Factors of sums and alternating sums involving binomial coefficients and powers of integers
|
|
- Joanna Marshall
- 5 years ago
- Views:
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
arxiv: v1 [math.nt] 28 Apr 2014
Proof of a supercogruece cojectured by Z.-H. Su Victor J. W. Guo Departmet of Mathematics, Shaghai Key Laboratory of PMMP, East Chia Normal Uiversity, 500 Dogchua Rd., Shaghai 0041, People s Republic of
More informationFactors of alternating sums of products of binomial and q-binomial coefficients
ACTA ARITHMETICA 1271 (2007 Factors of alteratig sums of products of biomial ad q-biomial coefficiets by Victor J W Guo (Shaghai Frédéric Jouhet (Lyo ad Jiag Zeg (Lyo 1 Itroductio I 1998 Cali [4 proved
More informationProof of a conjecture of Amdeberhan and Moll on a divisibility property of binomial coefficients
Proof of a cojecture of Amdeberha ad Moll o a divisibility property of biomial coefficiets Qua-Hui Yag School of Mathematics ad Statistics Najig Uiversity of Iformatio Sciece ad Techology Najig, PR Chia
More informationA q-analogue of some binomial coefficient identities of Y. Sun
A -aalogue of some biomial coefficiet idetities of Y. Su arxiv:008.469v2 [math.co] 5 Apr 20 Victor J. W. Guo ad Da-Mei Yag 2 Departmet of Mathematics, East Chia Normal Uiversity Shaghai 200062, People
More informationOn Divisibility concerning Binomial Coefficients
A talk give at the Natioal Chiao Tug Uiversity (Hsichu, Taiwa; August 5, 2010 O Divisibility cocerig Biomial Coefficiets Zhi-Wei Su Najig Uiversity Najig 210093, P. R. Chia zwsu@ju.edu.c http://math.ju.edu.c/
More informationarxiv: v1 [math.co] 6 Jun 2018
Proofs of two cojectures o Catala triagle umbers Victor J. W. Guo ad Xiuguo Lia arxiv:1806.02685v1 [math.co 6 Ju 2018 School of Mathematical Scieces, Huaiyi Normal Uiversity, Huai a 223300, Jiagsu, People
More informationProc. Amer. Math. Soc. 139(2011), no. 5, BINOMIAL COEFFICIENTS AND THE RING OF p-adic INTEGERS
Proc. Amer. Math. Soc. 139(2011, o. 5, 1569 1577. BINOMIAL COEFFICIENTS AND THE RING OF p-adic INTEGERS Zhi-Wei Su* ad Wei Zhag Departmet of Mathematics, Naig Uiversity Naig 210093, People s Republic of
More informationON MONOTONICITY OF SOME COMBINATORIAL SEQUENCES
Publ. Math. Debrece 8504, o. 3-4, 85 95. ON MONOTONICITY OF SOME COMBINATORIAL SEQUENCES QING-HU HOU*, ZHI-WEI SUN** AND HAOMIN WEN Abstract. We cofirm Su s cojecture that F / F 4 is strictly decreasig
More informationSome identities involving Fibonacci, Lucas polynomials and their applications
Bull. Math. Soc. Sci. Math. Roumaie Tome 55103 No. 1, 2012, 95 103 Some idetities ivolvig Fiboacci, Lucas polyomials ad their applicatios by Wag Tigtig ad Zhag Wepeg Abstract The mai purpose of this paper
More informationThe log-behavior of n p(n) and n p(n)/n
Ramauja J. 44 017, 81-99 The log-behavior of p ad p/ William Y.C. Che 1 ad Ke Y. Zheg 1 Ceter for Applied Mathematics Tiaji Uiversity Tiaji 0007, P. R. Chia Ceter for Combiatorics, LPMC Nakai Uivercity
More informationApplicable Analysis and Discrete Mathematics available online at ON JENSEN S AND RELATED COMBINATORIAL IDENTITIES
Applicable Aalysis ad Discrete Mathematics available olie at http://pefmathetfrs Appl Aal Discrete Math 5 2011, 201 211 doi:102298/aadm110717017g ON JENSEN S AND RELATED COMBINATORIAL IDENTITIES Victor
More informationA GENERALIZATION OF THE SYMMETRY BETWEEN COMPLETE AND ELEMENTARY SYMMETRIC FUNCTIONS. Mircea Merca
Idia J Pure Appl Math 45): 75-89 February 204 c Idia Natioal Sciece Academy A GENERALIZATION OF THE SYMMETRY BETWEEN COMPLETE AND ELEMENTARY SYMMETRIC FUNCTIONS Mircea Merca Departmet of Mathematics Uiversity
More informationSome p-adic congruences for p q -Catalan numbers
Some p-adic cogrueces for p q -Catala umbers Floria Luca Istituto de Matemáticas Uiversidad Nacioal Autóoma de México C.P. 58089, Morelia, Michoacá, México fluca@matmor.uam.mx Paul Thomas Youg Departmet
More informationBijective Proofs of Gould s and Rothe s Identities
ESI The Erwi Schrödiger Iteratioal Boltzmagasse 9 Istitute for Mathematical Physics A-1090 Wie, Austria Bijective Proofs of Gould s ad Rothe s Idetities Victor J. W. Guo Viea, Preprit ESI 2072 (2008 November
More informationAn analog of the arithmetic triangle obtained by replacing the products by the least common multiples
arxiv:10021383v2 [mathnt] 9 Feb 2010 A aalog of the arithmetic triagle obtaied by replacig the products by the least commo multiples Bair FARHI bairfarhi@gmailcom MSC: 11A05 Keywords: Al-Karaji s triagle;
More informationA generalization of Morley s congruence
Liu et al. Advaces i Differece Euatios 05 05:54 DOI 0.86/s366-05-0568-6 R E S E A R C H Ope Access A geeralizatio of Morley s cogruece Jiaxi Liu,HaoPa ad Yog Zhag 3* * Correspodece: yogzhag98@63.com 3
More informationarxiv: v1 [math.nt] 10 Dec 2014
A DIGITAL BINOMIAL THEOREM HIEU D. NGUYEN arxiv:42.38v [math.nt] 0 Dec 204 Abstract. We preset a triagle of coectios betwee the Sierpisi triagle, the sum-of-digits fuctio, ad the Biomial Theorem via a
More informationGENERALIZED HARMONIC NUMBER IDENTITIES AND A RELATED MATRIX REPRESENTATION
J Korea Math Soc 44 (2007), No 2, pp 487 498 GENERALIZED HARMONIC NUMBER IDENTITIES AND A RELATED MATRIX REPRESENTATION Gi-Sag Cheo ad Moawwad E A El-Miawy Reprited from the Joural of the Korea Mathematical
More informationA LIMITED ARITHMETIC ON SIMPLE CONTINUED FRACTIONS - II 1. INTRODUCTION
A LIMITED ARITHMETIC ON SIMPLE CONTINUED FRACTIONS - II C. T. LONG J. H. JORDAN* Washigto State Uiversity, Pullma, Washigto 1. INTRODUCTION I the first paper [2 ] i this series, we developed certai properties
More informationSuper congruences concerning Bernoulli polynomials. Zhi-Hong Sun
It J Numer Theory 05, o8, 9-404 Super cogrueces cocerig Beroulli polyomials Zhi-Hog Su School of Mathematical Scieces Huaiyi Normal Uiversity Huaia, Jiagsu 00, PR Chia zhihogsu@yahoocom http://wwwhytceduc/xsjl/szh
More informationThe log-concavity and log-convexity properties associated to hyperpell and hyperpell-lucas sequences
Aales Mathematicae et Iformaticae 43 2014 pp. 3 12 http://ami.etf.hu The log-cocavity ad log-covexity properties associated to hyperpell ad hyperpell-lucas sequeces Moussa Ahmia ab, Hacèe Belbachir b,
More informationk-generalized FIBONACCI NUMBERS CLOSE TO THE FORM 2 a + 3 b + 5 c 1. Introduction
Acta Math. Uiv. Comeiaae Vol. LXXXVI, 2 (2017), pp. 279 286 279 k-generalized FIBONACCI NUMBERS CLOSE TO THE FORM 2 a + 3 b + 5 c N. IRMAK ad M. ALP Abstract. The k-geeralized Fiboacci sequece { F (k)
More information#A51 INTEGERS 14 (2014) MULTI-POLY-BERNOULLI-STAR NUMBERS AND FINITE MULTIPLE ZETA-STAR VALUES
#A5 INTEGERS 4 (24) MULTI-POLY-BERNOULLI-STAR NUMBERS AND FINITE MULTIPLE ZETA-STAR VALUES Kohtaro Imatomi Graduate School of Mathematics, Kyushu Uiversity, Nishi-ku, Fukuoka, Japa k-imatomi@math.kyushu-u.ac.p
More informationSome families of generating functions for the multiple orthogonal polynomials associated with modified Bessel K-functions
J. Math. Aal. Appl. 297 2004 186 193 www.elsevier.com/locate/jmaa Some families of geeratig fuctios for the multiple orthogoal polyomials associated with modified Bessel K-fuctios M.A. Özarsla, A. Altı
More informationThe 4-Nicol Numbers Having Five Different Prime Divisors
1 2 3 47 6 23 11 Joural of Iteger Sequeces, Vol. 14 (2011), Article 11.7.2 The 4-Nicol Numbers Havig Five Differet Prime Divisors Qiao-Xiao Ji ad Mi Tag 1 Departmet of Mathematics Ahui Normal Uiversity
More informationA CONTINUED FRACTION EXPANSION FOR A q-tangent FUNCTION
Sémiaire Lotharigie de Combiatoire 45 001, Article B45b A CONTINUED FRACTION EXPANSION FOR A q-tangent FUNCTION MARKUS FULMEK Abstract. We prove a cotiued fractio expasio for a certai q taget fuctio that
More informationNEW FAST CONVERGENT SEQUENCES OF EULER-MASCHERONI TYPE
UPB Sci Bull, Series A, Vol 79, Iss, 207 ISSN 22-7027 NEW FAST CONVERGENT SEQUENCES OF EULER-MASCHERONI TYPE Gabriel Bercu We itroduce two ew sequeces of Euler-Mascheroi type which have fast covergece
More informationGAMALIEL CERDA-MORALES 1. Blanco Viel 596, Valparaíso, Chile. s: /
THE GELIN-CESÀRO IDENTITY IN SOME THIRD-ORDER JACOBSTHAL SEQUENCES arxiv:1810.08863v1 [math.co] 20 Oct 2018 GAMALIEL CERDA-MORALES 1 1 Istituto de Matemáticas Potificia Uiversidad Católica de Valparaíso
More informationMONOTONICITY OF SEQUENCES INVOLVING GEOMETRIC MEANS OF POSITIVE SEQUENCES WITH LOGARITHMICAL CONVEXITY
MONOTONICITY OF SEQUENCES INVOLVING GEOMETRIC MEANS OF POSITIVE SEQUENCES WITH LOGARITHMICAL CONVEXITY FENG QI AND BAI-NI GUO Abstract. Let f be a positive fuctio such that x [ f(x + )/f(x) ] is icreasig
More information1 Generating functions for balls in boxes
Math 566 Fall 05 Some otes o geeratig fuctios Give a sequece a 0, a, a,..., a,..., a geeratig fuctio some way of represetig the sequece as a fuctio. There are may ways to do this, with the most commo ways
More informationFLOOR AND ROOF FUNCTION ANALOGS OF THE BELL NUMBERS. H. W. Gould Department of Mathematics, West Virginia University, Morgantown, WV 26506, USA
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A58 FLOOR AND ROOF FUNCTION ANALOGS OF THE BELL NUMBERS H. W. Gould Departmet of Mathematics, West Virgiia Uiversity, Morgatow, WV
More informationJournal of Ramanujan Mathematical Society, Vol. 24, No. 2 (2009)
Joural of Ramaua Mathematical Society, Vol. 4, No. (009) 199-09. IWASAWA λ-invariants AND Γ-TRANSFORMS Aupam Saikia 1 ad Rupam Barma Abstract. I this paper we study a relatio betwee the λ-ivariats of a
More informationCSE 1400 Applied Discrete Mathematics Number Theory and Proofs
CSE 1400 Applied Discrete Mathematics Number Theory ad Proofs Departmet of Computer Scieces College of Egieerig Florida Tech Sprig 01 Problems for Number Theory Backgroud Number theory is the brach of
More informationIn number theory we will generally be working with integers, though occasionally fractions and irrationals will come into play.
Number Theory Math 5840 otes. Sectio 1: Axioms. I umber theory we will geerally be workig with itegers, though occasioally fractios ad irratioals will come ito play. Notatio: Z deotes the set of all itegers
More informationAppendix to Quicksort Asymptotics
Appedix to Quicksort Asymptotics James Alle Fill Departmet of Mathematical Scieces The Johs Hopkis Uiversity jimfill@jhu.edu ad http://www.mts.jhu.edu/~fill/ ad Svate Jaso Departmet of Mathematics Uppsala
More informationSOME TRIBONACCI IDENTITIES
Mathematics Today Vol.7(Dec-011) 1-9 ISSN 0976-38 Abstract: SOME TRIBONACCI IDENTITIES Shah Devbhadra V. Sir P.T.Sarvajaik College of Sciece, Athwalies, Surat 395001. e-mail : drdvshah@yahoo.com The sequece
More informationSeries with Central Binomial Coefficients, Catalan Numbers, and Harmonic Numbers
3 47 6 3 Joural of Iteger Sequeces, Vol. 5 (0), Article..7 Series with Cetral Biomial Coefficiets, Catala Numbers, ad Harmoic Numbers Khristo N. Boyadzhiev Departmet of Mathematics ad Statistics Ohio Norther
More informationMath 155 (Lecture 3)
Math 55 (Lecture 3) September 8, I this lecture, we ll cosider the aswer to oe of the most basic coutig problems i combiatorics Questio How may ways are there to choose a -elemet subset of the set {,,,
More informationFibonacci numbers and orthogonal polynomials
Fiboacci umbers ad orthogoal polyomials Christia Berg April 10, 2006 Abstract We prove that the sequece (1/F +2 0 of reciprocals of the Fiboacci umbers is a momet sequece of a certai discrete probability,
More informationAbstract. 1. Introduction This note is a supplement to part I ([4]). Let. F x (1.1) x n (1.2) Then the moments L x are the Catalan numbers
Abstract Some elemetary observatios o Narayaa polyomials ad related topics II: -Narayaa polyomials Joha Cigler Faultät für Mathemati Uiversität Wie ohacigler@uivieacat We show that Catala umbers cetral
More informationBertrand s Postulate
Bertrad s Postulate Lola Thompso Ross Program July 3, 2009 Lola Thompso (Ross Program Bertrad s Postulate July 3, 2009 1 / 33 Bertrad s Postulate I ve said it oce ad I ll say it agai: There s always a
More informationLecture 1. January 8, 2018
Lecture 1 Jauary 8, 018 1 Primes A prime umber p is a positive iteger which caot be writte as ab for some positive itegers a, b > 1. A prime p also have the property that if p ab, the p a or p b. This
More informationarxiv: v1 [math.fa] 3 Apr 2016
Aticommutator Norm Formula for Proectio Operators arxiv:164.699v1 math.fa] 3 Apr 16 Sam Walters Uiversity of Norther British Columbia ABSTRACT. We prove that for ay two proectio operators f, g o Hilbert
More informationarxiv: v3 [math.nt] 24 Dec 2017
DOUGALL S 5 F SUM AND THE WZ-ALGORITHM Abstract. We show how to prove the examples of a paper by Chu ad Zhag usig the WZ-algorithm. arxiv:6.085v [math.nt] Dec 07 Keywords. Geeralized hypergeometric series;
More informationTR/46 OCTOBER THE ZEROS OF PARTIAL SUMS OF A MACLAURIN EXPANSION A. TALBOT
TR/46 OCTOBER 974 THE ZEROS OF PARTIAL SUMS OF A MACLAURIN EXPANSION by A. TALBOT .. Itroductio. A problem i approximatio theory o which I have recetly worked [] required for its solutio a proof that the
More informationMAT1026 Calculus II Basic Convergence Tests for Series
MAT026 Calculus II Basic Covergece Tests for Series Egi MERMUT 202.03.08 Dokuz Eylül Uiversity Faculty of Sciece Departmet of Mathematics İzmir/TURKEY Cotets Mootoe Covergece Theorem 2 2 Series of Real
More informationAn enumeration of flags in finite vector spaces
A eumeratio of flags i fiite vector spaces C Rya Viroot Departmet of Mathematics College of William ad Mary P O Box 8795 Williamsburg VA 23187 viroot@mathwmedu Submitted: Feb 2 2012; Accepted: Ju 27 2012;
More informationSequences of Definite Integrals, Factorials and Double Factorials
47 6 Joural of Iteger Sequeces, Vol. 8 (5), Article 5.4.6 Sequeces of Defiite Itegrals, Factorials ad Double Factorials Thierry Daa-Picard Departmet of Applied Mathematics Jerusalem College of Techology
More informationChapter 8. Euler s Gamma function
Chapter 8 Euler s Gamma fuctio The Gamma fuctio plays a importat role i the fuctioal equatio for ζ(s that we will derive i the ext chapter. I the preset chapter we have collected some properties of the
More informationSOME TRIGONOMETRIC IDENTITIES RELATED TO POWERS OF COSINE AND SINE FUNCTIONS
Folia Mathematica Vol. 5, No., pp. 4 6 Acta Uiversitatis Lodziesis c 008 for Uiversity of Lódź Press SOME TRIGONOMETRIC IDENTITIES RELATED TO POWERS OF COSINE AND SINE FUNCTIONS ROMAN WITU LA, DAMIAN S
More informationCALCULATION OF FIBONACCI VECTORS
CALCULATION OF FIBONACCI VECTORS Stuart D. Aderso Departmet of Physics, Ithaca College 953 Daby Road, Ithaca NY 14850, USA email: saderso@ithaca.edu ad Dai Novak Departmet of Mathematics, Ithaca College
More informationRandom Models. Tusheng Zhang. February 14, 2013
Radom Models Tusheg Zhag February 14, 013 1 Radom Walks Let me describe the model. Radom walks are used to describe the motio of a movig particle (object). Suppose that a particle (object) moves alog the
More informationHoggatt and King [lo] defined a complete sequence of natural numbers
REPRESENTATIONS OF N AS A SUM OF DISTINCT ELEMENTS FROM SPECIAL SEQUENCES DAVID A. KLARNER, Uiversity of Alberta, Edmoto, Caada 1. INTRODUCTION Let a, I deote a sequece of atural umbers which satisfies
More informationMATH 304: MIDTERM EXAM SOLUTIONS
MATH 304: MIDTERM EXAM SOLUTIONS [The problems are each worth five poits, except for problem 8, which is worth 8 poits. Thus there are 43 possible poits.] 1. Use the Euclidea algorithm to fid the greatest
More informationAN ALMOST LINEAR RECURRENCE. Donald E. Knuth Calif. Institute of Technology, Pasadena, Calif.
AN ALMOST LINEAR RECURRENCE Doald E. Kuth Calif. Istitute of Techology, Pasadea, Calif. form A geeral liear recurrece with costat coefficiets has the U 0 = a l* U l = a 2 " ' " U r - l = a r ; u = b, u,
More information1, if k = 0. . (1 _ qn) (1 _ qn-1) (1 _ qn+1-k) ... _: =----_...:... q--->1 (1- q) (1 - q 2 ) (1- qk) - -- n! k!(n- k)! n """"' n. k.
Abstract. We prove the ifiite q-biomial theorem as a cosequece of the fiite q-biomial theorem. 1. THE FINITE q-binomial THEOREM Let x ad q be complex umbers, (they ca be thought of as real umbers if the
More informationOn a general q-identity
O a geeral -idetity Aimi Xu Istitute of Mathematics Zheiag Wali Uiversity Nigbo 3500, Chia xuaimi009@hotmailcom; xuaimi@zwueduc Submitted: Dec 2, 203; Accepted: Apr 24, 204; Published: May 9, 204 Mathematics
More informationGeneral Properties Involving Reciprocals of Binomial Coefficients
3 47 6 3 Joural of Iteger Sequeces, Vol. 9 006, Article 06.4.5 Geeral Properties Ivolvig Reciprocals of Biomial Coefficiets Athoy Sofo School of Computer Sciece ad Mathematics Victoria Uiversity P. O.
More informationConcavity of weighted arithmetic means with applications
Arch. Math. 69 (1997) 120±126 0003-889X/97/020120-07 $ 2.90/0 Birkhäuser Verlag, Basel, 1997 Archiv der Mathematik Cocavity of weighted arithmetic meas with applicatios By ARKADY BERENSTEIN ad ALEK VAINSHTEIN*)
More informationq-lucas polynomials and associated Rogers-Ramanujan type identities
-Lucas polyomials associated Rogers-Ramauja type idetities Joha Cigler Faultät für Mathemati, Uiversität Wie johacigler@uivieacat Abstract We prove some properties of aalogues of the Fiboacci Lucas polyomials,
More informationOn Generalized Fibonacci Numbers
Applied Mathematical Scieces, Vol. 9, 215, o. 73, 3611-3622 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/1.12988/ams.215.5299 O Geeralized Fiboacci Numbers Jerico B. Bacai ad Julius Fergy T. Rabago Departmet
More informationOn the Inverse of a Certain Matrix Involving Binomial Coefficients
It. J. Cotemp. Math. Scieces, Vol. 3, 008, o. 3, 5-56 O the Iverse of a Certai Matrix Ivolvig Biomial Coefficiets Yoshiari Iaba Kitakuwada Seior High School Keihokushimoyuge, Ukyo-ku, Kyoto, 60-0534, Japa
More informationWorksheet on Generating Functions
Worksheet o Geeratig Fuctios October 26, 205 This worksheet is adapted from otes/exercises by Nat Thiem. Derivatives of Geeratig Fuctios. If the sequece a 0, a, a 2,... has ordiary geeratig fuctio A(x,
More informationCERTAIN GENERAL BINOMIAL-FIBONACCI SUMS
CERTAIN GENERAL BINOMIAL-FIBONACCI SUMS J. W. LAYMAN Virgiia Polytechic Istitute State Uiversity, Blacksburg, Virgiia Numerous writers appear to have bee fasciated by the may iterestig summatio idetitites
More informationSome Extensions of the Prabhu-Srivastava Theorem Involving the (p, q)-gamma Function
Filomat 31:14 2017), 4507 4513 https://doi.org/10.2298/fil1714507l Published by Faculty of Scieces ad Mathematics, Uiversity of Niš, Serbia Available at: http://www.pmf.i.ac.rs/filomat Some Extesios of
More informationSubject: Differential Equations & Mathematical Modeling-III
Power Series Solutios of Differetial Equatios about Sigular poits Subject: Differetial Equatios & Mathematical Modelig-III Lesso: Power series solutios of differetial equatios about Sigular poits Lesso
More informationHarmonic Number Identities Via Euler s Transform
1 2 3 47 6 23 11 Joural of Iteger Sequeces, Vol. 12 2009), Article 09.6.1 Harmoic Number Idetities Via Euler s Trasform Khristo N. Boyadzhiev Departmet of Mathematics Ohio Norther Uiversity Ada, Ohio 45810
More informationApplied Mathematics Letters. On the properties of Lucas numbers with binomial coefficients
Applied Mathematics Letters 3 (1 68 7 Cotets lists available at ScieceDirect Applied Mathematics Letters joural homepage: wwwelseviercom/locate/aml O the properties of Lucas umbers with biomial coefficiets
More informationTHE N-POINT FUNCTIONS FOR INTERSECTION NUMBERS ON MODULI SPACES OF CURVES
THE N-POINT FUNTIONS FOR INTERSETION NUMBERS ON MODULI SPAES OF URVES KEFENG LIU AND HAO XU Abstract. We derive from Witte s KdV equatio a simple formula of the -poit fuctios for itersectio umbers o moduli
More informationLecture 7: Properties of Random Samples
Lecture 7: Properties of Radom Samples 1 Cotiued From Last Class Theorem 1.1. Let X 1, X,...X be a radom sample from a populatio with mea µ ad variace σ
More informationInteresting Series Associated with Central Binomial Coefficients, Catalan Numbers and Harmonic Numbers
3 47 6 3 Joural of Iteger Sequeces Vol. 9 06 Article 6.. Iterestig Series Associated with Cetral Biomial Coefficiets Catala Numbers ad Harmoic Numbers Hogwei Che Departmet of Mathematics Christopher Newport
More informationMATH 324 Summer 2006 Elementary Number Theory Solutions to Assignment 2 Due: Thursday July 27, 2006
MATH 34 Summer 006 Elemetary Number Theory Solutios to Assigmet Due: Thursday July 7, 006 Departmet of Mathematical ad Statistical Scieces Uiversity of Alberta Questio [p 74 #6] Show that o iteger of the
More informationPUTNAM TRAINING INEQUALITIES
PUTNAM TRAINING INEQUALITIES (Last updated: December, 207) Remark This is a list of exercises o iequalities Miguel A Lerma Exercises If a, b, c > 0, prove that (a 2 b + b 2 c + c 2 a)(ab 2 + bc 2 + ca
More informationSum of cubes: Old proofs suggest new q analogues
Sum of cubes: Old proofs suggest ew aalogues Joha Cigler Faultät für Mathemati, Uiversität Wie ohacigler@uivieacat Abstract We show how old proofs of the sum of cubes suggest ew aalogues 1 Itroductio I
More informationCERTAIN CONGRUENCES FOR HARMONIC NUMBERS Kotor, Montenegro
MATHEMATICA MONTISNIGRI Vol XXXVIII (017) MATHEMATICS CERTAIN CONGRUENCES FOR HARMONIC NUMBERS ROMEO METROVIĆ 1 AND MIOMIR ANDJIĆ 1 Maritie Faculty Kotor, Uiversity of Moteegro 85330 Kotor, Moteegro e-ail:
More informationDIVISIBILITY PROPERTIES OF GENERALIZED FIBONACCI POLYNOMIALS
DIVISIBILITY PROPERTIES OF GENERALIZED FIBONACCI POLYNOMIALS VERNER E. HOGGATT, JR. Sa Jose State Uiversity, Sa Jose, Califoria 95192 ad CALVIN T. LONG Washigto State Uiversity, Pullma, Washigto 99163
More informationShivley s Polynomials of Two Variables
It. Joural of Math. Aalysis, Vol. 6, 01, o. 36, 1757-176 Shivley s Polyomials of Two Variables R. K. Jaa, I. A. Salehbhai ad A. K. Shukla Departmet of Mathematics Sardar Vallabhbhai Natioal Istitute of
More informationON SOME DIOPHANTINE EQUATIONS RELATED TO SQUARE TRIANGULAR AND BALANCING NUMBERS
Joural of Algebra, Number Theory: Advaces ad Applicatios Volume, Number, 00, Pages 7-89 ON SOME DIOPHANTINE EQUATIONS RELATED TO SQUARE TRIANGULAR AND BALANCING NUMBERS OLCAY KARAATLI ad REFİK KESKİN Departmet
More informationOn a class of convergent sequences defined by integrals 1
Geeral Mathematics Vol. 4, No. 2 (26, 43 54 O a class of coverget sequeces defied by itegrals Dori Adrica ad Mihai Piticari Abstract The mai result shows that if g : [, ] R is a cotiuous fuctio such that
More informationThe Riemann Zeta Function
Physics 6A Witer 6 The Riema Zeta Fuctio I this ote, I will sketch some of the mai properties of the Riema zeta fuctio, ζ(x). For x >, we defie ζ(x) =, x >. () x = For x, this sum diverges. However, we
More information(I.C) THE DISTRIBUTION OF PRIMES
I.C) THE DISTRIBUTION OF PRIMES I the last sectio we showed via a Euclid-ispired, algebraic argumet that there are ifiitely may primes of the form p = 4 i.e. 4 + 3). I fact, this is true for primes of
More information62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 +
62. Power series Defiitio 16. (Power series) Give a sequece {c }, the series c x = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + is called a power series i the variable x. The umbers c are called the coefficiets of
More informationChapter 8. Euler s Gamma function
Chapter 8 Euler s Gamma fuctio The Gamma fuctio plays a importat role i the fuctioal equatio for ζ(s) that we will derive i the ext chapter. I the preset chapter we have collected some properties of the
More informationANTI-LECTURE HALL COMPOSITIONS AND ANDREWS GENERALIZATION OF THE WATSON-WHIPPLE TRANSFORMATION
ANTI-LECTURE HALL COMPOSITIONS AND ANDREWS GENERALIZATION OF THE WATSON-WHIPPLE TRANSFORMATION SYLVIE CORTEEL, JEREMY LOVEJOY AND CARLA SAVAGE Abstract. For fixed ad k, we fid a three-variable geeratig
More informationOn the distribution of coefficients of powers of positive polynomials
AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 49 (2011), Pages 239 243 O the distributio of coefficiets of powers of positive polyomials László Major Istitute of Mathematics Tampere Uiversity of Techology
More informationMatrix representations of Fibonacci-like sequences
NTMSCI 6, No. 4, 03-0 08 03 New Treds i Mathematical Scieces http://dx.doi.org/0.085/tmsci.09.33 Matrix represetatios of Fiboacci-like sequeces Yasemi Tasyurdu Departmet of Mathematics, Faculty of Sciece
More informationFibonacci polynomials, generalized Stirling numbers, and Bernoulli, Genocchi and tangent numbers
Fiboacci polyomials, geeralied Stirlig umbers, ad Beroulli, Geocchi ad taget umbers Joha Cigler oha.cigler@uivie.ac.at Abstract We study matrices hich trasform the sequece of Fiboacci or Lucas polyomials
More informationWeek 5-6: The Binomial Coefficients
Wee 5-6: The Biomial Coefficiets March 6, 2018 1 Pascal Formula Theorem 11 (Pascal s Formula For itegers ad such that 1, ( ( ( 1 1 + 1 The umbers ( 2 ( 1 2 ( 2 are triagle umbers, that is, The petago umbers
More informationGENERALIZED LEGENDRE POLYNOMIALS AND RELATED SUPERCONGRUENCES
J. Nuber Theory 0, o., 9-9. GENERALIZED LEGENDRE POLYNOMIALS AND RELATED SUPERCONGRUENCES Zhi-Hog Su School of Matheatical Scieces, Huaiyi Noral Uiversity, Huaia, Jiagsu 00, PR Chia Eail: zhihogsu@yahoo.co
More informationInduction proofs - practice! SOLUTIONS
Iductio proofs - practice! SOLUTIONS 1. Prove that f ) = 6 + + 15 is odd for all Z +. Base case: For = 1, f 1) = 41) + 1) + 13 = 19. Sice 19 is odd, f 1) is odd - base case prove. Iductive hypothesis:
More informationBINOMIAL PREDICTORS. + 2 j 1. Then n + 1 = The row of the binomial coefficients { ( n
BINOMIAL PREDICTORS VLADIMIR SHEVELEV arxiv:0907.3302v2 [math.nt] 22 Jul 2009 Abstract. For oegative itegers, k, cosider the set A,k = { [0, 1,..., ] : 2 k ( ). Let the biary epasio of + 1 be: + 1 = 2
More informationHankel determinants of some polynomial sequences. Johann Cigler
Hael determiats of some polyomial sequeces Joha Cigler Faultät für Mathemati, Uiversität Wie ohacigler@uivieacat Abstract We give simple ew proofs of some Catala Hael determiat evaluatios by Ömer Eğecioğlu
More informationREGULARIZATION OF CERTAIN DIVERGENT SERIES OF POLYNOMIALS
REGULARIZATION OF CERTAIN DIVERGENT SERIES OF POLYNOMIALS LIVIU I. NICOLAESCU ABSTRACT. We ivestigate the geeralized covergece ad sums of series of the form P at P (x, where P R[x], a R,, ad T : R[x] R[x]
More informationDominant of Functions Satisfying a Differential Subordination and Applications
Domiat of Fuctios Satisfyig a Differetial Subordiatio ad Applicatios R Chadrashekar a, Rosiha M Ali b ad K G Subramaia c a Departmet of Techology Maagemet, Faculty of Techology Maagemet ad Busiess, Uiversiti
More informationand each factor on the right is clearly greater than 1. which is a contradiction, so n must be prime.
MATH 324 Summer 200 Elemetary Number Theory Solutios to Assigmet 2 Due: Wedesday July 2, 200 Questio [p 74 #6] Show that o iteger of the form 3 + is a prime, other tha 2 = 3 + Solutio: If 3 + is a prime,
More informationA symbolic approach to multiple zeta values at the negative integers
A symbolic approach to multiple zeta values at the egative itegers Victor H. Moll a, Li Jiu a Christophe Vigat a,b a Departmet of Mathematics, Tulae Uiversity, New Orleas, USA Correspodig author b LSS/Supelec,
More informationSquare-Congruence Modulo n
Square-Cogruece Modulo Abstract This paper is a ivestigatio of a equivalece relatio o the itegers that was itroduced as a exercise i our Discrete Math class. Part I - Itro Defiitio Two itegers are Square-Cogruet
More informationChapter 7 COMBINATIONS AND PERMUTATIONS. where we have the specific formula for the binomial coefficients:
Chapter 7 COMBINATIONS AND PERMUTATIONS We have see i the previous chapter that (a + b) ca be writte as 0 a % a & b%þ% a & b %þ% b where we have the specific formula for the biomial coefficiets: '!!(&)!
More informationEXPANSION FORMULAS FOR APOSTOL TYPE Q-APPELL POLYNOMIALS, AND THEIR SPECIAL CASES
LE MATEMATICHE Vol. LXXIII 208 Fasc. I, pp. 3 24 doi: 0.448/208.73.. EXPANSION FORMULAS FOR APOSTOL TYPE Q-APPELL POLYNOMIALS, AND THEIR SPECIAL CASES THOMAS ERNST We preset idetities of various kids for
More informationProof of Fermat s Last Theorem by Algebra Identities and Linear Algebra
Proof of Fermat s Last Theorem by Algebra Idetities ad Liear Algebra Javad Babaee Ragai Youg Researchers ad Elite Club, Qaemshahr Brach, Islamic Azad Uiversity, Qaemshahr, Ira Departmet of Civil Egieerig,
More information