Using Watson's terminating
Citation: Victor J. W. Guo. A family of q-congruences modulo the square of a cyclotomic polynomial[J]. Electronic Research Archive, 2020, 28(2): 1031-1036. doi: 10.3934/era.2020055
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Using Watson's terminating
In 1997, Van Hamme [19,(H.2)] proved the following supercongruence: for any prime
(p−1)/2∑k=0(12)3kk!3≡0(modp2), | (1.1) |
where
mp−1∑k=0(12)3kk!3≡0(modp2). | (1.2) |
The first purpose of this paper is to prove the following
Theorem 1.1. Let
mn−1∑k=0(1+q4k+1)(q2;q4)3k(1+q)(q4;q4)3kqk≡0(modΦn(q)2), | (1.3) |
[5pt]mn+(n−1)/2∑k=0(1+q4k+1)(q2;q4)3k(1+q)(q4;q4)3kqk≡0(modΦn(q)2). | (1.4) |
Here and in what follows, the
Φn(q)=∏1≤k≤ngcd(n,k)=1(q−ζk), |
where
The
In 2016, Swisher [18,(H.3) with
(p2−1)/2∑k=0(12)3kk!3≡p2(modp5), | (1.5) |
The second purpose of this paper is to prove the following
Theorem 1.2. Let
(n2−1)/2∑k=0(1+q4k+1)(q2;q4)3k(1+q)(q4;q4)3kqk≡[n2]q2(q3;q4)(n2−1)/2(q5;q4)(n2−1)/2q(1−n2)/2, | (1.6) |
[5pt]n2−1∑k=0(1+q4k+1)(q2;q4)3k(1+q)(q4;q4)3kqk≡[n2]q2(q3;q4)(n2−1)/2(q5;q4)(n2−1)/2q(1−n2)/2. | (1.7) |
Let
limq→1(q3;q4)(p2−1)/2(q5;q4)(p2−1)/2=(p2−1)/2∏k=14k−14k+1=(34)(p2−1)/2(54)(p2−1)/2. |
Therefore, we obtain the following conclusion.
Corollary 1. Let
(p2−1)/2∑k=0(12)3kk!3≡p2(34)(p2−1)/2(54)(p2−1)/2(modp4), | (1.8) |
[5pt]p2−1∑k=0(12)3kk!3≡p2(34)(p2−1)/2(54)(p2−1)/2(modp4). | (1.9) |
Comparing (1.5) and (1.8), we would like to propose the following conjecture, which was recently confirmed by Wang and Pan [20].
Conjecture 1. Let
(p2r−1)/2∏k=14k−14k+1≡1(modp2). | (1.10) |
Note that the
We need to use Watson's terminating
8ϕ7[a,qa12,−qa12,b,c,d,e,q−na12,−a12,aq/b,aq/c,aq/d,aq/e,aqn+1;q,a2qn+2bcde]=(aq;q)n(aq/de;q)n(aq/d;q)n(aq/e;q)n4ϕ3[aq/bc, d, e, q−naq/b,aq/c,deq−n/a;q,q], | (2.1) |
where the basic hypergeometric
r+1ϕr[a1,a2,…,ar+1b1,…,br;q,z]:=∞∑k=0(a1;q)k(a2;q)k…(ar+1;q)k(q;q)k(b1;q)k⋯(br;q)kzk. |
The left-hand side of (1.4) with
8ϕ7[q2,q5,−q5,q2,q,q2,q4+(4m+2)n,q2−(4m+2)nq,−q,q4,q5,q4,q2−(4m+2)n,q4+(4m+2)n;q4,q]. | (2.2) |
By Watson's transformation formula (2.1) with
(q6;q4)mn+(n−1)/2(q−(4m+2)n;q4)mn+(n−1)/2(q4;q4)mn+(n−1)/2(q2−(4m+2)n;q4)mn+(n−1)/2×4ϕ3[q3, q2,q4+(4m+2)n, q2−(4m+2)nq4,q5,q6;q4,q4]. | (2.3) |
It is not difficult to see that there are exactly
(q3;q4)k(q2;q4)k(q4+(4m+2)n;q4)k(q2−(4m+2)n;q4)k(q4;q4)2k(q5;q4)k(q6;q4)kq4k |
in the
It is easy to see that
The author and Zudilin [11,Theorem 1.1] proved that, for any positive odd integer
(n−1)/2∑k=0(1+q4k+1)(q2;q4)3k(1+q)(q4;q4)3kqk≡[n]q2(q3;q4)(n−1)/2(q5;q4)(n−1)/2q(1−n)/2(modΦn(q)2), | (3.1) |
which is also true when the sum on the left-hand side of (3.1) is over
It is easy to see that, for
[n2]q2(q3;q4)(n2−1)/2(q5;q4)(n2−1)/2q(1−n2)/2≡0(modΦn(q)2) |
because
Swisher's (H.3) conjecture also indicates that, for positive integer
(p2r−1)/2∑k=0(12)3kk!3≡p2r(modp2r+3). | (4.1) |
Motivated by (4.1), we shall give the following generalization of Theorem 1.2.
Theorem 4.1. Let
(n2r−1)/2∑k=0(1+q4k+1)(q2;q4)3k(1+q)(q4;q4)3kqk≡[n2r]q2(q3;q4)(n2r−1)/2(q5;q4)(n2r−1)/2q(1−n2r)/2, | (4.2) |
[5pt]n2r−1∑k=0(1+q4k+1)(q2;q4)3k(1+q)(q4;q4)3kqk≡[n2r]q2(q3;q4)(n2r−1)/2(q5;q4)(n2r−1)/2q(1−n2r)/2. | (4.3) |
Proof. Replacing
[n2r]q2(q3;q4)(n2r−1)/2(q5;q4)(n2r−1)/2q(1−n2r)/2≡0(modr∏j=1Φn2j−1(q)2). |
Further, by Theorem 1.1, we can easily deduce that the left-hand sides of (4.2) and (4.3) are also congruent to
Letting
Corollary 2. Let
(p2r−1)/2∑k=0(12)3kk!3≡p2r(34)(p2r−1)/2(54)(p2r−1)/2(modp2r+2), | (4.4) |
[5pt]p2r−1∑k=0(12)3kk!3≡p2r(34)(p2r−1)/2(54)(p2r−1)/2(modp2r+2). | (4.5) |
In light of (1.10), the supercongruence (4.4) implies that (4.1) holds modulo
It is known that
Conjecture 2 (Guo and Zudilin). Let
mn−1∑k=0(q;q2)2k(q2;q4)k(q2;q2)2k(q4;q4)kq2k≡0(modΦn(q)2),mn+(n−1)/2∑k=0(q;q2)2k(q2;q4)k(q2;q2)2k(q4;q4)kq2k≡0(modΦn(q)2). | (4.6) |
The author and Zudilin [10,Theorem 2] themselves have proved (4.6) for the
Conjecture 3. Let
(n2−1)/2∑k=0(q;q2)2k(q2;q4)k(q2;q2)2k(q4;q4)kq2k≡[n2](q3;q4)(n2−1)/2(q5;q4)(n2−1)/2,n2−1∑k=0(q;q2)2k(q2;q4)k(q2;q2)2k(q4;q4)kq2k≡[n2](q3;q4)(n2−1)/2(q5;q4)(n2−1)/2. |
There are similar such new
The author is grateful to the two anonymous referees for their careful readings of this paper.
[1] |
G. Gasper and M. Rahman, Basic Hypergeometric Series, Encyclopedia of Mathematics and Its Applications, 96, Cambridge University Press, Cambridge, 2004. doi: 10.1017/CBO9780511526251
![]() |
[2] |
C.-Y. Gu and V. J. W. Guo, q-Analogues of two supercongruences of Z.-W. Sun, Czechoslovak Math. J., in press. doi: 10.21136/CMJ.2020.0516-18
![]() |
[3] |
V. J. W. Guo, Common q-analogues of some different supercongruences, Results Math., 74 (2019), 15pp. doi: 10.1007/s00025-019-1056-1
![]() |
[4] |
V. J. W. Guo, Proof of a generalization of the (B.2) supercongruence of Van Hamme through a q-microscope, Adv. in Appl. Math., 116 (2020), 19pp. doi: 10.1016/j.aam.2020.102016
![]() |
[5] |
V. J. W. Guo, q-Analogues of Dwork-type supercongruences, J. Math. Anal. Appl., 487 (2020), 9pp. doi: 10.1016/j.jmaa.2020.124022
![]() |
[6] |
q-Analogues of two Ramanujan-type formulas for 1/π. J. Difference Equ. Appl. (2018) 24: 1368-1373. ![]() |
[7] |
V. J. W. Guo and M. J. Schlosser, Some new q-congruences for truncated basic hypergeometric series: Even powers, Results Math., 75 (2020), 15pp. doi: 10.1007/s00025-019-1126-4
![]() |
[8] |
Some q-supercongruences for truncated basic hypergeometric series. Acta Arith. (2015) 171: 309-326. ![]() |
[9] |
A q-microscope for supercongruences. Adv. Math. (2019) 346: 329-358. ![]() |
[10] |
On a q-deformation of modular forms. J. Math. Anal. Appl. (2019) 475: 1636-1646. ![]() |
[11] |
V. J. W. Guo and W. Zudilin, A common q-analogue of two supercongruences, Results Math., 75 (2020), 11pp. doi: 10.1007/s00025-020-1168-7
![]() |
[12] |
Some supercongruences on truncated 3F2 hypergeometric series. J. Difference Equ. Appl. (2018) 24: 438-451. ![]() |
[13] |
On Van Hamme's (A.2) and (H.2) supercongruences. J. Math. Anal. Appl. (2019) 471: 613-622. ![]() |
[14] |
Some supercongruences occurring in truncated hypergeometric series. Adv. Math. (2016) 290: 773-808. ![]() |
[15] |
H.-X. Ni and H. Pan, Some symmetric q-congruences modulo the square of a cyclotomic polynomial, J. Math. Anal. Appl., 481 (2020), 12pp. doi: 10.1016/j.jmaa.2019.07.062
![]() |
[16] |
Generalized Legendre polynomials and related supercongruences. J. Number Theory (2014) 143: 293-319. ![]() |
[17] |
On sums of Apéry polynomials and related congruences. J. Number Theory (2012) 132: 2673-2699. ![]() |
[18] |
H. Swisher, On the supercongruence conjectures of van Hamme, Res. Math. Sci., 2 (2015), 21pp. doi: 10.1186/s40687-015-0037-6
![]() |
[19] | L. Van Hamme, Some conjectures concerning partial sums of generalized hypergeometric series, in p-Adic Functional Analysis, Lecture Notes in Pure and Appl. Math., 192, Dekker, New York, 1997,223–236. |
[20] | C. Wang and H. Pan, On a conjectural congruence of Guo, preprint, arXiv: 2001.08347. |
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