Factors of sums and alternating sums involving binomial coefficients and powers of integers

Transcription

1 Factors of sums ad alteratig sums ivolvig biomial coefficiets ad powers of itegers Victor J. W. Guo 1 ad Jiag Zeg 2 1 Departmet of Mathematics East Chia Normal Uiversity Shaghai People s Republic of Chia jwguo@math.ecu.edu.c 2 Uiversité de Lyo; Uiversité Lyo 1; Istitut Camille Jorda UMR 5208 du CNRS; 43 boulevard du 11 ovembre 1918 F Villeurbae Cedex Frace zeg@math.uiv-lyo1.fr Abstract. We study divisibility properties of certai sums ad alteratig sums ivolvig biomial coefficiets ad powers of itegers. For example we prove that for all positive itegers 1... m m+1 1 ad ay oegative iteger r there holds 1 m ε k 2k + 1 2r+1 i + i m m + 1 i k ad cojecture that for ay oegative iteger r ad positive iteger s such that r + s is odd where ε ±1. s 2 ε k 2k + 1 r 2 0 k k 1 2 Keywords: biomial coefficiets divisibility properties Chu-Vadere formula Lucas theorem AMS Subject Classificatios: 05A10 11B65 11A07 1 Itroductio There have bee lastig iterests i biomial sums. Although some biomial sums have o closed formulas it is still possible to show that they have some ice factors. For example a result of Calki [2] reads m k 0 for m 1. + k k Geeralizig Calki s result Guo Jouhet ad Zeg [8] proved amog other thigs that 1 k 1 1 k i + i+1 0 i + k 1 + m for all 1... m 1 ad m+1 1. Recetly motivated by the momets of the Catala triagle see [311] Guo ad Zeg [10] were led to study some differet biomial sums ad proved the cogruece 1 m 2 k 2r+1 i + i+1 0 i + k k m

2 for all 1... m 1 ad m+1 1. I this paper we will prove some divisibility properties of aother kid of sums ad alteratig sums ivolvig biomial coefficiets ad powers of itegers. Let N deote the set of oegative itegers ad Z + the set of positive itegers. Oe of our mai results may be stated as follows. Theorem 1.1. For all 1... m Z + m+1 1 ad r N there holds 1 m ε k 2k + 1 2r+1 i + i m m + 1 i k where ε ±1. Actually we shall derive Theorem 1.1 from the followig more geeral result. Theorem 1.2. For all 1... m Z + m+1 1 ad r N there hold 1 k r k + 1 r i + i k + 1 i k 1 + m m + 1 mi{1r} 1 mi{1r 2} m 1 Ideed by the expasio 1 k k r k + 1 r i + i k + 1 i k 1 + m m + 1 mi{1r} 1 mi{1r} m k + 1 2r 4k 2 + 4k + 1 r r i0 r 4 i k i k + 1 i i it is clear that Theorem 1.2 ifers Theorem 1.1. Recetly Miaa ad Romero [12 Theorem 10] evaluated the momets Ψ r : 2k + 1r A 2 k where A k 0 k are the ballot umbers defied by A k 2k k k k 1 By the formulas of Ψ 1 Ψ 3 ad Ψ 5 see [12 Remark 11] we cojecture that Ψ 2r+1 is divisible by 2. More geerally we have Cojecture 1.3. For all r N ad s Z + such that r + s 1 2 there holds where ε ±1. ε k 2k + 1 r A s k As a check let 7 for all r 0 ad s 1 such that r + s 1 2 the sum 7 2k + 1 r A s 7k 429 s + 3 r 1001 s + 5 r 1001 s + 7 r 637 s + 9 r 273 s + 11 r 77 s + 13 r+s + 15 r is obviously divisible by I this paper we shall cofirm Cojecture 1.3 i some special cases. Theorem 1.4. The cogruece 1.1 holds if is a prime power or s 1. I the ext three sectios we shall provide the proof of Theorem 1.2 correspodig respectively to the cases m 1 m 2 ad m 3. We the prove Theorem 1.4 i Sectio 5. Fially we give some further cosequeces of Theorem 1.1 ad related cojectures i Sectio 6. 2

3 2 Proof of Theorem 1.2 for m 1 Let P r : Q r : k r k + 1 r 2k + 1 k k k r k + 1 r 2k + 1. k The m 1 case of Theorem 1.2 may be stated as follows. Theorem 2.1. For all Z + ad r N there hold 2 P r mi{2r} 2 Q r mi{22r}. Proof of Theorem 2.1. We proceed by iductio o r. For r 0 we have P k k 1 2 Q k 2 k k 1 For r 1 observig that kk k we have k { 0 if > 0 1 if k 1 P r + 1P r P r Q r + 1Q r Q r for 1. For the above recurreces we derive immediately that 2 2 P P ad Q Q Q Q ad Q 1 Q 2 0 for 3. Therefore Theorem 2.1 is true for r Now suppose that r 3 ad Theorem 2.1 holds for r 1. The P r 1 is divisible by ad P r 1 1 is divisible by By 2.1 we see that P r is also divisible by This completes the iductive step for P r. Similarly by 2.1 we ca prove the case for Q r. We may also cosider the followig sums: U r : 2k + 1 2r k V r : 1 k 2k + 1 2r. k 3

4 It is easy to see that k 2 k k k 1 k k k k 1 Similarly to 2.1 ad 2.2 we have U r U r U r V r V r V r By we immediately obtai the followig result. Corollary 2.2. For Z + ad r N there hold k + 1 2r 1 2 k α 1 k 2k + 1 2r 1 2 k where α deotes the umber of 1 s i the biary expasio of. 3 Proof of Theorem 1.2 for m 2 We first give two combiatorial idetities. Lemma 3.1. For all 1 2 N there hold k k 2 k k k k 2 k Proof. It is easy to see that the left-had side of 3.1 may be writte as k k 2 k k k 1 2 k which is equal to the right-had side of 3.1. Replacig k by k 1 i the left-had side of 3.2 we observe that 1 1 k k k 2 k 1 1 k k k 2 k k

5 It follows that 3.2 is equivalet to 1 k k k k 2 k 1 Sice k k 2 k 1 k 2 k k 2 k 1 to prove 3.3 it suffices to establish the followig two idetities: 1 k k 2 k 1 1 k k 2 k 1 1 k k which follow immediately by comparig the coefficiets of x 22 i the expasio of This completes the proof. 1 x x x1 x Remark. We ca give aother proof of 3.1 ad 3.2 by computig their geeratig fuctios ad usig the followig idetity m0 m + + α m + + β m x m y 2 α+β 1 x + y + α 1 + x y + β 3.4 where : 1 2x 2y 2xy + x 2 + y 2. The idetity 3.4 is equivalet to the geeratig fuctio of Jacobi polyomials. See [1 p. 298] [13 p. 271] or [4] for a proof of this idetity ad [9] for a applicatio to prove some double-sum biomial coefficiet idetities. As a example we compute the geeratig fuctio for the left-had side of 3.2: mi{ 1 2} x 1 y 2 1 k k k 2 k 2k + 1 xy k x 1 k y 2 k k 2 k k 2 4k+2 2k + 1 xy k 1 x + y + 2k x y + 2k+1 2 2k+1 2k + 1 xy k 1 x y + 2k xy1 x y x y xy1 x y x y 2. 5

6 Clearly the last expressio is the geeratig fuctio for the right-had side of 3.2. Let P r 1 2 : k r k + 1 r 2k k 2 k 1 Q r 1 2 : 1 k k r k + 1 r 2k k 2 k The the m 2 case of Theorem 1.2 may be stated as follows. Theorem 3.2. For all 1 2 Z + ad r N there hold P r Q r mi{1r} 1 mi{1r 2} 2 mi{1r} 1 mi{1r} 2. Proof of Theorem 3.2. We proceed by iductio o r. Writig kk k 2 k 1 k 2 k k 1 2 k we derive P r P r P r Q r Q r Q r r From the above recurreces ad Lemma 3.1 we immediately get P P Q Q Therefore Theorem 3.2 is true for r Now suppose that the statemet is true for r 1 r 3. The 1 P r is divisible by ad P r is divisible by By 3.5 we see that P r 1 2 is divisible by ad we complete the iductive step for P r 1 2. The case for Q r 1 2 is exactly the same. Remark. Similarly if we set 1 P r 1 2 : k k 2r+1 2 k 1 6

7 the P P r P r P r r 1 from which we ca deduce that 2P r 1 2 is divisible by by iductio o r. This result is the base of the iductive proof of [10 Theorem 1.3] though it was ot explicitly stated there. 4 Proof of Theorem 1.2 for m 3 For all oegative itegers a 1... a l ad k let where a l+1 a 1 ad let S r 1... m T r 1... m Observe that for m 3 we have C 1... m ; k Ca 1... a l ; k 1! m! 1 + m + 1! 1! m! 1 + m + 1! 1 1 l ai + a i a i k k r k + 1 r 2k + 1C 1... m ; k k k r k + 1 r 2k + 1C 1... m ; k ! m ! 1 + k + 1! 2 k! m ! ad by the Chu-Vadere formula see [1 p. 67] we have k 1 k s k Substitutig 4.3 ad 4.4 ito the right-had side of 4.1 we get S r 1... m ! 1! m! m ! ! 1! m! m ! C 3... m ; k k + 1! 2 k! s!s + 2k + 1! 1 k s! 2 k s! k s0 1 l l0 where l s + k. Now i the last sum makig the substitutio 2k + 1 2r+1 C 3... m ; k s!s + 2k + 1! 1 k s! 2 k s! 2k + 1 2r+1 C 3... m ; k l k!l + k + 1! 1 l! 2 l! C 3... m ; k l k!l + k + 1! m ! 3 + l + 1! m + l + 1! Cl 3... m ; k we obtai the followig recurrece relatio 1 S r 1... m l S r l 3... m m l 2 l 7

8 Similarly we have 1 T r 1... m l T r l 3... m m l 2 l We ow proceed by iductio o m. By Theorem 3.2 we suppose that Theorem 1.2 is true for m 1 m 3. If r 0 the by 4.5 ad 4.6 Theorem 1.2 is true for m. If r 1 the by defiitio ad by the iductio hypothesis S r m T r m 0 S r l 3... m 0 l mi{1r} mi{1r 2} m T r l 3... m 0 l mi{1r} mi{1r} m for l 1. Hece by 4.5 ad 4.6 ad oticig that 1 l l l 1 we get S r 1... m 0 T r 1... m 0 mi{1r} 1 mi{1r 2} m mi{1r} 1 mi{1r} m Namely Theorem 1.2 is true for m 3. This completes the proof. By repeatedly usig 4.5 ad 4.6 for r 0 1 we obtai the followig results. Corollary 4.1. For all m 3 ad 1... m Z + there hold 1 i + i k + 1 i k 1 + m m 2 λm 2 + m 1 + m λ λ m 2 i + i kk + 12k + 1 i k 1 + m λm 2 + m 1 m 1 + m + 1 λ m λ λi 1 λ i i+1 + i i+1 λ i m 2 λi 1 λ i i+1 + i i+1 λ i where m+1 λ 0 1 ad the sums are over all sequeces λ λ 1... λ m 2 of oegative itegers such that 1 λ 1 λ m 2. Corollary 4.2. For all m 3 ad 1... m Z + there hold 1 1 k i + i k + 1 i k 1 + m m 2 λi 1 i+1 + i m λ λ i i+1 λ i 1 1 k+1 i + i kk + 12k + 1 i k 1 + m m 2 λi 1 i+1 + i m 1 + m + 1 λ m 2 1 λ i i+1 λ i λ where m+1 λ 0 1 ad the sums are over all sequeces λ λ 1... λ m 2 of oegative itegers such that 1 λ 1 λ m 2. 8

9 Note that the above idetities show that the alteratig sums i the left-had sides are positive. Whe m 3 applyig the Chu-Vadere formula we derive the followig idetities from Corollary 4.2: 1 k k ! 1 k 2 k 3 k 1! 2! 3! 1 k kk + 12k k 2 k 3 k Proof of Theorem ! 1 1! 2 1! 3 1!. I this sectio we shall use the followig theorem of Lucas see for example [7]. We refer the reader to [ ] for recet applicatios of Lucas theorem. Lemma 5.1 Lucas theorem. Let p be a prime ad let a 0 b 0... a m b m {0... p 1}. The a0 + a 1 p + + a m p m b 0 + b 1 p + + b m p m i0 ai b i p. Suppose that r + s 1 2 ad s 1. Puttig m s ad 1 s i Theorem 1.1 we see that s s k + 1 r+s 1 k 2k + 1 r+s k k Therefore by the defiitio of A k we have 2k + 1 r A s k 2k + 1 r 1 k A s k 0 2 gcd s 1 ad so the cogruece 1.1 holds if s 1. Now suppose that p a p 3 is a prime power. By Lucas theorem we have 2 p a a 1 p 1 p a 1 ap 1/2 p 1/2 p 1/2 which meas that gcd s This completes the proof of Theorem 1.4. Remark. It is atural to woder which umbers satisfy 5.1 besides those we just metioed. Via Maple we fid that all such umbers less tha 300 are as follows: That is to say Theorem 1.4 is also true for these umbers. O the other had it is easy to see from Lucas theorem that if p ad p 2 p + 1 are both odd primes the p 3 p 2 + p 1/2 satisfies 5.1. Via Maple we fid that there are 5912 such primes p amog the first 10 5 primes. Here we list the first 20 such primes:

10 6 Cosequeces of Theorem 1.1 ad ope problems For coveiece let ε ±1 throughout this sectio. We shall give several iterestig cosequeces of Theorem 1.1 i this sectio. Note that we ca also give the correspodig cosequeces of Theorem 1.2 i the same way. Lettig 1 m i Theorem 1.1 we have Corollary 6.1. For all m Z + ad r N there holds m ε k 2k + 1 2r k For m 2a we propose the followig cojecture: 2. Cojecture 6.2. For all a Z + ad r N there hold 1 2 2a if 2 b 1 2k + 1 2r+1 2 k 0 otherwise 1 2 2a if 2 b + 2 c b c N 1 k 2k + 1 2r+1 k 0 otherwise 2. Lettig 2i 1 m ad 2i for 1 i a i Theorem 1.1 we obtai Corollary 6.3. For all a m Z + ad r N there holds m a a m m ε k 2k + 1 2r+1 0 m m k k Lettig 3i 2 l 3i 1 m ad 3i for 1 i a i Theorem 1.1 we obtai Corollary 6.4. For all a l m Z + ad r N there holds l ε k 2k + 1 2r+1 l + m + 1 l k a m m k m + m a a + l l + m + 1 k. l + m Lettig m 2a + b 1 3 2a 1 ad lettig all the other i be 1 i Theorem 1.1 we get Corollary 6.5. For all a Z + ad b r N there holds a a 2 ε k 2k + 1 2r k k 1 k 1 b 0 It is easy to see that Theorem 1.1 ca be restated i the followig form. 2 l. Theorem 6.6. For all 1... m Z + ad r N the expressio 1! i + i+1 + 1! 2 i + 1! 1 ε k 2k + 1 2r+1 m 2i + 1 i k m+1 1 is always a iteger. 10

11 It is easy to see cf. [15] that for all m N the umbers 2m+1!2+1! m++1!m!! ad 2m!2! m+!m!! are itegers by cosiderig the p-adic order of a factorial. Lettig 1 a m ad a+1 a+b i Theorem 6.6 we obtai Corollary 6.7. For all a b m Z + ad r N there holds m a b 2m ε k 2k + 1 2r+1 0 m k k 2m + 1!2 + 1! m + + 1!m!!. For example we have a b ε k 2k + 1 2r k + 1 k a b ε k 2k + 1 2r k k a b ε k 2k + 1 2r !2 + 1! 0 3 k k 4 + 1!3!!. Similarly to the proof of Theorem 1.4 we ca deduce the followig results. Corollary 6.8. Let r N ad s t Z + such that r + s + t 1 2 ad let Z +. If s t 1 or gcd the ε k 2k + 1 r A s +1kA t k I particular if ad are both prime powers the 6.2 holds Corollary 6.9. Let r N ad s t Z + such that r + s + t 1 2 ad let Z +. If s t 1 or gcd the ε k 2k + 1 r A s 2kA t k I particular if ad are both prime powers the 6.3 holds Cojecture The cogrueces 6.2 ad 6.3 hold for all Z + ad r s t give i Corollary 6.8. Cojecture There are ifiitely may umbers Z + such that gcd Cojecture There are ifiitely may umbers Z + such that gcd

12 f g Table 1: Some values of fx ad gx. Let fx ad gx deote the umbers of positive itegers x satisfyig 6.4 ad 6.5 respectively. Via Maple we fid that fx ad gx grow a little slowly with respect to x. Table 1 gives some values of fx ad gx. Usig Lucas theorem it is easy to see that if ad are both prime powers the 6.4 holds. It is well kow that the twi prime cojecture states that there are ifiitely may primes p such that p + 2 is also prime. Therefore if the twi prime cojecture is true the so is Cojecture Sice the twi prime cojecture is rather difficult we hope that Cojecture 6.11 might be tackled i aother way. I fact we have the followig more geeral cojecture: Cojecture For all r N ad m s t Z + such that r + s + t 1 2 there holds m m 2m!2! m+!m!! ε k 2k + 1 r A s mka t k 0 2m!2! m +!m!!. Note that the umbers are called super Catala umbers of which o combiatorial iterpretatios are kow for geeral m ad util ow. See Gessel [6] ad Georgiadis et al. [5]. From Theorem 6.6 it is easy to see that for all a 1... a m Z + 1! i + i+1 + 1! 2 i + 1! 1 ε k 2k + 1 2r+1 m ai 2i + 1 m i k is a iteger. For m 3 lettig be or we obtai the followig three corollaries. Corollary For all a b c Z + ad r N there holds a b c ε k 2k + 1 2r k k + 1 k + 2 Corollary For all a b c Z + ad r N there hold a b c ε k 2k + 1 2r k 2 k k a b c ε k 2k + 1 2r k 2 k k 3 Corollary For all a b c Z + ad r N there hold a b c ε k 2k + 1 2r k 2 k k a b c ε k 2k + 1 2r k 3 k 2 k 3 12

13 Note that Z Cojecture For all r s t Z + such that r + s + t 1 2 there hold ε k A r 3kA s 2kA t k 0 ε k A r 3kA s 2kA t k 0 ε k A r 4kA s 2kA t k 0 ε k A r 4kA s 3kA t 2k Fially for geeral m 2 i 6.6 takig 1... m to be m 1 + m 2 + m 4 + m if m is odd m 1 + m 2 + m 4 + m 6... if m is eve we get the followig geeralizatio of cogrueces 6.1 ad 6.7. Corollary Let m 2 ad let a 1... a m Z + ad r N. The m ai 2 + 2i ε k 2k + 1 2r m 1. + i k 1 We ed this paper with the followig challegig cojecture related to Corollary Cojecture For all r 1... r m Z + such that r r m 1 2 there holds ε k m A ri +i 1k Note that the m 1 case of Cojecture 6.19 is a little weaker tha our previous Cojecture where the ulus is replaced by 2. By Corollary 6.18 it is easy to see that Cojecture 6.19 is true for 2 or m 6 ad with the help of Maple. Ackowledgmets. This work was partially supported by the Fudametal Research Fuds for the Cetral Uiversities Shaghai Risig-Star Program #09QA Shaghai Leadig Academic Disciplie Project #B407 ad the Natioal Sciece Foudatio of Chia # Refereces [1] G. E. Adrews R. Askey ad R. Roy Special Fuctios Ecyclopedia of Mathematics ad its Applicatios Vol. 71 Cambridge Uiversity Press Cambridge [2] N. J. Calki Factors of sums of powers of biomial coefficiets Acta Arith [3] X. Che ad W. Chu Momets o Catala umbers J. Math. Aal. Appl [4] D. Foata ad P. Leroux Polyômes de Jacobi iterprétatio combiatoire et foctio géératrice Proc. Amer. Math. Soc

14 [5] E. Georgiadis A. Muemasa ad H. Taaka A ote o super Catala umbers [6] I.M. Gessel Super ballot umbers J. Symbolic Comput [7] A. Graville Arithmetic properties of biomial coefficiets I Biomial coefficiets ulo prime powers i: Orgaic Mathematics Burady BC 1995 CMS Cof. Proc. 20 Amer. Math. Soc. Providece RI 1997 pp [8] V. J. W. Guo F. Jouhet ad J. Zeg Factors of alteratig sums of products of biomial ad q-biomial coefficiets Acta Arith [9] V. J. W. Guo ad J. Zeg The umber of covex polyomioes ad the geeratig fuctio of Jacobi polyomials Discrete Appl. Math [10] V. J. W. Guo ad J. Zeg Factors of biomial sums from the Catala triagle J. Number Theory [11] J. M. Gutiérrez M. A. Herádez P.J. Miaa ad N. Romero New idetities i the Catala triagle J. Math. Aal. Appl [12] P. J. Miaa ad N. Romero Momets of combiatorial ad Catala umbers J. Number Theory [13] E. D. Raiville Special Fuctios Chelsea Publishig Co. Brox New York [14] Z.-W. Su ad W. Zhag Biomial coefficiets ad the rig of p-adic itegers Proc. Amer. Math. Soc [15] S. O. Waraar ad W. Zudili A q-rious positivity Aequat. Math [16] L. L. Zhao H. Pa ad Z.-W. Su Some cogrueces for the secod-order Catala umbers Proc. Amer. Math. Soc