Some New q-Congruences for Truncated Basic Hypergeometric Series

Victor Guo,Michael Schlosser

doi:10.3390/sym11020268

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.

Highlights

In 1914, Ramanujan [1] stated rather mysteriously a number of formulas for 1/π, including ∞( )3k (6k + 1) k = . π k! k =0 ∑In 1997, Van Hamme [2] conjectured interesting p-adic analogues of Ramanujan’s or Ramanujan-type formulas for 1/π, such as p −1 p −1∑ k!3 (6k + 1) 4k ≡ p(−1) 2 k =0, (1)where ( a)n = a( a + 1) · · · ( a + n − 1) denotes the Pochhammer symbol and p is an odd prime
All of the 13 supercongruences have been confirmed by different techniques up to now
During the past few years, q-analogues of congruences and supercongruences have caught the interests of many authors

Summary

Introduction

In 1914, Ramanujan [1] stated rather mysteriously a number of formulas for 1/π, including. We shall prove the following result, which was originally conjectured by the first author [28]. Where 3 denotes the Legendre symbol modulo 3 This q-congruence was originally conjectured in [10] when n = p is an odd prime. We shall prove Theorems 1 and 2 in Sections 2 and 3 by using the creative microscoping method developed by the first author and Zudilin [22]. We prove these by first establishing their parametric generalizations modulo (1 − aqn )( a − qn ) and letting a → 1.

Proof of Theorem 1

Proof of Theorem 2

Proof of Theorem 3

Proof of Theorem 4

Concluding Remarks and Open Problems