Chuanan Wei

In 1987, Andrews and Baxter presented six kinds of q-trinomial coefficients in exploring the solution of a model in statistical mechanics. In this paper, we are going to give some q-supercongruences for the truncated forms of these polynomials.

In terms of several summation and transformation formulas for basic hypergeometric series, two forms of the Chinese remainder theorem for coprime polynomials, the creative microscoping method introduced by Guo and Zudilin, Guo and Li's lemma, and El Bachraoui's lemma, we establish some $q$-supercongruences for multiple basic hypergeometric series modulo the fifth and sixth powers of a cyclotomic polynomial. In detail, we generalize Guo and Li's two $q$-supercongruences for double basic hypergeometric series, which are related to $q$-analogues of Van Hamme's (C.2) supercongruence and Long's supercongruence, respectively. In addition, we also present two conclusions for double and triple hypergeometric series associated with Van Hamme's (D.2) supercongruence.

With the help of El Bachraoui's lemma, the creative microscoping method, and a new form of the Chinese remainder theorem for coprime polynomials, we prove a $q$-supercongruence for double series and a $q$-supercongruence for triple series modulo the sixth power of a cyclotomic polynomial. As conclusions, two corresponding supercongruences for double and triple series, which are associated with the (D.2) supercongruence of Van Hamme, are given.