A FAMILY OF -SUPERCONGRUENCES MODULO THE CUBE OF A CYCLOTOMIC POLYNOMIAL

VICTOR J. W. GUO,MICHAEL J. SCHLOSSER

doi:10.1017/s0004972721000630

VICTOR J. W. GUO, MICHAEL J. SCHLOSSER

Open Access

https://doi.org/10.1017/s0004972721000630

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Abstract

AbstractWe establish a family of q-supercongruences modulo the cube of a cyclotomic polynomial for truncated basic hypergeometric series. This confirms a weaker form of a conjecture of the present authors. Our proof employs a very-well-poised Karlsson–Minton type summation due to Gasper, together with the ‘creative microscoping’ method introduced by the first author in recent joint work with Zudilin.

Highlights

In 1914, Ramanujan [11] mysteriously stated some representations of 1/π, such as ∞ k=0 (6k + 1) ( )3k k!3 4k =4 π, where (a)n = a(a + 1) · · · (a + n − 1) denotes the rising factorial
We prove the following q-supercongruence, which is a generalisation of the respective second cases of (1.2) and (1.3)
We apply the method of creative microscoping, recently introduced in a paper by the first author with Zudilin [6], to prove Theorem 1.1

Summary

Introduction

In 1914, Ramanujan [11] mysteriously stated some representations of 1/π, such as. 4 π, where (a)n = a(a + 1) · · · (a + n − 1) denotes the rising factorial. In [3], the present authors proved that, for odd integers d 5, n−1. We prove the following q-supercongruence, which is a generalisation of the respective second cases of (1.2) and (1.3). We apply the method of creative microscoping, recently introduced in a paper by the first author with Zudilin [6], to prove Theorem 1.1. These consist of a lemma about an elementary q-congruence modulo a cyclotomic polynomial Φn(q), and a very-well-poised Karlsson–Minton type summation by Gasper of which we need a special case.

Preliminaries

Concluding remarks