Some new q -congruences for truncated basic hypergeometric series: even powers
aa r X i v : . [ m a t h . N T ] N ov SOME NEW q -CONGRUENCES FOR TRUNCATEDBASIC HYPERGEOMETRIC SERIES: EVEN POWERS VICTOR J. W. GUO AND MICHAEL J. SCHLOSSER
Abstract.
We provide several new q -congruences for truncated basic hypergeometricseries with the base being an even power of q . Our results mainly concern congruencesmodulo the square or the cube of a cyclotomic polynomial and complement correspond-ing ones of an earlier paper containing q -congruences for truncated basic hypergeometricseries with the base being an odd power of q . We also give a number of related conjec-tures including q -congruences modulo the fifth power of a cyclotomic polynomial and acongruence for a truncated ordinary hypergeometric series modulo the seventh power ofa prime greater than 3. Introduction
In his first letter to Hardy from 1913, Ramanujan announced that (cf. [2, p. 25, Equa-tion (2)]) ∞ X k =0 (8 k + 1) ( ) k k ! = 2 √ √ π Γ( ) , (1.1)along with similar hypergeometric identities. Here ( a ) n = a ( a + 1) · · · ( a + n −
1) denotesthe Pochhammer symbol. He did not provide proofs. This identity was eventually provedby Hardy in [19, p. 495].In 1997, Van Hamme [32] proposed 13 interesting p -adic analogues ofRamanujan-type formulas for 1 /π [26], such as ( p − / X k =0 (8 k + 1) ( ) k k ! ≡ p Γ p ( )Γ p ( )Γ p ( ) (mod p ) , if p ≡ , (1.2)where p is an odd prime and Γ p is the p -adic gamma function [22]. Van Hamme [32] himselfproved three of them. Nowadays all of the 13 supercongruences have been confirmedby different techniques (see [20, 21, 23, 25, 30]). For some informative background onRamanujan-type supercongruences, we refer the reader to Zudilin’s paper [35]. During Mathematics Subject Classification.
Primary 33D15; Secondary 11A07, 11B65.
Key words and phrases. basic hypergeometric series; supercongruences; q -congruences; cyclotomicpolynomial; Andrews’ transformation.The first author was partially supported by the National Natural Science Foundation of China (grant11771175).The second author was partially supported by FWF Austrian Science Fund grant P 32305. the past few years, congruences and supercongruences have been generalized to the q -world by many authors (see, for example, [6–18, 24, 27, 31]). As explained in [18], q -supercongruences are closely related to studying the asymptotic behaviour of q -series atroots of unity.Recently, the authors [15, Theorems 1 and 2] proved that for odd d > n − X k =0 [2 dk + 1] ( q ; q d ) dk ( q d ; q d ) dk q d ( d − k ≡ ( n ( q ) ) , if n ≡ − d ),0 (mod Φ n ( q ) ) , if n ≡ − (mod d ), (1.3)and for odd d > n > n − X k =0 [2 dk −
1] ( q − ; q d ) dk ( q d ; q d ) dk q d ( d − k ≡ ( n ( q ) ) , if n ≡ d ),0 (mod Φ n ( q ) ) , if n ≡ (mod d ). (1.4)Here and throughout the paper, we adopt the standard q -notation: For an indeterminate q , let ( a ; q ) n = (1 − a )(1 − aq ) · · · (1 − aq n − )be the q -shifted factorial . For convenience, we compactly write( a , a , . . . , a m ; q ) n = ( a ; q ) n ( a ; q ) n · · · ( a m ; q ) n for a product of q -shifted factorials. Moreover,[ n ] = [ n ] q = 1 + q + · · · + q n − denotes the q -integer , which can be defined by [ n ] = ( q n − / ( q −
1) to hold for anyinteger n , including negative n , which in particular gives [ −
1] = − /q (which is neededin the k = 0 terms of (1.6) and (1.8) and at other places in this paper). Furthermore,Φ n ( q ) denotes the n -th cyclotomic polynomial in q , which may be defined asΦ n ( q ) = Y k n gcd( n,k )=1 ( q − ζ k ) , where ζ is an n -th primitive root of unity.In this paper, we shall prove results similar to (1.3) and (1.4) for even d . The firstresult concerns the case d = 2. Theorem 1.
Let n be an odd integer greater than . Then n − X k =0 [4 k + 1] ( q ; q ) k ( q ; q ) k q − k ≡ q [ n ] (mod [ n ] Φ n ( q )) , (1.5) n − X k =0 [4 k −
1] ( q − ; q ) k ( q ; q ) k q k ≡ − [ n ] (mod [ n ] Φ n ( q )) . (1.6) OME q -CONGRUENCES FOR TRUNCATED BASIC SERIES: EVEN POWERS 3 Theorem 2.
Let d > be an even integer and let n be a positive integer with n ≡ − d ) . Then n − X k =0 [2 dk + 1] ( q ; q d ) dk ( q d ; q d ) dk q d ( d − k ≡ n ( q ) ) . (1.7) Theorem 3.
Let d > be an even integer and let n > be an integer with n ≡ d ) . Then n − X k =0 [2 dk −
1] ( q − ; q d ) dk ( q d ; q d ) dk q d ( d − k ≡ n ( q ) ) . (1.8)Although neither (1.7) nor (1.8) holds modulo Φ n ( q ) in general, we have the followingcommon refinement of (1.3) and (1.7). Theorem 4.
Let d > be an integer and let n be a positive integer with n ≡ − d ) .Then n − X k =0 [2 dk + 1] ( q ; q d ) dk ( q d ; q d ) dk q d ( d − k ≡ n ( q ) Φ dn − n ( q )) . (1.9)Let n = p be an odd prime and d = p + 1 in (1.9). Then letting q →
1, we are led to p − X k =0 (2 p + 2 k + 1) ( p +1 ) p +1 k k ! p +1 ≡ p ) . (1.10)Note that Sun [28, Theorem 1.2] proved that for any prime p > p − X k =0 ( p +1 ) p +1 k k ! p +1 ≡ p ) , (1.11)which also holds modulo p for p = 3. Substituting (1.11) into (1.10), we arrive at thefollowing conclusion. Corollary 5.
Let p be an odd prime. Then p − X k =0 k ( p +1 ) p +1 k k ! p +1 ≡ p ) . This result is actually a special case of p − X k =0 k ( p +1 ) p +1 k k ! p +1 ≡ p − p p ) , (1.12)that was conjectured by Sun and was subsequently proved by Gao [4] in her masterthesis. See the discussion around Equation (1.3) in Wang’s paper [33] where (1.12) isfurther generalized to a congruence modolo p for p > VICTOR J. W. GUO AND MICHAEL J. SCHLOSSER
In Section 5 we propose an extension of Corollary 5 which contains additional factors inthe summand (see Conjecture 1).The paper is organized as follows. We shall prove Theorem 1 in Section 2 based on two q -series identities. Theorems 2 and 3 will be proved by giving a common generalization ofthem in Section 3. To accomplish this we shall make a careful use of Andrews’ multiseriesgeneralization of the Watson transformation [1, Theorem 4] (which was already usedin [12] to prove some q -analogues of Calkin’s congruence [3], and which was also appliedin [15] for proving some analogous results involving the base being odd powers of q ). Weshall prove Theorem 4 by using a certain anti-symmetry of the k -th summand on the left-hand side of (1.9) in Section 4. Finally, in Section 5 we give some concluding remarks andstate some open problems. These include some conjectural q -congruences modulo the fifthpower of a cyclotomic polynomial and congruences for truncated ordinary hypergeometricseries, one of them, see (5.7), modulo the seventh power of a prime greater than 3.We would like to thank the two anonymous referees for their comments. We espe-cially thank the second referee for her or his detailed list of constructive suggestions forimprovement of the paper. 2. Proof of Theorem 1
It is easy to prove by induction on n that n − X k =0 [4 k + 1] ( q ; q ) k ( q ; q ) k q − k = [ n ] (1 + q n ) ( q ; q ) n ( q ; q ) n q − n = [ n ] (cid:20) n − n − (cid:21) q − n ( − q ; q ) n − . Since q n ≡ n ( q )), the proof of (1.5) then follows from the fact (cid:20) n − n − (cid:21) = n − Y k =1 − q n − k − q k ≡ q − ( n )( − n − ≡ n ( q ))for odd n and ( − q ; q ) n − ≡ n ( q )) (see, for example, [8, Equation (2.3)]).Similarly, we can prove by induction that n − X k =0 [4 k −
1] ( q − ; q ) k ( q ; q ) k q k = − [ n ] (1 + q n ) ( q − ; q ) n ( q ; q ) n q n . The proof of (1.6) then follows from that of (1.5) and the following relation( q − ; q ) n = ( q ; q ) n − q − − q n − . We point out that, using the congruence ( − q ; q ) n − / ≡ q ( n − / (mod Φ n ( q )) for odd n (see, for example, [8, Lemma 2.1]), we can prove the following similar congruences: forany odd positive integer n > ( n − / X k =0 [4 k + 1] ( q ; q ) k ( q ; q ) k q − k ≡ q [ n ] (mod [ n ] Φ n ( q )) , (2.1) OME q -CONGRUENCES FOR TRUNCATED BASIC SERIES: EVEN POWERS 5 ( n +1) / X k =0 [4 k −
1] ( q − ; q ) k ( q ; q ) k q k ≡ − [ n ] (mod [ n ] Φ n ( q )) . (2.2)The details of the proof are left to the interested reader.3. Proof of Theorems 2 and 3
We shall first prove the following unified generalization of Theorems 2 and 3 for d = 4. Theorem 6.
Let r be an odd integer. Let n > be an odd integer with n ≡ − r (mod 4) and n > max { r, − r } . Then n − X k =0 [8 k + r ] ( q r ; q ) k ( q ; q ) k q (4 − r ) k ≡ n ( q ) ) . (3.1) Proof.
Let α , j and r be integers. It is easy to see that(1 − q αn − dj + d − r )(1 − q αn + dj − d + r ) + (1 − q dj − d + r ) q αn − dj + d − r = (1 − q αn ) and 1 − q αn ≡ n ( q )), and so(1 − q αn − dj + d − r )(1 − q αn + dj − d + r ) ≡ − (1 − q dj − d + r ) q αn − dj + d − r (mod Φ n ( q ) ) . It follows that ( q r − αn , q r + αn ; q d ) k ≡ ( q r ; q d ) k (mod Φ n ( q ) ) . (3.2)It is clear that 3 n ≡ r (mod 4). Therefore, by (3.2) and the q q , a q r , b q r , c q r +3 n , n (3 n − r ) / φ summation (see [5, Appendix(II.21)]): φ (cid:20) a, qa , − qa , b, c, q − n a , − a , aq/b, aq/c, aq n +1 ; q, aq n +1 bc (cid:21) = ( aq, aq/bc ; q ) n ( aq/b, aq/c ; q ) n , where the basic hypergeometric series r +1 φ r (see [5]) is defined as r +1 φ r (cid:20) a , a , . . . , a r +1 b , b , . . . , b r ; q, z (cid:21) = ∞ X k =0 ( a , a , . . . , a r +1 ; q ) k z k ( q, b , . . . , b r ; q ) k , modulo Φ n ( q ) , the left-hand side of (3.1) is congruent to (3 n − r ) / X k =0 [8 k + r ] ( q r , q r , q r +3 n , q r − n ; q ) k ( q , q , q − n , q n ; q ) k q (4 − r ) k = [ r ] ( q r +4 , q − n − r ; q ) (3 n − r ) / ( q , q − n ; q ) (3 n − r ) / . (3.3)Note that (3 n − r ) / n − n > − r . It is clear that ( q r +4 ; q ) (3 n − r ) / has the factor 1 − q n , and ( q − r − n ; q ) (3 n − r ) / has the factor (1 − q − n ) since (3 n − r ) / > ( n + r ) / n > r . Therefore the numerator on the right-hand side of (3.3)is divisible by Φ n ( q ) , while the denominator is relatively prime to Φ n ( q ). This completesthe proof. (cid:3) We need the following lemma in our proof of Theorems 2 and 3 for d > VICTOR J. W. GUO AND MICHAEL J. SCHLOSSER
Lemma 1.
Let d > be an integer and let r be an integer with gcd( d, r ) = 1 . Let n = ad − r > r with a > . Suppose that r + kd ≡ n ) for some k > . Then k > a ( d − / .Proof. Since gcd( d, r ) = 1, we have gcd( n, d ) = 1 for n = ad − r . Noticing that2 r + kd = ( k + 2 a ) d − ad − r ) = ( k + 2 a ) d − n, we conclude that ( k + 2 a ) d ≡ n ). It follows that k + 2 a is a multiple of n and so k + 2 a > n , i.e., k + 2 a > ad − r. By the condition ad − r > r in the lemma, we get r ad/
2. Substituting this into theabove inequality, we obtain the desired result. (cid:3)
We now give a common generalization of Theorems 2 and 3.
Theorem 7.
Let d > be an even integer and let r be an integer with gcd( d, r ) = 1 . Let n > be an integer with n ≡ − r (mod d ) and n > max { r, d − r } . Then n − X k =0 [2 dk + r ] ( q r ; q d ) dk ( q d ; q d ) dk q d ( d − r − k ≡ n ( q ) ) . (3.4) Proof.
The d = 4 case is just Theorem 6. We now suppose that d >
6. The proof of thiscase is intrinsically the same as that of Theorem 6. Here we need to use a complicatedtransformation formula due to Andrews [1, Theorem 4]: X k > ( a, q √ a, − q √ a, b , c , . . . , b m , c m , q − N ; q ) k ( q, √ a, −√ a, aq/b , aq/c , . . . , aq/b m , aq/c m , aq N +1 ; q ) k (cid:18) a m q m + N b c · · · b m c m (cid:19) k = ( aq, aq/b m c m ; q ) N ( aq/b m , aq/c m ; q ) N X l ,...,l m − > ( aq/b c ; q ) l · · · ( aq/b m − c m − ; q ) l m − ( q ; q ) l · · · ( q ; q ) l m − × ( b , c ; q ) l . . . ( b m , c m ; q ) l + ··· + l m − ( aq/b , aq/c ; q ) l . . . ( aq/b m − , aq/c m − ; q ) l + ··· + l m − × ( q − N ; q ) l + ··· + l m − ( b m c m q − N /a ; q ) l + ··· + l m − ( aq ) l m − + ··· +( m − l q l + ··· + l m − ( b c ) l · · · ( b m − c m − ) l + ··· + l m − , (3.5)which is a multiseries generalization of Watson’s φ transformation formula (see [5, Ap-pendix (III.18)]): φ (cid:20) a, qa , − qa , b, c, d, e, q − n a , − a , aq/b, aq/c, aq/d, aq/e, aq n +1 ; q, a q n +2 bcde (cid:21) = ( aq, aq/de ; q ) n ( aq/d, aq/e ; q ) n φ (cid:20) aq/bc, d, e, q − n aq/b, aq/c, deq − n /a ; q, q (cid:21) . (3.6) OME q -CONGRUENCES FOR TRUNCATED BASIC SERIES: EVEN POWERS 7 It is easy to see that ( d − n ≡ r (mod d ). Hence, by (3.2), modulo Φ n ( q ) , theleft-hand side of (3.4) is congruent to ( dn − n − r ) /d X k =0 [ r ] ( q r , q d √ q r , − q d √ q r , ( d − q r z }| { q r , . . . , q r , q r +( d − n , q r − ( d − n ; q d ) k ( q d , √ q r , −√ q r , q d , . . . , q d , q d − ( d − n , q d +( d − n ; q d ) k q d ( d − r − k , where we have used the fact ( dn − n − r ) /d n − n > d − r . Furthermore,by the q q d , a q r , b i q r , c i q r , for 1 i m − b m q r , c m q r +( d − n , N (( d − n − r ) /d case of Andrews’ transformation (3.5), the above summation canbe written as[ r ] ( q d + r , q d + n − dn − r ; q d ) ( dn − n − r ) /d ( q d , q d + n − dn ; q d ) ( dn − n − r ) /d X l ,...,l m − > ( q d − r ; q d ) l · · · ( q d − r ; q d ) l m − ( q d ; q d ) l · · · ( q d ; q d ) l m − × ( q r , q r ; q d ) l . . . ( q r , q r +( d − n ; q d ) l + ··· + l m − ( q d , q d ; q d ) l . . . ( q d , q d ; q d ) l + ··· + l m − × ( q r − ( d − n ; q d ) l + ··· + l m − ( q r ; q d ) l + ··· + l m − q ( d + r )( l m − + ··· +( m − l ) q d ( l + ··· + l m − ) q rl · · · q r ( l + ··· + l m − ) , (3.7)where m = ( d − / q d + r ; q d ) ( dn − n − r ) /d contains the factor 1 − q ( d − n . Similarly,( q d + n − dn − r ; q d ) ( dn − n − r ) /d contains the factor 1 − q (2 − d ) n since ( dn − n − r ) /d > ( n + r ) /d by the conditions d > n > r . Thus, the expression ( q d + r , q d + n − dn − r ; q d ) ( dn − n − r ) /d inthe fraction before the multiple summation is divisible by Φ n ( q ) .Note that the non-zero terms in the multiple summation of (3.7) are just those indexedby ( l , . . . , l m − ) with l + · · · + l m − ( dn − n − r ) /d n − q r − ( d − n ; q d ) l + ··· + l m − in the numerator. This immediately implies that all the other q -factorials in the denominator of the multiple summation of (3.7) do not contain factors ofthe form 1 − q αn (and are therefore relatively prime to Φ n ( q )), except for ( q r ; q d ) l + ··· + l m − .If n = d − r , then it is clear that at least one ( q d − r ; q d ) l i contains the factor 1 − q n for l + · · · + l m − >
0. We now assume that n > d − r and so n > max { d, r } in thiscase. Thus, if ( q r ; q d ) l + ··· + l m − has a factor 1 − q kn , then the number k is unique since l + · · · + l m − n − n, d ) = 1. Moreover, if such a k exists, then we musthave k > a ( d − / n = ad − r . It follows that l + · · · + l m − > k and at least one l i is greater than or equal to k/ ( m −
1) = 2 k/ ( d − > a and so( q d − r ; q d ) l i contains the factor 1 − q n in this case. This proves that the denominator ofthe reduced form of the multiple summation of (3.7) is always relatively prime to Φ n ( q ),which completes the proof of (3.4). (cid:3) VICTOR J. W. GUO AND MICHAEL J. SCHLOSSER Proof of Theorem 4
We shall prove n − X k =0 [2 dk + 1] ( q ; q d ) dk ( q d ; q d ) dk q d ( d − k ≡ dn − n ( q )) , which is equivalent to ( dn − n − /d X k =0 [2 dk + 1] ( q ; q d ) dk ( q d ; q d ) dk q d ( d − k ≡ dn − n ( q )) , (4.1)because ( q ; q d ) k has the factor 1 − q dn − n and is therefore divisible by Φ dn − n ( q ) for ( dn − n − /d < k n −
1, while ( q d ; q d ) k is coprime with Φ dn − n ( q ) for these k .Since q dn − n ≡ dn − n ( q )), we have( q ; q d ) ( dn − n − /d ( q d ; q d ) ( dn − n − /d = (1 − q )(1 − q d +1 ) · · · (1 − q dn − n − d )(1 − q d )(1 − q d ) · · · (1 − q dn − n − ) ≡ (1 − q )(1 − q d +1 ) · · · (1 − q dn − n − d )(1 − q − ( dn − n − d ) )(1 − q − ( dn − n − d ) ) · · · (1 − q − )= ( − dn − n − d q ( d − n − dn − n − d (mod Φ dn − n ( q )) . (4.2)Furthermore, for 0 k ( dn − n − /d , we have( q ; q d ) ( dn − n − /d − k ( q d ; q d ) ( dn − n − /d − k = ( q ; q d ) ( dn − n − /d ( q d ; q d ) ( dn − n − /d (1 − q dn − n − − ( k − d )(1 − q dn − n − − ( k − d ) · · · (1 − q dn − n − )(1 − q dn − n − kd )(1 − q dn − n − ( k − d ) · · · (1 − q dn − n − d ) ≡ ( − dn − n − d q ( d − n − dn − n − d (1 − q − − ( k − d )(1 − q − − ( k − d ) · · · (1 − q − )(1 − q − kd )(1 − q − ( k − d ) · · · (1 − q − d )= ( − dn − n − d q ( d − n − dn − n − d +( d − k ( q ; q d ) k ( q d ; q d ) k (mod Φ dn − n ( q )) . Taking the most left- and right-hand sides of this congruence to the power d , it follows,using q dn − n ≡ dn − n ( q )), that for 0 k ( dn − n − /d there holds[2 d (( dn − n − /d − k ) + 1] ( q ; q d ) d ( dn − n − /d − k ( q d ; q d ) d ( dn − n − /d − k q d ( d − dn − n − /d − k )2 ≡ ( − dn − n q ( dn − n )( dn − n − [2 dk + 1] ( q ; q d ) dk ( q d ; q d ) dk q d ( d − k (mod Φ dn − n ( q )) . OME q -CONGRUENCES FOR TRUNCATED BASIC SERIES: EVEN POWERS 9 It is easy to check that ( − dn − n q ( dn − n )( dn − n − ≡ − dn − n ( q )) whenever dn − n isodd or even. This proves that the k -th and (( dn − n − /d − k )-th terms of the left-handside of (4.1) cancel each other modulo Φ dn − n ( q ) and therefore (4.1) holds. Equivalently,(1.9) holds modulo Φ dn − n ( q ). Moreover, by (1.3) and (1.7), one sees that (1.9) also holdsmodulo Φ n ( q ) for d >
4. The proof then follows from the fact Φ n ( q ) and Φ dn − n ( q ) arerelatively prime polynomials.5. Concluding remarks and open problems
Having establishing ( q -)congruences for truncated (basic) hypergeometric series, onecan wonder what their ‘archimedian’ analogues are, i.e. whether the infinite sums from k = 0 to ∞ have known evaluations, just as (1.1) is such an archimedian analogue for(1.2).In many cases of our results, especially when dealing with arbitrary exponents d , weare not aware of explicit evaluations in the archimedian case. However for small d wecan easily find corresponding evaluations by suitably specializing known summations for(basic) hypergeometric series, such as Rogers’ nonterminating φ summation (cf. [5,Appendix (II.20)]), φ (cid:20) a, qa , − qa , b, c, da , − a , aq/b, aq/c, aq/d ; q, aqbcd (cid:21) = ( aq, aq/bc, aq/bd, aq/cd ; q ) ∞ ( aq/b, aq/c, aq/d, aq/bcd ; q ) ∞ , (5.1)where | aq/bcd | < q by q , and letting a = b = c = d = q r , we obtain from (5.1),after multiplying both sides by [ r ], the following identity: X k ≥ [8 k + r ] ( q r ; q ) k ( q ; q ) k q (4 − r ) k = [ r ] ( q r , q − r , q − r , q − r ; q ) ∞ ( q , q , q , q − r ; q ) ∞ = [ r ] Γ q (1 − r )Γ q (1 + r )Γ q (1 − r ) , (5.2)valid for r <
2. In the last equation we have rewritten the product using the q -Gammafunction Γ q ( x ) = ( q ; q ) ∞ ( q x ; q ) ∞ (1 − q ) − x , defined for 0 < q < r q → − Γ q ( x ) = Γ( x ), we obtain that in the q → − limit (5.2) becomes X k ≥ (8 k + r ) ( r ) k k ! = r Γ(1 − r )Γ(1 + r )Γ(1 − r ) = 4 sin( rπ ) Γ(1 − r ) π Γ(1 − r ) , (5.3)where we have used the well-known reflection formula for the Gamma function. It is nowimmediate that for r = 1 we get (1.1) while for r = − evaluation ∞ X k =0 (8 k −
1) ( − ) k k ! = − √ √ π Γ( ) . (5.4)Many other identities involving π can similarly be obtained. At this place, in passing, wewould like to point out that by replacing q by q in (5.1) and putting a = q , b = c = d = q one readily obtains X k ≥ q k +1 q (1 − q ) (1 − q k +1 ) q k = ( q , q , q , q ; q ) ∞ ( q , q , q , q ; q ) ∞ , (5.5)which, as recently noted by Sun [29, Equation (1.3)] (who derived this identity by com-pletely different means) is easily seen to be a q -analogue of Euler’s identity X k ≥ k + 1) = π , used to prove his famous evaluation ζ (2) = π /
6. See [34] for recent new samples ofexpansions involving π , obtained by suitably specializing q -series identities.We turn to discussing whether some of the results obtained in the paper can be furtherstrengthened. We have proved Theorems 2 and 3 by establishing a common generaliza-tion of them, namely Theorem 7. However, we are unable to prove a similar commongeneralization of (1.3) and (1.4). Numerical calculation for q = 1 suggests that there areno congruences for the left-hand side of (3.4) with odd d > d = 5 and r = 3 appears to be such a counterexample).Nevertheless, we would like to give the following result being similar to Theorem 7. Theorem 8.
Let d > be an odd integer and let r be an even integer with gcd( d, r ) = 1 .Let n > be an odd integer with n ≡ − r (mod d ) and n > max { r, d − r } . Then n − X k =0 [2 dk + r ] q ( q r ; q d ) dk ( q d ; q d ) dk q d ( d − r − k ≡ n ( q ) ) . (5.6)The proof of Theorem 8 is similar to that of Theorem 7. In this case we need toapply Andrews’ transformation (3.5) with m = ( d − / q → q d , a = q r , b = q d + r , b = . . . = b m = q r , c = · · · = c m − = q r , c m = q r +2( d − n and N = (( d − n − r ) /d .The details of the proof are omitted here.We can also prove the following refinement of (1.4) and (1.8). However, we are unableto deduce any interesting conclusion similar to (1.10) from this result by letting q → Theorem 9.
Let d > be an integer and let n > be an integer with n ≡ d ) .Then n − X k =0 [2 dk −
1] ( q − ; q d ) dk ( q d ; q d ) dk q d ( d − k ≡ n ( q ) Φ dn − n ( q )) . OME q -CONGRUENCES FOR TRUNCATED BASIC SERIES: EVEN POWERS 11 We would like to propose the following three conjectures which are similar to Corol-lary 5.
Conjecture 1.
Let r be a positive integer and let p be a prime with p > r + 1 . Then p − X k =0 k r (cid:18) k + 1 p + 1 (cid:19) r ( p +1 ) p +1 k k ! p +1 ≡ p ) . Conjecture 2.
Let p > be a prime. Then p − X k =0 (2 pk + 2 k + 1) ( p +1 ) p +2 k k ! p +2 ≡ p ) . (5.7) More generally, if p > is a prime and r a positive integer, then p r − X k =0 (cid:16) k p r +1 − p r − (cid:17) (cid:0) p r − p r +1 − (cid:1) pr +1 − p − k k ! pr +1 − p − ≡ p r +5 ) . (5.8) Conjecture 3.
Let p > be a prime. Then p − X k =0 (2 pk − k −
1) ( − p − ) p − k k ! p − ≡ p ) . (5.9) More generally, if p > is a prime and r a positive integer, then p r − X k =0 (2 kp r − k −
1) ( − p r − ) p r − k k ! p r − ≡ p r +3 ) . (5.10)Conjectures 2 and 3 are quite remarkable as they concern supercongruences modulohigh prime powers. We now give two partial q -analogues of (5.7) as follows. Conjecture 4.
Let n be an integer greater than . Then n − X k =0 [2 nk + 2 k + 1] ( q ; q n +1 ) n +2 k ( q n +1 ; q n +1 ) n +2 k q ( n +1)( n − k ≡ n ] Φ n ( q ) Φ n ( q )) . Conjecture 5.
Let p be a prime. Then p − X k =0 [2 pk + 2 k + 1] ( q ; q p +1 ) p +2 k ( q p +1 ; q p +1 ) p +2 k q ( p +1)( p − k ≡ − (2 p + 1)( p + 1) p ( p − q (1 − q ) [ p ] Φ p ( q ) (mod [ p ] Φ p ( q )) . (5.11)It is clear that the q → p . We would like toemphasize that (5.11), while still conjectural, appears to be the first example of a basichypergeometric supercongruence in the existing literature, that in the limit q → to a supercongruence (for a hypergeometric series being truncated after a number of termsthat is linear in p ) modulo p .Finally, we give a partial and a complete q -analogue of (5.9) as follows. Conjecture 6.
Let n be an integer greater than . Then n − X k =0 [2 nk − k −
1] ( q − ; q n − ) n − k ( q n − ; q n − ) n − k q ( n − k ≡ n ] Φ n ( q ) ) . Conjecture 7.
Let p be a prime. Then p − X k =0 [2 pk − k −
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School of Mathematics and Statistics, Huaiyin Normal University, Huai’an 223300,Jiangsu, People’s Republic of China
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