Some new $q$-congruences for truncated basic hypergeometric series

Abstract

We provide several new $q$-congruences for truncated basic hypergeometric series, mostly of arbitrary order. Our results include congruences modulo the square or the cube of a cyclotomic polynomial, and in some instances, parametric generalizations thereof. These are established by a variety of techniques including polynomial argument, creative microscoping (a method recently introduced by the first author in collaboration with Zudilin), Andrews' multiseries generalization of the Watson transformation, and induction. We also give a number of related conjectures including congruences modulo the fourth power of a cyclotomic polynomial.

14 pages, more background and references added

References in corpus (9)

• A $q$-microscope for supercongruences
• Ramanujan-type supercongruences
• On the (K.2) supercongruence of Van Hamme
• Some q-analogues of supercongruences of Rodriguez-Villegas
• On a $q$-deformation of modular forms
• Proof of a basic hypergeometric supercongruence modulo the fifth power of a cyclotomic polynomial
• Some new $q$-congruences for truncated basic hypergeometric series
• Factors of Alternating Sums of Products of Binomial and q-Binomial Coefficients
• q-Analogs of some congruences involving Catalan numbers

Cited by in corpus (9)

• Some new $q$-congruences for truncated basic hypergeometric series: even powers
• On a $q$-deformation of modular forms
• Proof of a basic hypergeometric supercongruence modulo the fifth power of a cyclotomic polynomial
• Some new $q$-congruences for truncated basic hypergeometric series
• Symbolic summation methods and hypergeometric supercongruences
• Congruence properties for the trinomial coefficients
• Some congruences related to a congruence of Van Hamme
• ($q$-)Supercongruences hit again
• On a congruence involving $q$-Catalan numbers