q -Analogues of Two Supercongruences of Z.-W. Sun
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Czechoslovak Mathematical Journal
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q-ANALOGUES OF TWO SUPERCONGRUENCES OF Z.-W. SUN Cheng-Yang Gu, Victor J. W. Guo, Huai’an Received November 26, 2018. Published online January 22, 2020.
Abstract. Z.-W. Sun: (pr −1)/2
X
k=0
We give several different q-analogues of the following two congruences of
1 8k
2k k
!
2 ≡ (mod p2 ) pr
(pr −1)/2
and
X
k=0
1 16k
2k k
!
≡
3 (mod p2 ), pr
where p is an odd prime, r is a positive integer, and ( m n ) is the Jacobi symbol. The proofs of them require the use of some curious q-series identities, two of which are related to Franklin’s involution on partitions into distinct parts. We also confirm a conjecture of the latter author and Zeng in 2012. Keywords: congruences; q-binomial coefficient; cyclotomic polynomial; Franklin’s involution MSC 2010 : 11B65, 05A10, 05A30, 11A07
1. Introduction Among other things, Sun in [14], (1.7) and (1.8) proved the congruences (pr −1)/2
(1.1)
X
k=0 (pr −1)/2
(1.2)
X
k=0
2 1 2k ≡ r (mod p2 ), k 8 k p
1 2k 3 ≡ r (mod p2 ), k 16 k p
The latter author was partially supported by the National Natural Science Foundation of China (grant 11771175). DOI: 10.21136/CMJ.2020.0516-18
1
where p is an odd prime, r is a positive integer, and ( m n ) is the Jacobi symbol. Recently, the latter author and Liu in [6], Theorem 1.2 gave the following q-analogue of (1.1): for odd n, (n−1)/2
X
(1.3)
k=0
2
2 (q; q 2 )k q k ≡ (−q)(1−n )/8 (mod Φn (q)2 ). (q 4 ; q 4 )k
Here and in what follows, (a; q)n = (1−a)(1−aq) . . . (1−aq n−1 ) and Φn (q) is the nth cyclotomic polynomial in q. The first aim of this paper is to give q-analogues of (1.1) and (1.2) as follows. Theorem 1.1. Let n be a positive odd integer. Then (n−1)/2
(1.4)
X
2 2 (q; q 2 )k q 2k q 2⌊(n+1)/4⌋ (mod Φn (q)2 ), ≡ 2 2 2 (q ; q )k (−q; q )k n
X
(q 4 ; q 4 )
k=0 (n−1)/2
(1.5)
k=0
3 2 (q; q 2 )k q 2k q (n −1)/12 (mod Φn (q)2 ), ≡ 2 n k (−q; q )k
where ⌊x⌋ denotes the largest integer not exceeding x. It is easy to see that the congruences (1.4) and (1.5) reduce to (1.1) and (1.2), respectively, when q → 1 and n = pr . Recall that the q-binomial coefficients nk are defined by (q; q)n n n = = (q; q)k (q; q)n−k k k q 0
if 0 6 k 6 n, otherwise.
Moreover, the q-integer is defined as [n] = [n]q = (1 − q n )/(1 − q). The second aim of this paper is to give the following result, which in the case n = pr confirms a conjecture of the latter author and Zeng, see [8], Conjecture 5.13. Theorem 1.2. Let n be a positive integer. Then
(1.6)
n−1 X k=0
2
q
(n−k)2
n+k k
2
2 n−1 ≡ q[n] (mod Φn (q)2 ). k
Online first
Note that, exactly similarly to the proof of Theorem 5.3 in [8], we can show that n−1 X
(1.7)
q
k=0
(n−k)2
n+k k
2
2 n−1 ≡ 0 (mod [n]). k
Therefore, combining the congruences (1.6) and (1.7), we see that the congruence (1.6) also holds modulo [n]Φn (q). We refer the reader to [7] and references therein for other congruences on sums involving q-binomial coefficients. Suggested by the
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