Proof of two divisibility properties of binomial coefficients conjectured by Z.-W. Sun

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1 Proof of two ivisibility properties of bioial coefficiets cojecture by Z.-W. Su Victor J. W. Guo Departet of Matheatics Shaghai Key Laboratory of PMMP East Chia Noral Uiversity 500 Dogchua Roa Shaghai People s Republic of Chia jwguo@ath.ecu.eu.c Subitte: May ; Accepte: Ju ; Publishe: Ju Matheatics Subject Classificatios: 11B65 05A10 05A30 Abstract For all positive itegers we prove the followig ivisibility properties: ( ( ( ( ( ( (2 3 3 a ( This cofirs two recet cojectures of Z.-W. Su. Soe siilar ivisibility properties are give. Moreover we show that for all positive itegers a the prouct a ( ( ab 1 ab a a is ivisible by. I fact the latter result ca be further geeralize to the -bioial coefficiets a -itegers case which geeralizes the positivity of -Catala ubers. We also propose several relate cojectures. Keywors: cogrueces bioial coefficiets p-aic orer -Catala ubers reciprocal a uioal polyoials 1 Itrouctio I [18 19] Z.-W. Su prove soe ivisibility properties of bioial coefficiets such as ( ( ( (2 1 3 (1.1 ( ( ( ( (1.2 the electroic joural of cobiatorics 21(2 (2014 #P2.54 1

2 Soe siilar ivisibility results were later obtaie by Guo [10] a Guo a Krattethaler [11]. A geeralizatio of (1.1 was recetly give by Sepaski [15]. It is worth etioig that Bober [6] has copletely escribe whe ratios of factorial proucts of the for (a 1! (a k! (b 1! (b k1! with a 1 a k b 1 b k1 are always itegers. Let ( 6 3 ( 15 ( 5 1 S 3( 2(2 1 ( a t (10 1 ( 3. I this paper we first prove the followig two results cojecture by Z.-W. Su [18 19]. Theore 1.1 (see [18 Cojecture 3(i] Let be a positive iteger. The 3S 0 (o 2 3. (1.3 Theore 1.2 [19 Cojecture 1.3] Let be a positive iteger. The 21t 0 (o We shall also give ore cogrueces for S a t as follows. Theore 1.3 Let be a positive iteger. The 105S 0 (o 2 5 ( S 0 (o 2 7 ( S 0 (o 2 9 ( t 0 (o 2 1 ( t 0 (o 10 7 ( t 0 (o (1.9 Let Z eote the set of itegers. Aother result i this paper is the followig. Theore 1.4 Let a b be positive itegers. The ( ( ab a b a b a ( ( a b 1 a b Z. (a b( a a a a (1.10 Lettig a b 1 i (1.10 we get the followig result of which a cobiatorial iterpretatio was give by Gessel [9 Sectio 7]. Corollary 1.5 Let be positive itegers. The ( ( 2 2 Z. (1.11 2( I the ext sectio we give soe leas. The proofs of Theores will be give i Sectios 3 5 respectively. A proof of the -aalogue of Theore 1.4 will be give i Sectio 6. We close our paper with soe further rearks a ope probles i Sectio 7. the electroic joural of cobiatorics 21(2 (2014 #P2.54 2

3 2 Soe leas For the p-aic orer of! there is a kow forula or p! (2.1 where x eotes the greatest iteger ot exceeig x. I this sectio we give soe results o the floor fuctio x. Lea 2.1 For ay real uber x we have i1 6x x 3x 2 2x (2.2 15x 2x 10x 4x 3x. (2.3 Proof. See [6 Theore 1.1] a oe of the 52 sporaic step fuctios give i [6 Table 2 lie# 32]. Lea 2.2 Let a be two positive itegers such that 2 3 a 5. The (2.4 Proof. Let {x} x x be the fractioal part of x. The (2.4 is euivalet to { } 6 { { } { } (2.5 } Now suppose that 2 3 a 5. We have { } 2 3 > a 2 3 It follows that { } 6 { } 3 2 { } if if if if 9. 2 Therefore the ietity (2.5 is true for ay positive iteger (o 2. the electroic joural of cobiatorics 21(2 (2014 #P2.54 3

4 Lea 2.3 Let a be two positive itegers such that 10 3 a 9. The (2.6 Proof. It is easy to see that (2.6 is euivalet to { } { } { } { } 4 Now suppose that 10 3 a 9. We have { } a A : It is easy to check that { } ( ( 6 6 ({ } { } { } ( ( { } 3 1. ( (o 10. if A 0 (o 10 if A 2 (o 10 if A 6 (o 10 if A 8 (o 10 a so the ietity (2.7 hols. 3 Proofs of Theore 1.1 First Proof. Let gc(a b eote the greatest coo ivisor of two itegers a a b. For ay positive iteger sice gc( to prove Theore 1.1 it is eough to show that 3 ( 6 3 (2 3 3( ( 2. (3.1 By (2.1 for ay o prie p the p-aic orer of ( 6 3 3( (2 3 ( (2 2!(6!(! 2 (2 3!(3!(2! 2 the electroic joural of cobiatorics 21(2 (2014 #P2.54 4

5 is give by ( (3.2 i1 Note that { if otherwise. By Leas 2.1 a 2.2 for p 5 the suatio (3.2 is clearly greater tha or eual to 0. For p 3 we have (3.2 1 because if the positive iteger i satisfies 3 i 2 3 a 3 i < 5 the we ust have i 1. This proves that 3 ( 6 3 3( (2 3 ( 2 is always a iteger. Hece (3.1 hols. Seco Proof (provie by T. Aeberha a V.H. Moll. Replacig by 1 i (1.1 we see that (after soe rearrageet ( ( (2 3 ( (6 5(6 1S Z. ( 1(2 3 Hece (2 3 6(6 5(6 1S. Sice gc(2 3 2 gc( gc( we ust have (2 3 3S. Reark. Z.-W. Su [18 Cojecture 3(i] also cojecture that S is o if a oly if is a power of 2. After reaig a previous versio of this paper Qua-Hui Yag tol e that it is easy to show that or 2 ((6!!/(3!(2! 2 euals the uber of 1 s i the biary expasio of by oticig or 2 (6! 3 or 2 (3! or 2 (2! or 2! a usig Legere s theore. T. Aeberha a V.H. Moll also poite out this. 4 Proof of Theore 1.2 For ay positive iteger sice gc( to prove Theore 1.2 it is eough to show that 21 ( ( (10 3 ( 3. (4.1 the electroic joural of cobiatorics 21(2 (2014 #P2.54 5

6 Furtherore sice gc( a ( ( oe sees that (4.1 is euivalet to 21 ( ( (10 3 ( 3. (4.2 By (2.1 for ay o prie p the p-aic orer of ( 15 ( 5 5 is give by i1 (10 3 ( 3 (10 2!(15!(2! (10 3!(10!(4!(3! ( (4.3 Note that { if otherwise. By Leas 2.1 a 2.3 for p 11 or p 5 the suatio (4.3 is clearly greater tha or eual to 0. For p 3 7 we have (4.3 1 because there is at ost oe iex i 1 satisfyig 10 3 a < 9 i this case. This proves that 21 ( ( (10 3 ( 3 is always a iteger. Naely (4.2 is true. 5 Proof of Theore 1.3 Lea 5.1 Let a be two positive itegers. The (5.1 if 2 5 a 9 or 2 7 a 11 or 2 9 a 15. Proof. The proof is siilar to that of Lea 2.2. We oly cosier the case whe 25 a 9. I this case we have { } 2 5 > a (o 2. the electroic joural of cobiatorics 21(2 (2014 #P2.54 6

7 It follows that a so This proves (5.1. { } 6 { } 5 2 { } 3 { } 6 { } 2 15 if if if if 15 2 { } 3 2 { } 2 1. Lea 5.2 Let a be two positive itegers. The (5.2 if 2 1 a 15 or 10 7 a 21 or 10 9 a 27. Proof. The proof is siilar to that of Lea 2.3. We oly cosier the case whe 10 9 a 27. I this case we have { } a A : (o It follows that ({ } 2 { } 4 { } { } 3 ( ( ( ( if A 0 (o 10 if A 2 (o 10 if A 6 (o 10 if A 8 (o 10. the electroic joural of cobiatorics 21(2 (2014 #P2.54 7

8 Hece { } 15 { } 2 { } 10 { } 4 { } 3 1 which eas that (5.2 hols. Proof of Theore 1.3. Sice the proofs of the cogrueces (1.4 (1.9 are siilar i view of Leas 5.1 a 5.2 we oly give proofs of (1.5 a (1.9. Noticig that gc( or 3 to prove (1.5 it suffices to show that 105 ( 6 3 (2 7 3( ( 2. (5.3 Let X : ( 6 3 3( (2 7 ( 2 By (2.1 for ay o prie p we have or p X i1 (2 6!(6!(! (2 7!(3!(2! 2. ( Note that (5.1 is also true for 3 a 1 (o 3 a By Leas 2.1 a 5.1 we obtai { if otherwise. { orp X 0 if p 11 or p X 1 if p This proves (5.3. Siilarly sice gc( gc( the cogruece (1.9 is euivalet to ( ( (10 9 ( 3. (5.4 Let Y : ( 15 ( 5 5 (10 9 ( 3 (10 8!(15!(2! (10 9!(10!(4!(3!. the electroic joural of cobiatorics 21(2 (2014 #P2.54 8

9 The for ay o prie p or p Y is give by ( i1 Note that (5.2 also hols for a ay positive iteger such that 109. Siilarly as before we have or p Y 0 if p or p 29 or p Y 1 if p or p Y 2 if p 3. Observig that we coplete the proof of ( A -aalogue of Theore 1.4 Recall that the -bioial coefficiets are efie by [ ] (1 (1 1 (1 k1 if 0 k (1 (1 k 2 (1 k 0 otherwise. We begi with the aouce stregtheig of Theore 1.4. Theore 6.1 Let a b 1. The [ ] [ ] 1 gc(a a b 1 a b 1 a a (6.1 is a polyoial i with o-egative iteger coefficiets. Corollary 6.2 Let a b 1. The [ ] [ ] 1 a a b 1 a b 1 a a (6.2 is a polyoial i with o-egative iteger coefficiets. It is easily see that Theore 1.4 ca be obtaie upo lettig 1 i Corollary 6.2. Moreover whe a b 1 the ubers (6.2 reuce to the -Catala ubers C ( 1 [ ] the electroic joural of cobiatorics 21(2 (2014 #P2.54 9

10 It is well kow that the -Catala ubers C ( are polyoials with o-egative iteger coefficiets (see [ ]. There are ay ifferet -aalogues of the Catala ubers (see Fürliger a Hofbauer [7]. For the so-calle t-catala ubers see [ ]. Recall that a polyoial P ( i0 i i of egree is calle reciprocal if p i for all i a that it is calle uioal if there is a iteger r with 0 r a 0 p 0 p r p 0. A eleetary but crucial property of reciprocal a uioal polyoials is the followig. Lea 6.3 If A( a B( are reciprocal a uioal polyoials the so is their prouct A(B(. Lea 6.3 is well kow a its proof ca be fou e.g. i [1] or [16 Propositio 1]. Siilarly to the proof of [11 Theore 3.1] the followig lea plays a iportat role i the proof of Theore 6.1. It is a slight geeralizatio of [14 Propositio 10.1.(iii] which extracts the essetials out of Arews [4 Proof of Theore 2]. Lea 6.4 Let P ( be a reciprocal a uioal polyoial a a positive itegers with. Furtherore assue that A( 1 P ( is a polyoial i. 1 The A( has o-egative coefficiets. Proof. See [11 Lea 5.1]. Proof of Theore 6.1. It is well kow that the -bioial coefficiets are reciprocal a uioal polyoials i (cf. [17 Ex ] a by Lea 6.3 so is the prouct of two -bioial coefficiets. I view of Lea 6.4 for provig Theore 6.1 it is eough to show that the expressio (6.1 is a polyoial i. We shall accoplish this by a cout of cyclotoic polyoials. Recall the well-kow fact that 1 Φ ( where Φ ( eotes the -th cyclotoic polyoial i. Coseuetly [ ] [ ] 1 gc(a a b 1 a b 1 a a i{ab 1 ab} 2 Φ ( e with a b 1 a b e χ( gc(a χ( a b 1 a b (6.3 the electroic joural of cobiatorics 21(2 (2014 #P

11 where χ(s 1 if S is true a χ(s 0 otherwise. This is clearly o-egative uless a gc(a. So let us assue that a gc(a which eas that a a therefore a b 1 a b (a b( 1 a( a a b 1 b 1 b( 1 a so e 0 is also o-egative i this case. This copletes the proof of polyoiality of (6.1. Proof of Corollary 6.2. This follows ieiately fro Theore 6.1 a the fact that gc(a a. 7 Cocluig rearks a ope probles O Jauary T. Aeberha a V.H. Moll (persoal couicatio fou the followig geeralizatio of Theore 1.1 which was soo prove by Q.-H. Yag [21] a C. Krattethaler. Cojecture 7.1 Let a b a be positive itegers with a > b. The ( ( ( 2b 2a a (2b 1(2b 3 3(a b(3a b. b a b Let []! (1 (1. By a result of Waraar a Zuili [20 Propositio 3] oe sees that for ay positive iteger the polyoial [6]![]! [3]![2]! 2 has o-egative iteger coefficiets. Siilarly as before we ca prove the followig geeralizatio of cogrueces (1.3 (1.5. Theore 7.2 Let be a positive iteger. The all of (1 [6]![]! (1 21 [3]![2]! (1 3 [6]![]! 2 (1 23 [3]![2]! (1 (1 3 [6]![]! 2 (1 21 (1 23 [3]![2]! 2 (1 3 (1 5 (1 7 [6]![]! (1 23 (1 25 (1 27 [3]![2]! 2 ( 2 (1 3 2 (1 5 (1 7 [6]![]! (1 21 (1 23 (1 25 (1 27 [3]![2]! 2 ( 2 are polyoials i. the electroic joural of cobiatorics 21(2 (2014 #P

12 We have the followig two relate cojectures. Cojecture 7.3 All the polyoials i Theore 7.2 have o-egative iteger coefficiets. Cojecture 7.4 Let 2. The the polyoial [6]![]! [6]![]! [3]![2]! 2 is uioal. It is obvious that the polyoial is reciprocal. If Cojecture 7.4 is true the [3]![2]! 2 applyig Lea 6.3 we coclue that the first two polyoials i Theore 7.2 have o-egative iteger coefficiets. It was cojecture by Waraar a Zuili (see [20 Cojecture 1] that [15]![2]! [10]![4]![3]! has o-egative iteger coefficiets. Siilarly we have the followig geeralizatio of Theore 1.2. Theore 7.5 Let be a positive iteger. The both (1 [15]![2]! (1 101 [10]![4]![3]! a (1 3 (1 7 [15]![2]! (1 (1 103 [10]![4]![3]! are polyoials i. We e the paper with the followig cojecture stregtheig the above theore. Cojecture 7.6 The two polyoials i Theore 7.5 have o-egative iteger coefficiets. Ackowlegets. The author thaks Qua-Hui Yag T. Aeberha a V.H. Moll for helpful coets o a previous versio of this paper. This work was partially supporte by the Fuaetal Research Fus for the Cetral Uiversities a the Natioal Natural Sciece Fouatio of Chia (grat Refereces [1] G.E. Arews A theore o reciprocal polyoials with applicatios to perutatios a copositios Aer. Math. Mothly 82 ( [2] G.E. Arews Catala ubers -Catala ubers a hypergeoetric series J. Cobi. Theory Ser. A 44 ( [3] G.E. Arews O the ifferece of successive Gaussia polyoials J. Statist. Pla. Iferece 34 ( the electroic joural of cobiatorics 21(2 (2014 #P

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