Proof of a q -supercongruence conjectured by Guo and Schlosser
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Proof of a q-supercongruence conjectured by Guo and Schlosser Long Li1 · Su-Dan Wang2 Received: 1 June 2020 / Accepted: 13 August 2020 © The Royal Academy of Sciences, Madrid 2020
Abstract In this paper, we confirm the following conjecture of Guo and Schlosser: for any odd integer n > 1 and M = (n + 1)/2 or n − 1, M (q −2 ; q 4 )4 [4k − 1]q 2 [4k − 1]2 4 4 4k q 4k ≡ (2q + 2q −1 − 1)[n]q4 2 (q ; q )k k=0
(mod [n]q4 2 n (q 2 )),
where [n] = [n]q = (1−q n )/(1−q), (a; q)0 = 1, (a; q)k = (1−a)(1−aq) · · · (1−aq k−1 ) for k ≥ 1 and n (q) denotes the n-th cyclotomic polynomial. Keywords Cyclotomic polynomial · q-binomial coefficients · Supercongruences · Identities Mathematics Subject Classification Primary 11B65; Secondary 11A07 · 11F33
1 Introduction In 1997, Van Hamme [18] conjectured 13 supercongruences involving truncated forms of Ramanujan’s and Ramanujan-like formulas for 1/π. In particular, the following supercongruence of Van Hamme [18, (B.2)], ( p−1)/2 4k + 1 2k 3 p−1 ≡ p(−1) 2 (mod p 3 ) k (−64) k k=0
Long Li was partially supported by the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (Grant 19KJB110006). Su-Dan Wang was partially supported by the Natural Science Foundation of Inner Mongolia, China (Grant 2020BS01012).
B
Su-Dan Wang [email protected] Long Li [email protected]
1
School of Mathematics and Statistics, Huaiyin Normal University, Huai’an 223300, Jiangsu, People’s Republic of China
2
College of Mathematics Science, Inner Mongolia Normal University, Inner Mongolia, Huhhot 010022, People’s Republic of China 0123456789().: V,-vol
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L. Li, S.-D. Wang
was first proved by Mortenson [15] using a 6 F5 transformation and a technical evaluation of a quotient of Gamma functions, where p is an odd prime. Recently, q-analogues of congruences and supercongruences have caught the interests of many authors (see, for example, [1,3–7,9– 13,16,17,19,20]). In [2, Conjecture 4.3], Guo conjectured: for any prime p > 3 and positive integer r , ( pr −1)/2 k=0 r −1 p
k=0
(4k + 1)3 2k 4 ≡ − pr k 256k
(mod pr +3 ),
(1.1)
(4k + 1)3 2k 4 ≡ − pr 256k k
(mod pr +3 ).
(1.2)
Later, Guo [4, Theorem 1.1] proved (1.1) and (1.2) by establishing the following complete q-analogues of them: for odd integer n > 1, modulo [n]q 2 n (q 2 )3 , (n−1)/2
[4k + 1]q 2 [4k + 1]2
k=0
(q 2 ; q 4 )4k (q 4 ; q 4 )4k
q −4k ≡ −[n]q 2
2q 2−n (n 2 − 1)(1 − q 2 )2 q 2−n −[n]q3 2 , 2 1+q 12(1 + q 2 )
n−1 2 2 2 2−n (q 2 ; q 4 )4 2q 2−n 3 (n − 1)(1 − q ) q [4k + 1]q 2 [4k + 1]2 4 4 4k q −4k ≡ −[n]q 2 −[n] . 2 q 1 + q2 12(1 + q 2 ) (q ; q )k k=0
Here and throughout the paper, (a; q)n = (1 − a)(1 − aq) . . . (1 − aq n−1 ) denotes the qshifted factorial, [n] = [n]q = 1 + q + · · · + q n−1 stands for the q-integer, and n (q) is the n-th cyclotomic polynomial in q, i.e.,
n (q) =
n
(q − ζ k )
1≤k≤n gcd(n,k)=1
with ζ being an n-th primitive root of unity. Guo [4, Theorem 1.2] also proved that, for any odd integer n > 1, (n+1)/2
(q −2 ; q 4 )4k
q 4k ≡
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