New q-Analogues of Van Hamme’s (E.2) Supercongruence and of a Supercongruence by Swisher

In this paper, a couple of q-supercongruences for truncated basic hypergeometric series are proved, most of them modulo the cube of a cyclotomic polynomial. One of these results is a new q-analogue of the (E.2) supercongruence by Van Hamme, another one is a new q-analogue of a supercongruence by Swisher, while the other results are closely related q-supercongruences. The proofs make use of special cases of a very-well-poised \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}_6\phi _5$$\end{document}6ϕ5 summation. In addition, the proofs utilize the method of creative microscoping (which is a method recently introduced by the first author in collaboration with Wadim Zudilin), and the Chinese remainder theorem for coprime polynomials.

Here and in what follows, (a; q) n = (1−a)(1−aq) · · · (1−aq n−1 ) is the q-shifted factorial, [n] = [n] q = (1 − q n )/(1 − q) is the q-integer, and Φ n (q) denotes the n-th cyclotomic polynomial in q, i.e., where ζ is an n-th primitive root of unity. The first author and Zudilin [10, Theorem 3.5 with r = 1] also gave another q-analogue of (1.2): for any positive integer n with n ≡ 1 (mod 6), One of the aims of this paper is to establish the following new q-analogue of (1.2). Theorem 1.1. Let n ≡ 1 (mod 6) be a positive integer. Then We shall also give the following similar result.
Note that the supercongruence (1.5) does not hold modulo [n]Φ n (q) 2 in general, even for n ≡ 1 (mod 6). We take this opportunity to point out that Theorems 1 and 2 in [8] only hold modulo Φ n (q) 3 and Φ n (q) 2 , respectively, but do not hold modulo [n], since Lemma 3 in [8] is not true (it only holds for even integers d).
Similarly, we have the following result.
Note that the q-supercongruence (1.8) does not hold modulo Φ n (q) 3 for n > 2. We shall prove Theorems 1.1, 1.2, and 1.3 modulo Φ n (q) 3 and Theorem 1.4 by using a summation for a very-well-poised 6 φ 5 series and the 'creative microscoping' method introduced by the first author in collaboration with Zudilin [9]. The proof of Theorems 1.1 and 1.3 also requires the use of a lemma previously given by the present authors.
From Theorems 1.2 and 1.4, we can deduce the following supercongruences.

Corollary 1.5. Let p ≡ 1 (mod 3) be a prime. Then
where Γ p (x) denotes the p-adic Gamma function.

Proof of Theorem 1.1
We first give the following result, which is due to the present authors [ If gcd(d, n) = 1, then the above q-congruence also holds for a = 1.
We also need the following result to prove the truth of (1.4) modulo [n].

Lemma 2.2.
Let n be a positive integer coprime with 6, and let a be an indeterminate. Then
Proof. Clearly, Lemma 2.2 is true for n = 1. We now assume that n > 1. By Lemma 2.1, we can easily deduce that the k-th and (m − k)-th terms on the left-hand side of (2.1) cancel each other modulo Φ n (q), i.e., Thus, we have proved that the q-congruence (2.1) holds modulo Φ n (q). Since the numerator contains the factor (q; q 3 ) k , it is easy to see that the k-th summand in (2.2) is congruent to 0 modulo Φ n (q) for m < k n − 1. This proves the q-congruence (2.2) modulo Φ n (q). Now we can prove (2.1) and (2.2) modulo [n]. Let ζ = 1 be an n-th root of unity, not necessarily primitive. In other words, ζ is a primitive root of unity of degree s satisfying s | n and s > 1. Let c q (k) stand for the k-th term on the left-hand side of (2.2), i.e., Taking n = s in the q-congruences (2.1) and (2.2) modulo Φ n (q), we get Therefore, Like most of the q-supercongruences in [9], we have the following parametric generalization of Theorem 1.1.
(2.5) (The infinite series in (2.5) converges for |q| < 1 and |aq/bcd| < 1.) Specializing (2.5) by letting q → q 3 , a = q, b = q 1−n , c = q 1+n , and d = −q 2 , we have This shows that both sides of (2.4) are equal for a = q −n and a = q n . This means that the congruence (2.4) holds modulo 1 − aq n and a − q n . Moreover, by Lemma 2.2, the left-hand side of (2.4) is congruent to 0 modulo [n]. Since 1 − q n (n is odd) is relatively prime to 1 + q k , we see that the right-hand side of (2.4) is also congruent to 0 modulo [n]. Noticing that 1 − aq n , a − q n , and [n] are pairwise coprime polynomials in q, we finish the proof of the theorem. Proof of Theorem 1.1. Since (1−q n ) 2 contains the factor Φ n (q) 2 and (q 3 ; q 3 ) M is coprime with Φ n (q), letting a = 1 in (2.4), we conclude that (1.4) is true modulo Φ n (q) 3 . Note that Lemma 2.2 also holds for a = 1. Namely, the qcongruence (1.4) is true modulo [n] and is therefore also true modulo [n]Φ n (q) 2 . This completes the proof.

Proof of Theorem 1.2
We first give the following parametric generalization of Theorem 1.2: for n ≡ 1 where M = (n − 1)/3 or M = n − 1. The proof of (3.1) is analogous to that of (2.4). This time, we make the substitutions q → q 3 , a = q, b = q 1−n , c = q 1+n , and d = q 2 in (2.5) to obtain Thus, the two sides of (3.1) are equal for a = q −n and a = q n . This means that the congruence (3.1) is true modulo 1 − aq n and a − q n . Moreover, by Lemma 2.1, for m = (n − 1)/3 we can deduce that the k-th and (m − k)-th terms on the left-hand side of (3.1) cancel each other modulo Φ n (q), i.e., (Note that we have utilized the fact that q n/2 ≡ −1 (mod Φ n (q)) for even n.) This proves (3.1) modulo Φ n (q). Finally, letting a = 1 in (3.1), we arrive at the q-supercongruence (1.5).

Proof of Corollaries 1.5 and 1.6
Proof of Corollary 1.5. Letting n = p, where p is a prime congruent to 1 (mod 3), and q → 1 in (1.5), we obtain Recall that the p-adic Gamma function has the properties: for any p-adic integer x, where a 0 (x) ∈ {1, 2, . . . , p} satisfies a 0 (x) ≡ x (mod p). Let Γ(x) be the classical Gamma function. Then (1) .