Abstract
Two q-supercongruences of truncated basic hypergeometric series containing two free parameters are established by employing specific identities for basic hypergeometric series. The results partly extend two q-supercongruences that were earlier conjectured by the same authors and involve q-supercongruences modulo the square and the cube of a cyclotomic polynomial. One of the newly proved q-supercongruences is even conjectured to hold modulo the fourth power of a cyclotomic polynomial.
Highlights
In 1914, Ramanujan [25] listed a number of representations of 1/π, including
We shall prove that the respective first cases of Conjectures 1 and 2 are true by establishing the following more general result
Note that the non-zero terms in the multi-summation in (2.7) are those indexed by ( j1, . . . , jm−1) that satisfy the inequality j1 + · · · + jm−1 ≤/d because the factor (qr−(d−1)n; qd ) j1+···+ jm−1 appears in the numerator
Summary
Q-congruences and q-supercongruences have been established by different authors (see, for example, [5,6,7,8,9,10,11,12,13,15,16,17,18,19,20,21,23,27,30,31,32,34]). The present authors [9] proved that, for any odd integer d ≥ 5, n−1. We shall prove that the respective first cases of Conjectures 1 and 2 are true by establishing the following more general result. The following generalization of the respective second cases of Conjectures 1 and 2 should be true. It should be pointed out that Andrews’ transformation plays an important part in combinatorics and number theory (see [7] and the introduction of [12] for more such examples)
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