theorem gives a way to identify block designs arising from codes. This result was broadened to matroids and weighted designs. In this work we present a further two-fold generalisation: first from matroids to polymatroids and also f
Two $q$-supercongruences of truncated basic hypergeometric series containing two free parameters are established by employing specific identities for basic hypergeometric series. The results partly extend two $q$-supercongruences that were earlier co