q-Analogues of two Ramanujan-type supercongruences☆
Introduction
In 1997, Van Hamme [27] developed several p-adic analogues of Ramanujan-type formulas for primes p. For example, the infinity series have the following nice p-adic analogues Here is the Pochhammer symbol and denotes the p-adic Gamma function [23]. Van Hamme [27] himself proved (1.1) and the modulus case of (1.2). In 2011, Long [21] established the following generalization of (1.1): and another interesting supercongruence: Later, Long and Ramakrishna [22, Theorem 2] generalized Van Hamme's conjecture (D.2) to modulus case. They [22, Proposition 25] also gave the following supercongruences In addition, He [15, Corollary 1.3] obtained the following Ramanujan-type supercongruences: for ,
During the past few years, finding q-analogues of known congruences or supercongruence has become an important thing for more and more scholars. Especially, most of Van Hamme's supercongruences have been generalized to the q-world with various techniques [2], [3], [4], [5], [6], [7], [8], [10], [11], [12], [13], [30], [31]. Some other recent progress on q-congruences can be found in [9], [14], [16], [17], [18], [19], [20], [24], [25], [28], [29], [32]. For example, Guo and Schlosser [11, Theorem 2.2] proposed the following partial q-analogue of (1.4), where n is a positive odd integer, and Guo and Wang [12] reproved (1.3) by finding the following q-congruence
By a result of Guo and Schlosser [10, Theorem 1.1], for , we have which is a q-analogue of He's supercongruence (1.7).
Here and in what follows, for an indeterminate q, is the q-shifted factorial and denotes a product of q-shifted factorials, is the q-integer. Moreover, the n-th cyclotomic polynomial is defined as the following product where ζ is an n-th primitive root of unity.
Inspired by the above work, we shall propose a new q-congruence modulo the fourth power of a cyclotomic polynomial, from which we can deduce a q-analogue of the (D.2) supercongruence of Van Hamme and a q-analogue of He's supercongruence (1.6).
The rest of the paper is organized as follows. We will give our main results in the next section. Then the proofs of our q-congruences will be given in Sections 3 and 4 by combining Guo and Zudilin's ‘creative microscoping’ method with the Chinese remainder theorem for coprime polynomials.
Section snippets
The main results
Our first result can be stated as follows. Theorem 1 Let be positive integer. Then, modulo , where or , and
In order to illustrate that our q-congruences are indeed q-analogues of the
Proof of Theorem 1
In order to prove Theorem 1, we need the following results. Lemma 1 Let be positive integers with . Let r be an integer and let a, b, c and e be indeterminates. Then, modulo , where and .
Proof Let denote the k-th term on the left-hand side in (3.1), i.e.,
Proof of Theorem 6 and Proposition 2
Proof of Theorem 6 At the end of [10], Guo and Schlosser proved that, for positive integers n and d satisfying , modulo , They pointed that, for , there is no concrete q-congruence modulo from (4.1) by letting and . Nevertheless, taking the specialization in (4.1), modulo
Conjectures
In 2015, Swisher [26] conjectured a series of Dwork-type congruences about Van Hamme's first 12 supercongruences, such as: for , According to numerical verification, we think that (5.1) have a companion, as Motivated by the above conjecture of Swisher, we put forward the following general congruence conjecture about He's supercongruence (1.6).
Conjecture 1
Acknowledgements
The authors thank the anonymous referees and the editor for many valuable comments on a previous version of this manuscript.
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