Some q-supercongruences for truncated basic hypergeometric series Victor J. W. Guo School of Mathematical Sciences Huaiyin Normal University Huaian, Jiangsu 223300 People’s Republic of China E-mail: [email protected] Jiang Zeng Universit´ede Lyon Universit´eLyon 1 Institut Camille Jordan, UMR 5208 du CNRS 43, boulevard du 11 novembre 1918 F-69622 Villeurbanne Cedex, France E-mail: [email protected] Abstract For any odd prime p we obtain q-analogues of van Hamme’s and Rodriguez-Villegas’ supercongruences involving products of three bi- nomial coefficients such as p−1 2 3 2k X 2k q 2 2 2 2 2 ≡ 0 (mod [p] ) for p ≡ 3 (mod 4), k 2 (−q ; q ) (−q; q) k=0 q k 2k p−1 2 3 2 3 3k X 2k (q; q )k(q ; q )kq 2 6 6 2 ≡ 0 (mod [p] ) for p ≡ 2 (mod 3), k 3 (q ; q ) k=0 q k p−1 n−1 where [p] = 1+q+···+q and (a; q)n = (1−a)(1−aq) ··· (1−aq ). We also prove q-analogues of the Sun brothers’ generalizations of the 2010 Mathematics Subject Classification: Primary 11B65; Secondary 05A10, 05A30. Key words and phrases: p-adic integer, congruences, supercongruences, basic hyper- geometric series, q-binomial theorem, q-Chu-Vandermonde formula. 1 2 V. J. W. Guo and J. Zeng above supercongruences. Our proofs are elementary in nature and use the theory of basic hypergeometric series and combinatorial q- binomial identities including a new q-Clausen-type summation for- mula. 1 Introduction We shall follow the standard q-notations from [4]. The q-shifted factorial n−1 is defined by (a; q)n = (1 − a)(1 − aq) ··· (1 − aq ) for n = 1, 2,..., and 1−qn (a; q)0 = 1, while the q-integer is denoted by [n] := 1−q . In a previous pa- per [6], we proposed several q-analogues of Rodriguez-Villegas and Morten- son type congruences for truncated hypergeometric series conjectured by Rodriguez-Villegas [11, 9] and proved the following q-analogue of one of their supercongruences: p−1 2 2 2 X (q; q )k −1 1−p 2 (1.1) ≡ q 4 (mod [p] ). (q2; q2)2 p k=0 k · Here and in what follows p denotes an odd prime, and ( p ) is the Legendre symbol modulo p. Recently, by using the properties of generalized Legendre polynomials, Z.-H. Sun [13, Theorem 2.5] proved the following remarkable congruence: p−1 X 2ka−1 − a 1 (1.2) ≡ 0 (mod p2), k k k 4k k=0 where a is a p-adic integer such that the least nonnegative residue of a mod- ulo p is odd. It is interesting to note that (1.2) is a common generalization of several congruences due to van Hamme and Rodriguez-Villegas. On the other hand, van Hamme [17] proved the following congruence (1.3) p−1 2 3 ( X 2k 1 4x2 − 2p (mod p2), if p = x2 + y2 with x odd, ≡ k 64k 0 (mod p2), if p ≡ 3 (mod 4), k=0 of which a generalization was recently conjectured by H. Swisher [16, (H.3)]. The aim of this paper is to prove some q-supercongruences for certain trun- cated basic hypergeometric series generalizing the above results. n Recall that the q-binomial coefficients k are then defined by n−k+1 (q ; q)k n n , if 0 6 k 6 n, = = (q; q)k k k q 0, otherwise. truncated basic hypergeometric series 3 The first aim of this paper is to give a unified q-analogue of (1.3) and Z.-W. Sun’s generalization [14, Theorem 1.1(i)] of (1.3). p−1 Theorem 1.1. Let 0 6 s 6 2 . Then the following congruence holds modulo [p]2: (1.4) p−1 2 2 X 2k 2k q2k k k + s (−q2; q2)2(−q; q)2 k=0 q2 q2 k 2k 2 2 2 2 2 p−1 p−2s−1 p+2s−1 p−1 2 (q ; q ) (q ; q ) s −s 2 2 2 p−1 (−1) q 2 , if s ≡ mod 2, p−2s−1 (q4; q4)2 2 ≡ 4 q4 p−1 2 0, otherwise. When s = 0, the congruence (1.4) reduces to the following result. Corollary 1.2. There holds p−1 2 3 X 2k q2k (1.5) k (−q2; q2)2(−q; q)2 k=0 q2 k 2k p−1 2 p−1 2 1 q 2 , if p ≡ 1 (mod 4), p−1 (−q2; q2)2 2 ≡ 4 q4 p−1 (mod [p] ). 2 0, otherwise, To see that (1.5) is a q-analogue of (1.3) we need to recall a known result. Let p be a prime such that p ≡ 1 (mod 4) and p = x2 + y2 with x ≡ 1 (mod 4), then we have the so-called Beukers-Chowla-Dwork-Evans congruence [3, 10]: p−1 2p−1 + 1 p 2 ≡ 2x − (mod p2), p−1 2 2x 4 which easily implies Sun’s congruence [12, Lemma 3.4]) 2 p−1 1 (1.6) 2 ≡ 4x2 − 2p (mod p2). p−1 2p−1 4 Using (1.6), it is clear that (1.5) reduces to (1.3) when q → 1. Generalizing a result of Mortenson [7, 8], Z.-W. Sun [15, (1.4)] obtained the following congruence p−1 2 2k2k+2s X −1 p − 1 (1.7) k k+s ≡ (mod p2), for 0 s . 42k+s p 6 6 2 k=0 4 V. J. W. Guo and J. Zeng For any p-adic integer x, let hxip denote the least nonnegative residue of x modulo p. The second aim of this paper is to give a unified q-analogue of (1.2) and (1.7). Theorem 1.3. Let m and r two positive integers with p - m. Let s 6 r m−r r min{h− m ip, h− m ip} be a nonnegative integer. If h− m ip ≡ s + 1 (mod 2), then p−1 2 m m r m m−r m mk X (q ; q )2k(q ; q )k(q ; q )kq 2 (1.8) m m m m 2m 2m 2 ≡ 0 (mod [p] ), (q ; q )k−s(q ; q )k+s(q ; q ) k=s k p−1 m m r m m−r m mk X (q ; q )2k(q ; q )k(q ; q )kq 2 (1.9) m m m m 2m 2m 2 ≡ 0 (mod [p] ). (q ; q )k−s(q ; q )k+s(q ; q ) k=s k r If h− m ip ≡ s (mod 2), then the following congruence holds: p−1 2 m m r m m−r m mk X (q ; q )2k(q ; q )k(q ; q )kq (1.10) m m m m 2m 2m 2 (q ; q )k−s(q ; q )k+s(q ; q ) k=s k (s+ p−1 )m m 2m m 2m 2 r ≡ q (q ; q ) h− ip−s (q ; q ) m−r m h− m ip−s 2 2 −mh− r i m −mh− m−r i m (q m p ; q ) (q m p ; q ) × s s (mod [p]). 2m 2m r 2m 2m (q ; q ) h− ip+s (q ; q ) m−r m h− m ip+s 2 2 r Letting s = 0, − m = a and q → 1 in (1.9), we obtain (1.2). On the other hand, it is not difficult to see that (see [6]), for any prime p > 5, h− 1 i −3 h− 1 i −2 h− 1 i −1 (−1) 3 p = , (−1) 4 p = , (−1) 6 p = . p p p Taking r = 1 and m = 3, 4, 6 in (1.8), we obtain Corollary 1.4. Let > 5 be a prime and let s be a nonnegative integer. Then the following congruences hold modulo [p]2: p−1 2 3 2 3 3k −3 X 2k (q; q )k(q ; q )kq p − 1 1 + ( p ) ≡ 0, if s and s ≡ mod 2, k + s (q6; q6)2 6 3 2 k=s q3 k p−1 2 4 3 4 4k −2 X 2k (q; q )k(q ; q )kq p − 1 1 + ( p ) ≡ 0, if s and s ≡ mod 2, k + s (q8; q8)2 6 4 2 k=s q4 k p−1 2 6 5 6 6k −1 X 2k (q; q )k(q ; q )kq p − 1 1 + ( p ) ≡ 0, if s and s ≡ mod 2. k + s (q12; q12)2 6 6 2 k=s q6 k truncated basic hypergeometric series 5 The proof of Theorem 1.3 is based on the following q-Clausen type sum- mation formula, which seems new and interesting in its own right. Theorem 1.5. Let n and s be nonnegative integers with s 6 n. Then n −2n 2 k ! n −2n 2 k ! X (q ; q )k(x; q)kq X (q ; q )k(q/x; q)kq (1.11) (q; q)k−s(q; q)k+s (q; q)k−s(q; q)k+s k=s k=s n 2 2 2 −n2 n k 2 2 k2−2nk (−1) (q ; q )nq X (−1) (q ; q )n+k(x; q)k(q/x; q)kq = 2 2 2 2 2 2 . (q ; q )n−s(q ; q )n+s (q ; q )n−k(q; q)k−s(q; q)k+s(q; q)2k k=s We also have the following q-analogue of (1.7), which reduces to (1.1) when s = 0.