Applicable Analysis and Discrete Mathematics available online at ON JENSEN S AND RELATED COMBINATORIAL IDENTITIES

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1 Applicable Aalysis ad Discrete Mathematics available olie at Appl Aal Discrete Math , doi:102298/aadm g ON JENSEN S AND RELATED COMBINATORIAL IDENTITIES Victor J W Guo Motivated by the recet wor of Chu [Electro J Combi , #N24], we give simple proofs of Jese s idetity x+z y z x+y z, ad Chu s ad Mohaty-Hada s geeralizatios of Jese s idetity We also give a simple proof of a equivalet form of Graham-Kuth- Patashi s idetity m+r + x m y m 0 r + x m x+y, m 0 which was rediscovered, respectively, by Su i 2003 ad Muarii i 2005 Fially we give a multiomial coefficiet geeralizatio of this idetity 1 INTRODUCTION Abel s idetity see, for example, [8, 31] xx+z 1 y z x+y 2010 Mathematics Subject Classificatio Primary 05A10 Secodly 05A19 Keywords ad Phrases Jese s idetity, Chu s idetity, Mohaty-Hada s idetity, Graham- Kuth-Patashi s idetity, multiomial coefficiet 201

2 202 Victor J W Guo ad Rothe s idetity [23] or Hage-Rothe s idetity, see, for example, [9, 54] x x z x z y +z x+y are famous i literature ad play a importat role i eumerative combiatorics Recetly, Chu [6] gave elemetary proofs of Abel s idetity ad Rothe s idetity by usig the biomial theorem ad the Chu-Vadermode covolutio formula, respectively Motivated by Chu s wor, we study Jese s idetity [17], which is closely related to Rothe s idetity, ad ca be stated as follows: 1 x+z y z, x+y z Jese s idetity 1 has attracted much attetio by differet authors Gould [11] obtaied the followig Abel-type aalogue: 2 x+z! y z! x+y z! Carlitz [1] gave two iterestig theorems related to 1 ad 2 by mathematical iductio With the help of geeratig fuctios, Gould [12] derived the followig variatio of Jese s idetity 1: x+z y z x+y x+y z z x+y E G-Rodeja F [10] deduced Gould s idetity 2 from 1 by establishig a idetity which icludes both Cohe ad Su [7] also gave a expressio which uifies 1 ad 2 Chu [4] geeralized Jese s idetity 1 to a multi-sum form: si1 xi + i z i +s 2 x1 + +x s +z z Moreover, the idetities 1 ad 3 were respectively geeralized by Mohaty ad Hada [19] ad Chu [5] to the case of multiomial coefficiets to be stated i Sectio 4 The primary purpose of this paper is to give simple proofs of Jese s idetity, Chu s idetity 3, Mohaty-Hada s idetity, ad Chu s geeralizatio of Mohaty-Hada s idetity We use the Chu-Vadermode covolutio formula x y x+y

3 O Jese s ad related combiatorial idetities 203 ad the well-ow idetity { 1 r 0, if 0 r 1, 4!, if r Equatio 4 may be easily deduced from the Stirlig umbers of the secod id [27, p 34, 24a] The first case of 4 was already utilized by the author [13] to give a simple proof of Dixo s idetity ad by Chu [6] i his proofs of Abel s ad Rothe s idetities It is iterestig that our proof of Chu s idetity 3 also leads to a very short proof of Graham-Kuth-Patashi s idetity, which was rediscovered several times i the past few years The secodary purpose of this paper is to give a multiomial coefficiet geeralizatio of Graham-Kuth-Patashi s idetity 5 2 PROOF OF JENSEN S IDENTITY By the Chu-Vadermode covolutio formula, we have x+z y z x+z x+y +1 x z 1 i i Iterchagig the summatio order i 5 ad oticig that x+z x z 1 i x+z +i 1 i, i i we have x+z y z 6 i i x+y +1 i x+z +i 1 i i i x+y +1 z 1 i, i i0 i0 x+z +i where the secod equality holds because is a polyomial i of i degree i with leadig coefficiet z 1 i /i! ad we ca apply 4 to simplify We ow substitute x x 1, y y + 1 ad z z +1 i 6 ad observe that 7 The we obtai x x+ 1 1 x+z y z x+y i z i, i i0

4 204 Victor J W Guo as desired Combiig 1 ad 6, we get the followig idetity: x x+1 z z 1, which is equivalet to the followig idetity i Graham et al [9, p 218]: m+r x y m r x x+y m m m 3 PROOFS OF CHU S AND GRAHAM-KNUTH-PATASHNIK S IDENTITIES Comparig the coefficiets of x i both sides of the equatio by the biomial theorem, we have a1 + +a s 8 1+x a1+ +as 1+x a1 1+x as i 1+ +i s a1 i 1 Lettig a i x i i z 1 1 i s 1 ad a s x 1 + +x s +z +s 1 i 8, we have xs + s z xs + 1 s 1 z s 1 s 1 x1 + +x s +z +s 1 j j 1+ + s 1 j 1+ +j s 1j s 1 xi i z 1 j i i i1 where s It follows that xi + i z si1 i x1 + +x s +z +s 1 j, as i s 1+ + s 10j 1+ + s 1 j 1+ +j s 1j s 1 xi + i z xi i z 1 i1 i j i i Iterchagig the summatio order i 9 ad observig that xi + i z xi i z 1 ji xi + 1 ji i i z +j i i i j i i i j i

5 O Jese s ad related combiatorial idetities 205 xi + ad iz +j i i is a polyomial i j i of degree j i with leadig coefficiet i z 1 ji /j i!, by 4 we get xi + i z x1 + +x s +z +s si1 i j j0 j +s 2 z 1 j x1 + +x s +z +s 1 z 1 j j j j 1+ +j s 1j j0 Substitutig x i x i 1 i 1,,s ad z z+1 i 10 ad usig 7, we immediately get Chu s idetity 3 Comparig 3 with 10 ad replacig s with s+2, we obtai 11 +s x z +s x+s+1 z 1 It is easy to see that the idetity 11 is equivalet to each of the followig ow idetities: Graham-Kuth-Patashi s idetity [9, p 218] m+r + 12 x m y 0 0 r m m + x m x+y Su s idetity [29] m m + 1 m 1+x + a a Muarii s idetity [20] β α+ β x m+ a α x m+ a β + x For example, substitutig m, s, x r 1 ad z y/x i 11, we are led to 12 Replacig by m ad respectively i both sides of 13, we get m+ a 1 m m m+ a m+ a m+ a 1+x m+ a x m+ a,

6 206 Victor J W Guo which is equivalet to 11 by chagig to m+ a Moreover, the followig special case x + x was reproved by Simos [26], Hirschhor [15], Chapma [2], Prodiger [21], Wag ad Su [30] 4 MOHANTY-HANDA S IDENTITY AND CHU S GENERALIZATION Let m be a fixed positive iteger For a a 1,,a m N m ad b b 1,,b m C m, set a a 1 + +a m, a! a 1! a m!, a+b a 1 +b 1,,a m + b m, a b a 1 b a m b m, ad b a b a1 1 bam m For ay variable x ad 1,, m Z m x, the multiomial coefficiet is defied by { x xx 1 x +1/!, if N m, 0, otherwise Moreover, we let 0 0,,0 ad 1 1,,1 Note that the Chu-Vadermode covolutio formula has the followig trivial geeralizatio 16 x y x+y as metioed by Zeg [32], while 4 ca be easily geeralized as 17 1 r, { 0, if r i < i for some 1 i m!, if r, where : m i1 i i I 1969, Mohaty ad Hada [19] established the followig multiomial coefficiet geeralizatio of Jese s idetity 18 x+ z y z x+y z

7 O Jese s ad related combiatorial idetities 207 Here ad i what follows, 1,, m Twety years later, Mohaty- Hada s idetity was geeralized by Chu [5] as follows: si1 xi + i z i +s 2 x1 + +x s + z z, which is also a geeralizatio of 3 Here i i1,, im, i 1,,m Remar Note that the correspodig multiomial coefficiet geeralizatio of Rothe s idetity was already obtaied by Raey [22] for a special case ad Mohaty [18] The reader is referred to Strehl [28] for a historical ote o Raey-Mohaty s idetity We will give a elemetary proof of Chu s idetity 19 similar to that of 3 Lemma 41 For N m ad s 1, there holds si1 i +s 1 i Proof For ay oegative itegers a 1,,a s such that a 1 + +a s, by the Chu-Vadermode covolutio formula 16, the followig idetity holds si1 Moreover, for s, we have i1 ai i ai i 0 if ad oly if i a i i 1,,s Thus, the idetity 21 may be rewritte as It follows that i 1+ + si1 i 1+ + s i1 1 a 1,, s a s ai i a 1+ +a s 1+ + s i1 1 a 1,, s a s +s 1, ai i a 1+ +a s as desired

8 208 Victor J W Guo By repeatedly usig the covolutio formula 16, we may rewrite the lefthad side of 19 as x1 + +x s + z+m 1 22 j 1+ + s 10j 1+ + s 1 j 1+ +j s 1j s 1 xi + i z xi i z 1 i j i i i1 Iterchagig the summatio order i 22, observig that xi + i z xi i z 1 1 ji i ji xi + i z+ j i i i j i i i j i ad xi + i z+ j i i is a polyomial i i1,, im with the coefficiet of ji i beig applyig 17, we get xi + i z si1 i x1 + +x s + z+s 1 j0 j j i z 1 j j +s 2 x1 + +x s + z+s 1 j0 j j j 1+ +j s 1ji1 z 1 j, ji j i z 1 ji /j i!, m ji where the secod equality follows from 20 Substitutig x i x i 1 i x 1,,s ad z z+1 i 23 ad observig that 1 x+ 1, we immediately get 19 Comparig 19 with 23 ad replacig s with s+2, we obtai the followig result Theorem 42 For N m ad z C m, it holds that +s x +s 24 z x+s+1 j i z 1 It is easy to see that 24 is a multiomial coefficiet geeralizatio of 11 Substitutig s β, x α β 1 ad z 1+x i 24, we get β α+ β + α β x x,

9 O Jese s ad related combiatorial idetities 209 which is a geeralizatio of Muarii s idetity 14 If α β, the 25 reduces to x x, which is a geeralizatio of Simos idetity 15 Note that Shattuc [25] ad Che ad Pag [3] have give differet combiatorial proofs of 14 It is atural to as the followig problem Problem 43 Is there a combiatorial iterpretatio of 25? I fact, such a proof was recetly foud by Yag [31] 5 CONCLUDING REMARKS We ow that biomial coefficiet idetities usually have ice q-aalogues However, there are oly curious ot atural q-aalogues of Abel s ad Rothe s idetities see [24] ad refereces therei up to ow There seems to have o q-aalogues of Jese s idetity i the literature It is iterestig that Hou ad Zeg [16] gave a q-aalogue of Su s idetity 13: 26 m [ ][ m + 1 m a ] xq a ;q + a q [ +1 2 m+ a 2 ][ ] m+ x m+ a q m+ 2, a where a;q 1 a1 aq 1 aq 1 ad [ ] α q α +1 ;q, if 0, q;q 0, if < 0 Clearly, 26 may be writte as a q-aalogue of Muarii s idetity 14: 27 [ ][ ] β α+ β + 1 q 2 2 x;q [ ][ ] α β + +1 q 2 +β α x, as metioed by Guo ad Zeg [14] We ed this paper with the followig problem

10 210 Victor J W Guo Problem 51 Is there a q-aalogue of 25? Or equivaletly, is there a multi-sum geeralizatio of 27? Acowledgmets This wor was partially supported by the Fudametal Research Fuds for the Cetral Uiversities, Shaghai Leadig Academic Disciplie Project #B407, Shaghai Risig-Star Program #09QA , ad the Natioal Sciece Foudatio of Chia # REFERENCES 1 L Carlitz: Some formulas of Jese ad Gould Due Math J, , R Chapma: A curious idetity revisited Math Gazette, , W Y C Che, S X M Pag: O the combiatorics of the Pfaff idetity Discrete Math, , W Chu: O a extesio of a partitio idetity ad its Abel aalog J Math Res Expositio, , W Chu: Jese s theorem o multiomial coefficiets ad its Abel-aalog Appl Math J Chiese Uiv, , i Chiese 6 W Chu: Elemetary proofs for covolutio idetities of Abel ad Hage-Rothe Electro J Combi, , #N24 7 M E Cohe, H S Su: A ote o the Jese-Gould covolutios Caad Math Bull, , L Comtet: Advaced Combiatorics D Reidel Publishig Compay, Dordrecht- Hollad, R L Graham, D E Kuth, O Patashi: Cocrete Mathematics 2d Editio, Addiso-Wesley Publishig Compay, Readig, MA, E G-Rodeja F: O idetities of Jese, Gould ad Carlitz I: Proc Fifth Aual Reuio of Spaish Mathematicias Valecia, 1964, Publ Ist Jorge Jua Mat, Madrid, 1967, pp H W Gould: Geeralizatio of a theorem of Jese cocerig covolutios Due Math J, , H W Gould: Ivolvig sums of biomial coefficiets ad a formula of Jese Amer Math Mothly, , V J W Guo: A simple proof of Dixo s idetity Discrete Math, , V J W Guo, J Zeg: Combiatorial proof of a curious q-biomial coefficiet idetity Electro J Combi, , #N13 15 M Hirschhor: Commet o a curious idetity Math Gazette, , S J X Hou, J Zeg: A q-aalog of dual sequeces with applicatios Europea J Combi, , J L W V Jese: Sur ue idetité d Abel et sur d autres formules aalogues Acta Math, , S G Mohaty: Some covolutios with multiomial coefficiets ad related probability distributios SIAM Rev, , S G Mohaty, B R Hada: Extesios of Vadermode type covolutios with several summatios ad their applicatios, I Caad Math Bull, , E Muarii: Geeralizatio of a biomial idetity of Simos Itegers, , #A15 21 H Prodiger: A curious idetity proved by Cauchy s itegral formula Math Gazette, ,

11 O Jese s ad related combiatorial idetities G N Raey: Fuctioal compositio patters ad power series reversio Tras Amer Math Soc, , H A Rothe: Formulae de serierum reversioe demostratio uiversalis sigis localibus combiatorio-aalyticorum vicariis exhibita Leipzig, M Schlosser: Abel-Rothe type geeralizatios of Jacobi s triple product idetity i: Theory ad Applicatios of Special Fuctios Dev Math, 13, Spriger, New Yor, 2005, pp M Shattuc: Combiatorial proofs of some Simos-type biomial coefficiet idetities Itegers, , #A27 26 S Simos: A curious idetity Math Gazette, , R P Staley: Eumerative Combiatorics Vol 1, Cambridge Studies i Advaced Mathematics, 49, Cambridge Uiversity Press, Cambridge, V Strehl: Idetities of Rothe-Abel-Schläfli-Hurwitz-type Discrete Math, , Z-W Su: Combiatorial idetities i dual sequeces Europea J Combi, , X Wag, Y Su: A ew proof of a curious idetity Math Gazette, , D-M Yag: A combiatorial proof of Guo s multi-geeralizatio of Muarii s idetity Itegers, to appear 32 J Zeg: Multiomial covolutio polyomials Discrete Math, , Departmet of Mathematics, Received March 10, 2011 East Chia Normal Uiversity, Revised July 12, 2011 Shaghai , People s Republic of Chia jwguo@mathecueduc