SUPERCONGRUENCES INVOLVING PRODUCTS OF TWO BINOMIAL COEFFICIENTS

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1 Finite Fields Al. 013, SUPERCONGRUENCES INVOLVING PRODUCTS OF TWO BINOMIAL COEFFICIENTS Zhi-Wei Sun Deartent of Matheatics, Nanjing University Nanjing 10093, Peole s Reublic of China E-ail: zwsun@nju.edu.cn htt://ath.nju.edu.cn/ zwsun Abstract. In this aer we deduce soe new suercongruences odulo owers of a rie > 3. Let d {0, 1,..., 1/}. We show that 1/ 0 +d 8 0 od if d + 1 od, and 1/ 0 +d d 4 E 3 d + 1 od 3, where E 3 x denotes the Euler olynoial of degree 3, and stands for the Legendre sybol. The aer also contains soe other results such as / od. 1. Introduction Let be an odd rie and let be the Legendre sybol. For each d N {0, 1,... } and any rational -adic integer λ, we define xx 1x λ a d λ : x d. 1.1 x0 010 Matheatics Subject Classification. Priary 11B65; Secondary 05A10, 11A07, 11B39, 11B68, 11E5, 1E0. Keywords. Central binoial coefficients, suercongruences odulo rie owers. Suorted by the National Natural Science Foundation grant of China and the PAPD of Jiangsu Higher Education Institutions. 1

2 ZHI-WEI SUN Note that a 0 λ arises naturally fro counting the nuber of oints on the cubic curve E λ : y xx 1x λ over the finite field F Z/Z, where λ is the residue class λ od. The following theore in the case d 0 is a nown result cf. S. Ahlgren [A, Theore ]. Theore 1.1. Let be an odd rie and let d {0,..., 1/}. Then, for any rational -adic integer λ we have 1/ a d +1/ λd λ 1 4 d 0 +d +d 16 λ δ d, 1/ od, 1. where the Kronecer sybol δ s,t taes 1 or 0 according as s t or not. Rear 1.1. Let d {0,..., 1/} with an odd rie. Clearly x a d 1 x d 1 x d+ 1/ 1 δ d, 1/ 1 od. x0 x1 Thus 1. with λ 1 gives the congruence 1/ 0 +d +d 16 4 d 1 od. Soon we will see that this congruence even holds odulo. Recall that the Euler nubers E 0, E 1, E,... are integers defined by E 0 1 and n 0 n E n 0 for n Z + {1,, 3,... }. For each n N, the Euler olynoial of degree n is given by E n x n 0 Clearly E n 1/ E n / n. Now we state our second theore. n E x 1 n.

3 SUPERCONGRUENCES INVOLVING PRODUCTS OF BINOMIAL COEFFICIENTS 3 Theore 1.. Let > 3 be a rie and let d {0,..., 1/}. Then 1/ 0 +d d E 3 d od Rear 1.. Let > 3 be a rie. The suercongruence 1/ od was a conjecture of Rodriguez-Villegas [RV] confired by E. Mortenson [Mo1] via an advanced tool involving the -adic Gaa function and the Gross-Koblitz forula for character sus. 1.3 with d 0 yields the congruence 1/ E 3 od 3 which was first roved in [S4] with the hel of the software Siga. Corollary 1.1. Let > 3 be a rie. For any d 0,..., 1/, we have 1/ 0 +d +d 16 4 d 1 od. 1.4 Let 1 od 4 be a rie. It is well nown that x + y for soe x, y Z with x 1 od 4. A celebrated result of Gauss asserts that 1/ 1/4 x od see, e.g., [BEW, 9.0.1]. This was refined in [CDE] as follows: 1/ x 1/4 x od. Recently, J. B. Cosgrave and K. Dilcher [CD] even deterined 1/ 1/4 od 3. Recall that x +y with x 1 od 4. Z.-H. Sun [Su] confired the author s following conjecture cf. [S3, Conjecture 5.5]: 1/ 0 8 1/ 16 0 x x 1/ 3 0 od.

4 4 ZHI-WEI SUN In [S5] the author showed that 1/ 0 1/ / 0 x x od Note that those integers / are called Catalan nubers and they occur naturally in any enueration robles in cobinatorics see, e.g., [St,. 19 9]. Motivated by 1. in the cases λ 1, we obtain the following result. Theore 1.3. i If 3 od 4 is a rie, then 1/ / 0 1+1/4 1/ / + 1/4 od 8 1/ and 1/ 0 8 1/ /4 +1/ od /4 ii For any odd rie, we have 1/ 0 +d 8 0 od 1.7 for all d {0,..., 1/} with d + 1/ od. Rear 1.3. In 009 the author conjectured that /8 0 od for any rie 1 od 4 and this was confired by his student Yong Zhang in her PhD thesis. Besides 1.4 with d 0, Rodriguez-Villegas [RV] also ade the following siilar conjectures confired in [Mo] on suercongruences with a rie greater

5 SUPERCONGRUENCES INVOLVING PRODUCTS OF BINOMIAL COEFFICIENTS 5 than 3: od, 1.8 od, 1.9 od Note that the denoinators 7, 64, 43 coe fro the following observation via the Stirling forula: 3 3 7, π 4 64, π π. U to now no sile roofs of have been found. Motivated by the wor in [PS] and [ST], the author [S] deterined 1 0 / odulo in ters of Lucas sequences, where is an odd rie and is any integer not divisible by. In [S3] and [S4] the author osed any conjectures on sus of ters involving central binoial coefficients. For a sequence of a n n N of nubers, as in [S1] we introduce its dual sequence a n n N by defining a n : n 0 n 1 a n 0, 1,,.... It is well-nown that a n a n for all n N see, e.g., 5.48 of [GKP,. 19]. For Bernoulli nubers B 0, B 1, B,..., the sequence 1 n B n n N is self-dual. Theore 1.4. Let > 3 be a rie and let a n n N be any sequence of -adic integers. Then we have a 3 64 a a a od, a od, a od. 1.13

6 6 ZHI-WEI SUN Rear 1.4. Z.-H. Sun [Su] recently roved that 1/ 0 16 a 1 a 0 od for any odd rie via Legendre olynoials. We can also show, for any rie > 3, the following result siilar to 1.3 and 1.4: If d {0,..., /3 } then 1/ 1 4 d 0 3 +d 1/ 3 +d +d od ; if d {0,..., /4 } then 1/ 1 4 d 0 4 +d +d 64 1/ 0 4 +d 64 od. Let be a rie and let fx F [x] with degf <. Then fx is identically zero if fa 0 for all a F. Thus Theore 1.4 has the following consequence since 1 x j0 j 1 j x j for any N. Corollary 1.. Let > 3 be a rie and let Z be the ring of -adic integers. Then, in the ring Z [x] we have x x x 3 1 x 0 od, x 0 od, x 0 od Also, x x od, x 1 + x x 1 0 od, x 1 0 od. 1.19

7 SUPERCONGRUENCES INVOLVING PRODUCTS OF BINOMIAL COEFFICIENTS 7 Rear can be easily deduced fro by taing derivatives. Z.-H. Sun [Su, Theore.4] noted that for any rie > 3 we have / x 1 /3 1 x 0 od. Taing x 1/ in we iediately get the following result. Corollary 1.3. Let > 3 be a rie. Then od if 1 od 3, 54 0 od if od 3; od if 1, 3 od 8, od if 5, 7 od 8; od if 1 od 4, od if 3 od 4. Rear 1.6. The first and the second congruences od were obtained by Z.-H. Sun [Su]. The author [S4] and Z.-H. Sun [Su] conjectured the first and the second congruences resectively. Inutting FullSilify[Su[*Binoial[3,]Binoial[,]/54,{,0,Infty}]], we obtain fro Matheatica the exact result π 9Γ 4 3 Γ 7 6, which should follow fro certain algorith hidden in Matheatica and 1.17 in the case x 9/8, and 1.15 and 1.18 in the cases x 4/3, 8/9, 64/63, yield the following result.

8 8 ZHI-WEI SUN Corollary 1.4. Let > 3 be a rie. Then od, od. 1.1 Also, od, 4 19 od ; 576 od, od. If 7, then od, od. 0 Rear 1.7. Let > 3 be a rie. In [S4, Conjecture 5.13] the author conjectured that { 1/3 1/3 od if 1 od 3, / +1/3 +1/3 od if od 3. The author [S4] also ade conjectures on , 63, 7, / odulo with

9 SUPERCONGRUENCES INVOLVING PRODUCTS OF BINOMIAL COEFFICIENTS 9 For any rie > 3 and integer 0 od, we have od, od, due to the identities n n 0 n n 0 n n od, 1.4 n 1 n 1 n 1 n n 4n n 6n 3n 3n n n n 3n n which can be easily roved by induction on n. So, the following result follows fro Corollary 1.3 and 1.1. Corollary 1.5. Let > 3 be a rie. Then We also have od if 1 od 3, od if 1, 3 od 8, od if 1 od ,,, od. 1.8 Rear 1.8. Siilar to the identity in Rear 1.6, Matheatica version 7 also yields π 4 Γ 4 3 Γ 1 6, π Γ 1 8 Γ 11 8, 0 0

10 10 ZHI-WEI SUN and π Γ Γ 1. 0 Theore 1.5. Let > 3 be a rie. Then { 6 x od if x + y 1.9 with x 1 od 4, / +1/4 od if 3 od 4. Let A and B be integers. The Lucas sequences u n u n A, B n N and v n v n A, B n N are defined as follows: u 0 0, u 1 1, and u n+1 Au n Bu n 1 for n Z + ; v 0, v 1 A, and v n+1 Av n Bv n 1 for n Z +. When A 4B 0, by induction we see that u n A, B na/ n 1 and v n A, B A/ n for all n Z +. Our following theore is an analogue of Corollary 1.3 involving Lucas sequences with 0. Theore 1.6. Let A, B Z with A 0 and A 4B, and let u u A, B and v v A, B for all N. Let > 3 be a rie with A. i If 1 od 3, then 0 3 If od 3, then 0 7A u 3 7A v ii If 1, 3 od 8, then 0 4 If 5, 7 od 8, then 0 64A u 4 64A v A v 1 0 od A u 1 0 od A v 1 0 od A u 1 0 od. 1.33

11 SUPERCONGRUENCES INVOLVING PRODUCTS OF BINOMIAL COEFFICIENTS 11 iii If 1 od 4, then 0 If 3 od 4, then A u A v A v 1 0 od A u 1 0 od We will not list corollaries of Theore 1.6 with resect to soe secial Lucas sequences lie the Fibonacci sequence F n u n 1, 1 n N and its coanion L n v n 1, 1 n N. In the next section we are going to show Theores and Corollary 1.1. Sections 3 and 4 are devoted to our roofs of Theore 1.4 and Theores resectively.. Proofs of Theores and Corollary 1.1 Proof of Theore 1.1. Set n 1/. Then a d λ 0 1 0,l0 0 x d xx 1x λ n x n+d n 0 n 1 n x n l0 n λ l x n l l n n n 1 n λ l x 1+d+ l l x1 n n n 1 n λ l 1 l n 0 n 0 l n 1 l d+ n n 1 n λ d+ δ d,n d + 0 λ 0 od. Since 1/ 1/ 4 od for all 0,..., 1,.1 we iediately obtain 1. fro the above.

12 1 ZHI-WEI SUN Proof of Theore 1.. By induction, we have n n n n n n + 1 n. for each n, + 1,..., where N. Set n 1/. If 0 < n, then for the right-hand side R of. we have 1 1 R + 1 1/ 4 1 n n od 3 since 1 0<j j j 1 od for all 0,..., 1..3 As d n, we have n 16 + d 0 n <d <d 0 <d E <d 1 E d E 3 d E od 3. Note that n 0 +n 16 n n 16 n 1 1/ n od 3 by Morley s congruence [M], and that E 3 n + 1 E 3 E od.

13 SUPERCONGRUENCES INVOLVING PRODUCTS OF BINOMIAL COEFFICIENTS 13 It is well nown that E 0 0 for all Z +. Therefore n n 4 E 3 1 E 3 od 3 and hence 1.3 follows fro the above. Proof of Corollary 1.1. For 0, 1,..., we have + d + d d c d d + c d c by the Chu-Vanderonde identity cf. [GKP,. 169]. Note that +c is regarded as zero if + c < 0. In view of this and 1.3, 1/ 0 +d +d 16 d c d d c d 1/ d d c 0 1 d d c + c 16 1 d od. So 1.4 is valid and we are done. Proof of Theore 1.3. i For Z \ {0} and n N we have the cobinatorial identity n n + 1 n n + 1 n.4 n 0 which can be easily roved by induction on n. Now let n + 1 be a rie with 3 od 4. Setting n 1/ we obtain fro.4 that n n 0 for any integer 0 od. As n 1/ is odd, by a result of Z.-H. Sun [Su], n 0 16 x + 1 x fx od.5

14 14 ZHI-WEI SUN for soe olynoial fx of degree at ost 1/ with rational -adic integer coefficients. In articular, By integration, n 0 n x+1 Putting x 1 we obtain Since n 0 n n n 0 n 0 16 od x x 0 ftdt od. n + 1 n n 0 od, n + 1 n as observed by van Hae [vh] see also.5 with 16, we have n 0 With the hel of.1, n n n 0 0 1/ 1 Note that 1 n 1. Thus + 18 od..7 n n 1 0 od. 0 h0 n 0 h h 16 h 0 n j0 h h + j + 1 j j 16 j 4 n 0 16 n j0 j j j 16 j od. By [S5, Lea 3.], h0 h h h 0 h h 1 od.

15 SUPERCONGRUENCES INVOLVING PRODUCTS OF BINOMIAL COEFFICIENTS 15 Therefore 1 n 0 16 n In view of.5-.8, both 1.5 and 1.6 hold if n 0 For d {0, 1}, clearly a d /4 x1 + 1/ + 1/4 r0 od..8 od..9 xx 1x rr x d r + 1 d 1 and rr 1 r d 1 a d r0 r d+n r 1 n 1 r0 0 n 1 n+d+ n 0 n 1 n n 1 n r1 r n+d+ { 0 od if d 0, 1 3/4 n n 1/ δ,3 od if d 1. Thus we have a 0 a od and a 1 a a /4 n n 1/ δ,3 od. Alying Theore 1.1 with λ and d 0, 1, and noting that for all N, we get n δ,3 a 0 a 1 od.

16 16 ZHI-WEI SUN So.9 follows. ii Let n + 1 be an odd rie. Now we rove 1.7 for all d {0,..., n} with d n + 1 od. 1.7 is valid for d n 1 since n 0 Since Define +n 1 8 fd : n 1 n 1 8 n 1 + n n n 8 n n n 1 n 0 n 1 n n + for d 0, 1, d od. n + 16 od for 0,..., n.10 see, e.g., [Su, Lea.], we have fd n 0 +d 8 od for all d 0..., n. By alying the Zeilberger algorith cf. [PWZ, ] via Matheatica version 7, we find the recurrence relation n d 1n + d + d + 1fd + n + 1 d + 1fd + 1 n dn + d + 1d + 3fd. Note that n + 1. So, if 0 d n, then and hence fd n d 1n + d + d + 1 fd + od n dn + d + 1d + 3 fd + 0 od fd 0 od. Now it is clear that 1.7 holds for all d {0,..., n} with d n + 1 od. Proof of Observe that Proof of Theore a a 1 a 3 7.

17 SUPERCONGRUENCES INVOLVING PRODUCTS OF BINOMIAL COEFFICIENTS 17 So it suffices to show that 3 7 for all 0, 1,..., 1. For 0 < n define od f n n By Zeilberger s algorith via Matheatica version 7, we find that f n f n 3n 13n n 1 n 3n 3 7 n 1. n 1 n 1 Alying this with n > and noting that od 3.1 and we get od, 3. and hence f f od f f f f od 3.

18 18 ZHI-WEI SUN Thus, for every 0,...,, we have f od Since f + 1 f od od by 1.8, with the hel of 3.3 we obtain that f od for all 0, 1,..., 1. This concludes the roof. Proof of 1.1. Siilar to the roof of 1.11, we only need to show that od 3.3 for all 0, 1,..., 1. Since this congruence holds for 0 by 1.9, it suffices to rove that for any fixed 0 < 1 we have 4 g 64 od g od, where g n : n with n >. By the Zeilberger algorith, we find that Clearly g n g n 4n 14n 3 n 1 n 4n 4 64 n 1. n 1 n od 3.5

19 SUPERCONGRUENCES INVOLVING PRODUCTS OF BINOMIAL COEFFICIENTS 19 and hence od. In view of this and 3.1, fro 3.5 with n we get g g 3 1 od 3. This ilies 3.4 since We are done. Proof of For 0 < n define h n : n By the Zeilberger algorith, for, n N with < n 1, we have h n h n 6n 16n 5 n 1 3n 3 6n 6 43 n 1. n 1 3n 3 Recall the congruence 3. and note that if > 5 then od. So, no atter 5 or not, for every 0,..., we have h h 0 od. 3.6 For 0 < 1, since ,

20 0 ZHI-WEI SUN by 3.6 we have h 3 43 od h od. 3.7 This, together with 1.10, yields that h 3 43 od for all 0,..., 1. It follows that a h 0 This roves Proof of Theore 1.5. By 1.3, and hence 0 1 a Proofs of Theores a 3 43 od. 4 7 od od. So it suffices to deterine n 0 4 /7 od, where n 1/. Note that for n + 1,..., 1. The Legendre olynoial of degree n is given by P n x : n n n + x 1 0 It is nown see, e.g., [N] that 0 n n + x 1. 0 n n x 1 4x n 1 P n. 1 4x

21 SUPERCONGRUENCES INVOLVING PRODUCTS OF BINOMIAL COEFFICIENTS 1 Taing derivatives of both sides of this identity, we get n 0 n x 1 n n + n1 4x n/ 1 0 n n x n 3/ 0 1 4x 1/ 1 1 4x 1/ 1 1. In view of.1 and.10, by utting x /9 in the last equality we obtain 1 n n n n n 0 16 od and hence 3 n n n 0 16 od. Since n / 16 0 and n 0 / 16 odulo have been deterined cf. Theore 1.3i and the aragrah after 1.4, we finally obtain the desired result for n 0 4 /7 odulo. The roof of Theore 1.5 is now colete. Proof of Theore 1.6. We just show the first art in detail. Parts ii and iii can be roved siilarly. By 1.14 and 1.17, there are -adic integers a 0,..., a 1, b 0,..., b such that and x 3 7 x x 3 a x x 3 1 b x. 4. Let α and β be the two distinct roots of the equation x Ax + B 0. It is well nown that 0 u α β α β and v α + β for all N.

22 ZHI-WEI SUN As α/a + β/a 1, by 4.1 and 4. we have A 1 7A u + 3 7A v 3 7 A 1 u 1 v 1 + u 3 0 v 3 0 u a A u, a A v, b A u, v 1 3 b A v. Thus 1.30 holds when 1 od 3, and 1.31 holds when od 3. We are done. Acnowledgent. The author is grateful to the referee for helful coents. References [A] S. Ahlgren, Gaussian hyergeoetric series and cobinatorial congruences, in: Sybolic Coutation, Nuber Theory, Secial Functions, Physics and Cobinatorics Gainesville, FI, 1999,. 1-1, Dev. Math., Vol. 4, Kluwer, Dordrecht, 001. [BEW] B. C. Berndt, R. J. Evans and K. S. Willias, Gauss and Jacobi Sus, John Wiley & Sons, [CDE] S. Chowla, B. Dwor and R. J. Evans, On the od deterination of 1/ 1/4, J. Nuber Theory , [CD] J. B. Cosgrave and K. Dilcher, Mod 3 analogues of theores of Gauss and Jacobi on binoial coefficients, Acta Arith , [GKP] R. L. Graha, D. E. Knuth and O. Patashni, Concrete Matheatics, nd ed., Addison- Wesley, New Yor, [M] F. Morley, Note on the congruence 4n 1 n n!/n!, where n + 1 is a rie, Ann. Math , [Mo1] E. Mortenson, A suercongruence conjecture of Rodriguez-Villegas for a certain truncated hyergeoetric function, J. Nuber Theory , [Mo] E. Mortenson, Suercongruences between truncated F 1 by geoetric functions and their Gaussian analogs, Trans. Aer. Math. Soc , [N] T. D. Noe, On the divisibility of generalized central trinoial coefficients, J. Integer Seq , Article 06..7, 1. [PS] H. Pan and Z.-W. Sun, A cobinatorial identity with alication to Catalan nubers, Discrete Math , [PWZ] M. Petovše, H. S. Wilf and D. Zeilberger, A B, A K Peters, Wellesley, [RV] F. Rodriguez-Villegas, Hyergeoetric failies of Calabi-Yau anifolds, in: Calabi-Yau Varieties and Mirror Syetry Toronto, ON, 001,. 3-31, Fields Inst. Coun., 38, Aer. Math. Soc., Providence, RI, 003. [St] R. P. Stanley, Enuerative Cobinatorics, Vol., Cabridge Univ. Press, Cabridge,

23 SUPERCONGRUENCES INVOLVING PRODUCTS OF BINOMIAL COEFFICIENTS 3 [Su] Z.-H. Sun, Congruences concerning Legendre olynoials, Proc. Aer. Math. Soc , [S1] Z.-W. Sun, Cobinatorial identities in dual sequences, Euroean J. Cobin , [S] Z.-W. Sun, Binoial coefficients, Catalan nubers and Lucas quotients, Sci. China Math , [S3] Z.-W. Sun, On congruences related to central binoial coefficients, J. Nuber Theory , [S4] Z.-W. Sun, Suer congruences and Euler nubers, Sci. China Math , [S5] Z.-W. Sun, On sus involving roducts of three binoial coefficients, Acta Arith , [ST] Z.-W. Sun and R. Tauraso, New congruences for central binoial coefficients, Adv. in Al. Math , [vh] L. van Hae, Soe conjectures concerning artial sus of generalized hyergeoetric series, in: -adic Functional Analysis Nijegen, 1996,. 3 36, Lecture Notes in Pure and Al. Math., Vol. 19, Deer, 1997.