We give several different $q$-analogues of the following two congruences of Z.-W. Sun: $$\sum _{k=0}^{(p^r-1)/2}\frac{1}{8^k}\left(\genfrac{}{}{0pt}{}{2k}{k}\right)\equiv \left(\frac{2}{p^r}\right)(mod{p}^2)\text{and}\sum _{k=0}^{(p^r-1)/2}\frac{1}{{16}^k}\left(\genfrac{}{}{0pt}{}{2k}{k}\right)\equiv \left(\frac{3}{p^r}\right)(mod{p}^2),$$ where $p$ is an odd prime, $r$ is a positive integer, and $\left(\frac{m}{n}\right)$ is the Jacobi symbol. The proofs of them require the use of some curious $q$-series identities, two of which are related to Franklin’s involution on partitions into distinct parts. We also confirm a conjecture of the latter author and Zeng in 2012.

Nous explicitons la valeur de certains des coefficients binomiaux généralisés associés aux polynômes de Macdonald, c’est-à-dire la valeur en certains points particuliers des polynômes de Macdonald décalés. Ces expressions font intervenir les fonctions hypergéométriques de base $q$.

Binomial coefficients and central trinomial coefficients play important roles in combinatorics. Let p > 3 be a prime. We show that $T_{p-1}\equiv (p/3)3^{p-1}\left(modp²\right)$, where the central trinomial coefficient Tₙ is the constant term in the expansion of $(1+x+x^{-1})ⁿ$. We also prove three congruences modulo p³ conjectured by Sun, one of which is ${\sum }_{k=0}^{p-1}\left(\genfrac{}{}{0pt}{}{p-1}{k}\right)\left(\genfrac{}{}{0pt}{}{2k}{k}\right)({(-1)}^k-{(-3)}^{-k})\equiv (p/3)(3^{p-1}-1)\left(modp³\right)$. In addition, we get some new combinatorial identities.

We present some extensions of Chu's formulas and several striking generalizations of some well-known combinatorial identities. As applications, some new identities on binomial sums, harmonic numbers, and the generalized harmonic numbers are also derived.

We establish q-analogs for four congruences involving central binomial coefficients. The q-identities necessary for this purpose are shown via the q-WZ method.

For any odd prime p we obtain q-analogues of van Hamme’s and Rodriguez-Villegas’ supercongruences involving products of three binomial coefficients such as ${\sum }_{k=0}^{(p-1)/2}{\left[\genfrac{}{}{0pt}{}{2k}{k}\right]}_{q²}^3\left(q^{2k}\right)/\left((-q²;q²){²}_k(-q;q){²}_{2k}²\right)\equiv 0\left(mod\left[p\right]²\right)$ for p≡ 3 (mod 4), ${\sum }_{k=0}^{(p-1)/2}{\left[\genfrac{}{}{0pt}{}{2k}{k}\right]}_{q³}\left({(q;q³)}_k{(q²;q³)}_k{q}^{3k}\right)\left({(q⁶;q⁶)}_k²\right)\equiv 0\left(mod\left[p\right]²\right)$ for p≡ 2 (mod 3), where $\left[p\right]=1+q+\cdots +q^{p-1}$ and $(a;q)ₙ=(1-a)(1-aq)\cdots (1-a{q}^{n-1})$. We also prove q-analogues of the Sun brothers’ generalizations of the above supercongruences. Our proofs are elementary in nature and use the theory of basic hypergeometric series and combinatorial q-binomial identities including a new q-Clausen type summation formula.