A Common q-Analogue of Two Supercongruences

We give a q-congruence whose specializations q=-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q=-1$$\end{document} and q=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q=1$$\end{document} correspond to supercongruences (B.2) and (H.2) on Van Hamme’s list (in: p-Adic Functional Analysis (Nijmegen, 1996), Lecture Notes in Pure and Applied Mathematics, vol 192. Dekker, New York, pp 223–236, 1997): ∑k=0(p-1)/2(-1)k(4k+1)Ak≡p(-1)(p-1)/2(modp3)and∑k=0(p-1)/2Ak≡a(p)(modp2),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\sum _{k=0}^{(p-1)/2}(-1)^k(4k+1)A_k\equiv p(-1)^{(p-1)/2}\;({\text {mod}}p^3) \quad \text {and}\quad \\&\sum _{k=0}^{(p-1)/2}A_k\equiv a(p)\;({\text {mod}}p^2), \end{aligned}$$\end{document}where p>2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p>2$$\end{document} is prime, Ak=∏j=0k-1(1/2+j1+j)3=126k2kk3fork=0,1,2,…,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} A_k=\prod _{j=0}^{k-1}\biggl (\frac{1/2+j}{1+j}\biggr )^3=\frac{1}{2^{6k}}{\left( {\begin{array}{c}2k\\ k\end{array}}\right) }^3 \quad \text {for}\; k=0,1,2,\ldots , \end{aligned}$$\end{document}and a(p) is the pth coefficient of the modular form q∏j=1∞(1-q4j)6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q\prod _{j=1}^\infty (1-q^{4j})^6$$\end{document} (of weight 3). We complement our result with a general common q-congruence for related hypergeometric sums.


Introduction
The formula of Bauer [1] from 1859, 2k k 3 for k = 0, 1, 2, . . . , (1.1) is one of traditional targets for different methods of proofs of hypergeometric identities. Its special status is probably linked to the fact that it belongs to a family of series for 1/π of Ramanujan type, after Ramanujan [21] brought to life in 1914 a long list of similar looking equalities for the constant but with a faster convergence. Identity (1.1) is a particular instance of 4 F 3 hypergeometric summation (known to Ramanujan) but there are several proofs of it, including the original one [1] of Bauer, that do not require any knowledge of hypergeometric functions. One notable-computer-proof of (1.1) was given in 1994 by Ekhad and Zeilberger [2] using the Wilf-Zeilberger (WZ) method of creative telescoping. It was observed in 1997 by Van Hamme [28] that many Ramanujan's and Ramanujan-like evaluations have nice p-adic analogues; for example, the congruence (tagged (B.2) on Van Hamme's list) is valid for any prime p > 2 and corresponds to the equality (1.1). The congruence (1.2) was first proved by Mortenson [19] using a 6 F 5 hypergeometric transformation; it later received another proof by one of these authors [29] via the WZ method [in fact, using the very same 'WZ certificate' as in [2] for (1.1)]. Notice that (1.2) is an example of supercongruence meaning that it holds modulo a power of p greater than 1. Another entry on Van Hamme's 1997 list [28], tagged (H.2), is the congruence  Recently, these authors [14,Theorem 2] proved that, for any positive odd integer n, modulo Φ n (q) 2 , (1.5) Here and in what follows, Φ n (q) denotes the nth cyclotomic polynomial; the q-shifted factorial is given by (a; q) 0 = 1 and (a; q) n = (1 − a)(1 − aq) . . .
A goal of this note is to present the following new q-analogue of Van Hamme's supercongruence (1.3).
Recently, Mao and Pan [18] (see also Sun [25, In this note, we prove the following q-analogue of (1.8).

Theorem 1.2. Let n > 1 be an odd integer. Then
The n ≡ 1 (mod 4) case of Theorem 1.2 also confirms a conjecture of the first author and Schlosser [11,Conjecture 10.2].
For n prime, letting q → 1 in Theorem 1.2 we obtain the following generalization of (1.8).

Corollary 1.3. Let p be an odd prime. Then
On the other hand, for n prime and q → −1 in Theorem 1.2, we are led to the following result: Vol. 75 (2020) A Common q-Analogue Page 5 of 11 46 It should be mentioned that a different q-analogue of (1.9) was given in [13,Theorem 4.9] with r = −1, d = 2 and a = 1 (see also [11,Section 5]). Moreover, for the summation formula we have the following q-analogue.
Both Theorems 1.1 and 1.2 are particular cases of a more general result, which we state and prove in the next section, while Theorem 1.4 follows from a classical q-identity.
The following easily proved q-congruence (see [11,Lemma 3.1]) is necessary in our derivation of Theorem 2.1.
Like the proofs given in [13], we start with the following generalization of (1.7) with an extra parameter a.
Proof of Theorem 2.1. We assume that n > 1, since the n = 1 case (making = 0 only possible) is trivial. The limits of the denominators on both sides of (2.2) as a → 1 are relatively prime to Φ n (q 2 ), since k is in the range 0 k (n − 1)/2 + . On the other hand, the limit of (1 − aq 2n )(a − q 2n ) as a → 1 contains the factor Φ n (q 2 ) 2 .

Discussion
The method of creative microscoping used in our proofs indicates the origin of q-congruences from infinite q-hypergeometric identities; for example, the q-congruence (1.7) corresponds to the identity which is just a particular instance of (2.3). Note that the limiting cases as q → −1 and q → 1 of (3.1) give the formulas (1.1) and which originates from a q-analogue of Watson's 3 F 2 sum [3, Appendix (II. 16)]. When we choose a = q n (or a = q −n ) in (3.3), for n > 1 odd, we get the sum terminating after (n − 1)/2 terms on the left-hand side of (3.3), while the right-hand side vanishes if n is of the form 4m + 3 and it becomes equal to for any N ≥ (n − 1)/2. The limiting a → 1 case of the congruences can be shown to be

4)
Vol. 75 (2020) A Common q-Analogue Page 9 of 11 46 modulo Φ n (q) 2 Φ n (−q). This is quite similar in spirit to (1.5), though still far from constructing q-analogues for the coefficients a(p) in (1.6) of the modular form f (τ ). The latter means that a hunt for q-rational functions, which equal the left-hand side of (1.5) or (3.4) modulo Φ n (q) 2 and specialize to a(n) as q → 1 (at least for n prime), is still on its way. Such q-rational functions are also expected to be self-reciprocal, that is, invariant under the involution q → 1/q, as all the left-and right-hand sides in (1.5), (1.7), (3.4) and also (2.1) are.
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