#A18 INTEGERS 11 (2011) THE (EXPONENTIAL) BIPARTITIONAL POLYNOMIALS AND POLYNOMIAL SEQUENCES OF TRINOMIAL TYPE: PART I
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1 #A18 INTEGERS 11 (2011) THE (EXPONENTIAL) BIPARTITIONAL POLYNOMIALS AND POLYNOMIAL SEQUENCES OF TRINOMIAL TYPE: PART I Miloud Mihoubi 1 Uiversité des Scieces et de la Techologie Houari Bouediee Faculty of Matheatics El Alia Bab Ezzouar Algeria ihoubi@usthbdz iloudihoubi@gailco Hacèe Belbachir 2 Uiversité des Scieces et de la Techologie Houari Bouediee Faculty of Matheatics El Alia Bab Ezzouar Algeria hbelbachir@usthbdz haceebelbachir@gailco Received: 3/29/10 Revised: 10/3/10 Accepted: 12/31/10 Published: 2/4/11 Abstract The ai of this paper is to ivestigate ad preset the geeral properties of the (epoetial) bipartitioal polyoials After establishig relatios betwee bipartitioal polyoials ad polyoial sequeces of bioial ad trioial type a uber of idetities are deduced 1 Itroductio For ( ) N 2 with N {0 1 2 } the coplete bipartitioal polyoial A A ( ) with variables is defied by the su A :!! 01! 10!! 01 0!1! !0! 10 (1)!! where the suatio is carried out over all partitios of the bipartite uber ( ) that is over all oegative itegers which are solutio to the equatios i ij j ij (2) i1 j0 1 The research is partially supported by LAID3 Laboratory of USTHB Uiversity 2 The research is partially supported by LAID3 Laboratory ad CMEP project 09 MDU 765 j1
2 INTEGERS: 11 (2011) 2 The partial bipartitioal polyoial A A ( 01 ) with variables of degree N is defied by the su A :!! 01! 10!! 01 0!1! !0! 10 (3)!! where the suatio is carried out over all partitios of the bipartite uber ( ) ito parts that is over all oegative itegers which are solutio to the equatios i ij i1 j0 j ij j1 j0 ij with the covetio 00 0 (4) These polyoials were itroduced i [1 pp 454] with properties such as those give i (5) (6) (7) (8) below Ideed for all real ubers α β γ we have ad A βγ01 αγ 10 β 2 γ 02 α β γ β α γ A ( ) (5) A (α 01 β 10 α β ) α β A ( ) (6) Moreover we ca chec that the epoetial geeratig fuctios for A ad A are respectively provided by 1 + A ( 01 ) t u!! ep t i u j ij (7) ad A ( 01 ) t u!! 1 t i u j ij (8) Fro the above defiitios ad properties we attept to preset ore properties ad idetities for the partial ad coplete bipartitioal polyoials We also cosider the coectio betwee these polyoials ad polyoials (f ()) of trioial type defied by f ij (1) ti u j f () t u!! with f 00 () : 1 (9) ij 0 0 For the ivestigatio of polyoials of trioial ad ultioial type we ca refer to [8]
3 INTEGERS: 11 (2011) 3 We use the otatios A ( ij ) or A ( IJ ) for A ( ) ad A ( ij ) or A ( IJ ) for A ( 01 ) Moreover we represet respectively by B ( j ) ad B ( j ) the partial ad coplete Bell polyoials B ( 1 2 ) ad B ( 1 2 ) We recall that these polyoials are defied by their geeratig fuctios B ( j ) t! 1 t! B ( j ) t! ep t! 1 which adit the followig eplicit epressios: B ( j ) B ( j ) ! 1! 2! 1! 1! 2! 1 1! 1 1! ! ! For < 0 or < 0 we set f () 0 ad A ( ij ) 0 ad for < 0 or < 0 or < 0 we set A ( ij ) 0 For C where C is the set of cople ubers we set : 1 ( 1) ( + 1) for ad 0 which gives 0 for > N 0 otherwise 2 Properties of Bipartitioal Polyoials I this sectio we give soe recurrece relatios for the bipartitioal polyoials fro which we deduce coectios with Bell polyoials I the followig we cosider a sequece ( ) of real ubers with 00 0
4 INTEGERS: 11 (2011) 4 Theore 1 We start with i j i+j + +1 i j i+j + +1 i j i+j + +1 ij A i j 1 ( IJ ) A ( IJ ) (10) i ij A i j 1 ( IJ ) A ( IJ ) (11) j ij A i j 1 ( IJ ) A ( IJ ) (12) ad i j ij i j ij i ij A i j ( IJ ) A ( IJ ) (13) j ij A i j ( IJ ) A ( IJ ) (14) Proof Fro (8) we obtai d 1 t i ij d rs i! u j j! 1 t i ij ( 1)! i! u j j! 1 t r u s r! s! + r + s r s r s r+s+ 1 A 1 ( IJ ) t +r A r s 1 ( IJ ) t! u +s ( + r)! ( + s)! u! we also have d 1 t i u j ij d rs + d A ( IJ ) t u d rs!! Fro these two epressios we deduce that d A ( IJ ) A r s 1 ( IJ ) (15) d rs r s For α β 1 we tae the derivatives o both sides of (5) with respect to γ ad by eas of (15) we obtai (10) The sae goes for idetities (11) ad (12) Whe we su over all possible values of for both sides of each of the idetities (11) ad (12) we obtai idetities (13) ad (14) respectively
5 INTEGERS: 11 (2011) 5 The et theore provides the li betwee the bipartitioal polyoials ad the classical Bell polyoials Theore 2 We have j j0 A j j ( IJ ) B j j ij i (16) i A 0 ( IJ ) B ( 0j ) (17) A 0 ( IJ ) B ( j0 ) (18) ad j j0 A j j ( IJ ) B j j ij i (19) i A 0 ( IJ ) B ( 0j ) (20) A 0 ( IJ ) B ( j0 ) (21) Proof Fro (8) if we set t u we obtai 1 u i+j ij i!j! ie 1 s1 u s s! ij 0 + s s is i u s i s! s A ( IJ ) u+!! s s A s ( IJ ) which by idetificatio gives the idetity (16) The idetities (17) ad (18) result by settig respectivelly t 0 ad u 0 i (8) Suig over all possible values of o both sides of each of the idetities (16) (17) ad (18) we obtai idetities (19) (20) ad (21) respectively For a sequece of real ubers ( ) we deote by A ( i+j ) the quatity obtaied by replacig ij by i+j i A ( ij ) The sae goes for A ( i+j ) 0 Theore 3 Let ( ) be a sequece of real ubers The followig holds A ( i+j ) B + ( j ) (22) A ( i+j ) B + ( j ) (23) A + (i i+j ) B ++ (j j ) (24) A + (j i+j ) B ++ (j j ) (25)
6 INTEGERS: 11 (2011) 6 Proof Fro (8) we get for ij i+j that 1 t i i+j i! u j j! + A ( i+j ) t u!! ; we also get 1 t i i+j i! u j j! 1 s 1 s s! (t + u)s (t + u)s B s ( j ) s! s B + ( j ) t u!! + By idetificatio we obtai (22) Idetity (23) follows by suig over all possible values of o both sides of idetity (22)We have 1 t i i i+j i! u j j! 1 s s! s 1 t s 1 s s! i+js d dt i ti u j (t + u)s 1 t (t + u) s s s t + u s! s 1 t (t + u)r B r (j j ) r! r + B + (j j ) t! + + The by idetificatio with (8) ad aig use of the syetry of we obtai (24) ad (25) Note that for Theore 3 usig the idetities o partial ad coplete Bell polyoials give i [3] ad [4] we ca deduce several idetities for the partial ad coplete bipartitioal polyoials u!
7 INTEGERS: 11 (2011) 7 3 Polyoial Sequeces of Trioial Type We ow that the sequeces of bioial type have a strog relatioship with the partial ad coplete Bell polyoials If (f ()) is a sequece of bioial type the sequece (h ()) defied by h () : a+ f (a + ) is also of bioial type For sequeces (f ()) satisfyig the property of covolutio the followig is holds: f ( + y) 0 f () f (y) with f 0 () 1 Theore 4 Let (f ()) be a sequece of trioial type ad let a b be real ubers The the sequece (h ()) defied by is of trioial type h () : a + b + f (a + b + ) (26) Proof The defiitio of (f ()) give i (9) states that f ij (1) uj ti j! i! j0 which ca also be writte as follows: f () u t!! 0 0 u i (1) ti i! u () t! 0 with u () : f () u! 0 This iplies that the sequece (u ()) satisfies the property of covolutio Hece the sequece (U ()) give by U () : a + u (a + ) f (a + ) u a +! has the sae property of covolutio We have or siilarly v i (1) ui v () u i!! with v () : 0 0 U i (1) ti i! 0 0 U () t! a + f (a + ) t! This iplies that the sequece (v ()) satisfies the property of covolutio Moreover the sequece (V ()) give by V () : b + v (b + ) a + b + f (a + b + ) t! 0
8 INTEGERS: 11 (2011) 8 has the sae property of covolutio We have V j (1) uj or siilarly ij 0 u j h ij (1) ti 0 j0 h () t u!! j! with h 00 () 1 V () u! 0 Cosequetly the last idetity shows that the sequece (h ()) is of trioial type Rear Fro (9) we ca verify that if a sequece (f ()) is of trioial type the f ( + y) f ij () f i j (y) i j j0 If i (9) we set t 0 we deduce that the sequece (f ()) defied by f () : f 0 () is of bioial type Theore 5 Let (f ()) ad (g ()) be two sequeces of bioial type with f 0 () g 0 () 1 The the sequeces (p ()) ad (q ()) defied by are sequeces of trioial type p () : f + () q () : f () g () Proof We have p 00 () q 00 () 1 ad 0 p () t u!! 0 f () f () 0 f (1) 0 t u 0 (t + u) (t + u) 0 q () t u!! f () t g () u!! 0 0 f (1) t g (1) u!! 0 0
9 INTEGERS: 11 (2011) 9 Rear Let (f ()) ad (g ()) be two sequeces of bioial type with f 0 () g 0 () 1 The fro [3] the sequeces (F ()) ad (G ()) defied by ad F () : G () : (c a) + f ((c a) + ) (d b) + g ((d b) + ) are of bioial type for all real ubers a b c d This iplies accordig to Theore 4 that for all a b c d the sequeces ad P () : f + (a + b + ) a + b + Q () : (a + b + ) f (c + b + ) g (a + d + ) c + b + a + d + are of trioial type Eaple 6 We ow that sequeces ( )! ad (B ()) are of bioial type where B () is the sigle variable Bell polyoial The sequeces (a + ) 1! a + a + a + B (a + ) are also of bioial type Cobiig these sequeces Theore 5 states that the followig sequeces are also of trioial type: 2 (c + ) 1 (d + ) 1! 2 (c + ) 1 d + d + 2 c +!! (c + ) (d + ) d + 2 (c + ) 1 B (d + ) d + 2 c +! B (d + ) (c + ) (d + ) ( + )! + B + ()
10 INTEGERS: 11 (2011) 10 Cobiig these sequeces ad usig Theore 4 we deduce that the followig sequeces are of trioial type: a () (a + b + ) (c + b + ) 1 (a + d + ) 1 (a + b + ) (c + b + ) 1 a + d + b ()! a + d + (a + b + ) c + b + a + d + c ()!! (c + b + ) (a + d + ) (a + b + ) (c + b + ) 1 d () B (a + d + ) a + d + (a + b + ) c + b + e ()! B (a + d + ) (c + b + ) (a + d + ) ( + )! a + b + f () a + b + + g () a + b + B + (a + b + ) 4 Bipartitioal Polyoials ad Polyoials of Bioial ad Trioial Type Roa [7 p 82] proves that ay polyoial sequece of bioial type (f ()) with f 0 () 1 is coected with the partial Bell polyoials f () B (D α0 f j (α)) B (D α0 f j (α)) 1 (27) 1 Siilarly we establish that ay polyoial sequece of trioial type (f ()) with f 00 () 1 adits a coectio with the partial bipartitioal Bell polyoials Theore 7 Let (f ()) be a sequece of trioial type The + f () A (D α0 f ij (α)) A (D α0 f ij (α)) + 1 (28) 1 where D α0 d dα α0
11 INTEGERS: 11 (2011) 11 Proof Sice D α0 f 00 (α) 0 the for ij D α0 f ij (α) we have 0 t u + A ( ij )!! 0 A ( ij ) t u!! 0 + D α0 f ij (α) ti u j 0 ij 0 D α0 f ij (α) ti u j ij 0 α D α0 f ij (1) ti u j ij 0 l f ij (1) ti u j ij 0 f ij (1) ti u j ep ep ep ij 0 0 f () t u!! Therefore the desired idetity follows by idetificatio Soe idetities related to these polyoials are give by the followig theore Theore 8 Let (f ()) be a sequece of trioial type We have A ++ ij f i 1j 1 (a (i 1) + b (j 1) + ) a (i 1) + b (j 1) f (a + b + ) (29) a + b + Proof We have + A (ijf i 1j 1 ()) t t!!
12 INTEGERS: 11 (2011) 12 1 ijf i 1j 1 () ti u j t u t u f ij () ti u j ij 0 0 f () t u!! 1 ( + )! ( + )! t + u + f ()!! ( + )! ( + )! 0 f () t u!! which gives A (ijf i 1j 1 ()) f () (30) To fiish the proof we substitute the trioial type polyoials (h ()) give by (26) istead of (f ()) ad replace by + ad by + Eaple 9 Fro relatio (29) for r i 1 ad s j 1 we have 1 f () ( + )! + A ++ (r+1)(s+1)(r+s)! ar+bs+ ar+bs+ r+s ar+bs+ (+ 2 f () B + () A ++ (r + 1) (s + 1) Br+s(ar+bs+) ) ( 1)! (+)!(+)! a+b+ a+b+ + (+)!(+)! B +(a+b+) ( 1)!!! a+b+ 3 f () 2 ((c a) + ) 1 ((d b) + ) 1 A ++ (r + 1) (s + 1) (ar + bs + ) (cr + bs + ) r 1 (ar + ds + ) s 1 +!! 4 f () 2 ((c a)+)((d b)+) A ++ + (a+b+)(c+b+) 1 (a+d+) 1 (c a)+ (d b)+ (r+1)!(s+1)!(ar+bs+) (cr+bs+)(ar+ds+) (+)!(+)!(a+b+) ( 1)!(c+b+)(a+d+) cr+bs+ r c+b+ ar+ds+ s a+d+
13 INTEGERS: 11 (2011) 13 5 f ()! 2 ((c a)+) 1 (d b)+ (d b)+ A ++ (r+1)(s+1)!(ar+bs+)(cr+bs+) i 2 (ar+ds+) (+)!(+)!(a+b+)(c+b+) 1 ( 1)!!(a+d+) ar+ds+ s a+d+ 6 f () 2 ((c a)+) 1 (d b)+ B ((d b) + ) A (r+1)(s+1)(ar+bs+)(cr+bs+) r 1 ++ (ar+ds+) B s (ar + ds + ) + B (a + d + ) (a+b+) + (a+d+)(c+b+) 1 7 f ()! 2 ((c a)+)((d b)+) (c a)+ A ++ (s+1)(r+1)!(ar+bs+)bs(ar+ds+) (cr+bs+)(ar+ds+) (+)!(+)!(a+b+)b(a+d+) ( 1)!!(c+b+)(a+d+) B ((d b) + ) cr+bs+ r c+b+ Theore 10 Let (f ()) be a sequece of trioial type We have A + i f i 1j (a (i 1) + bj + ) a (i 1) + bj + + f (a + b + ) (31) a + b + A + j f ij 1 (ai + b (j 1) + ) ai + b (j 1) + + f (a + b + ) (32) a + b + Proof Siilarly to the proof i Theore 8 we obtai A (if i 1j ()) f () 0 A (jf ij 1 ()) f () 0 It will be sufficiet to use the trioial type polyoials (h ()) give i (26) istead of (f ()) ad replace by + ad by + respectively Eaple 11 Fro relatios (31) ad (32) we have the followig: 1 f () ( + )! A + i A + j + (i+j 1)! a(i 1)+bj+ (i+j 1)! ai+b(j 1)+ a(i 1)+bj+ i+j 1 ai+b(j 1)+ i+j 1 (+)! a+b+ (+)! a+b+ + + a+b+ + a+b+ +
14 INTEGERS: 11 (2011) 14 2 f () B + () A + A + i Bi+j 1(a(i 1)+bj+) a(i 1)+bj+ j Bi+j 1(ai+b(j 1)+) ai+b(j 1)+ + B+(a+b+) a+b+ + B+(a+b+) a+b+ 3 f () 2 ((c a) + ) 1 ((d b) + ) 1 A + i (a (i 1) + bj + ) (c (i 1) + bj + ) i 2 (a (i 1) + dj + ) j 2 + (a + b + ) (c + b + ) 1 (a + d + ) 1 A + j (ai + b (j 1) + ) (ci + b (j 1) + ) i 2 (ai + d (j 1) + ) j 2 + (a + b + ) (c + b + ) 1 (a + d + ) 1 4 f ()!! 2 ((c a)+)((d b)+) A i!(j 1)!(a(i 1)+bj+) + (+)!! ( 1)! A + (c(i 1)+bj+)(a(i 1)+dj+) a+b+ (c+b+)(a+d+) (i 1)!j!(ai+b(j 1)+) (ci+b(j 1)+)(ai+d(j 1)+)!(+)! ( 1)! a+b+ (c+b+)(a+d+) (c a)+ 5 f ()! 2 ((c a)+) 1 (d b)+ (d b)+ ij!(a(i 1)+bj+) A + c(i 1)+bj+ i 1 c+b+ ci+b(j 1)+ i c+b+ (d b)+ a(i 1)+dj+ j a+d+ ai+d(j 1)+ j 1 a+d+ a(i 1)+dj+ (a(i 1)+dj+)(c(i 1)+bj+) 2 i j (+)!!(a+b+)(c+b+) 1 a+d+ ( 1)!!(a+d+) A + i!j(ai+b(j 1)+) ai+d(j 1)+ (ai+d(j 1)+)(ci+b(j 1)+) 2 j j 1!(+)!(a+b+)(c+b+) 1 a+d+ ( 1)!!(a+d+) 6 f () 2 ((c a)+) 1 (d b)+ B ((d b) + ) A i(a(i 1)+bj+)(c(i 1)+bj+) i 2 + a(i 1)+dj+ B j (a (i 1) + dj + ) (a+b+)(c+b+) 1 + a+d+ B (a + d + ) A j(ai+b(j 1)+)(ci+b(j 1)+) i 1 + ai+d(j 1)+ B j 1 (ai + d (j 1) + ) + (a+b+)(c+b+) 1 a+d+ B (a + d + )
15 INTEGERS: 11 (2011) 15 7 f ()! 2 ((c a)+)((d b)+) A + i!(a(i 1)+bj+)Bj(a(i 1)+dj+) (c(i 1)+bj+)(a(i 1)+dj+) (+)!(a+b+) ( 1)!(c+b+)(a+d+) (c a)+ B ((d b) + ) c+b+ A + j!(ai+b(j 1)+)Bj 1(ai+d(j 1)+) (ci+b(j 1)+)(ai+d(j 1)+) (+)!(a+b+) ( 1)!(c+b+)(a+d+) c+b+ c(i 1)+bj+ i 1 B (a + d + ) ci+b(j 1)+ i B (a + d + ) Theore 12 Let (f ()) be a sequece of trioial type We have A ai + bj f ij (ai + bj) a + b + f (a + b + ) (33) Proof Fro (28) we get A (D α0 f ij (α)) f () which gives the desired idetity by replacig f () by give by (26) h () : a + b + f (a + b + ) Eaple 13 Fro relatio (33) we have the followig: 1 f () ( + )! + (i + j)! ai + bj ( + )! (a + b + ) A ai + bj i + j a + b f () B + () A ai + bj B i+j (ai + bj) a + b + B + (a + b + ) 3 f () 2 ((c a) + ) 1 ((d b) + ) 1 A (ai + bj) (ci + bj) i 1 (ai + dj) j 1 4 f ()!! 2 ((c a)+)((d b)+) (a + b + ) (c + b + ) 1 (a + d + ) 1 (c a)+ (d b)+ (ai + bj) ci + bj ai + dj A i!j! (ci + bj) (ai + dj) i j (a + b + ) c + b + a + d +!! (c + b + ) (a + d + )
16 INTEGERS: 11 (2011) 16 5 f ()! 2 ((c a)+) 1 (d b)+ A j! (ai + bj) (ci + bj)i 1 ai + dj! (d b)+ ai + dj j (a + b + ) (c + b + ) 1 a + d + 6 f () 2 ((c a)+) 1 (d b)+ B ((d b) + ) (ai + bj) (ci + bj) i 1 A B j (ai + dj) ai + dj a + d + (a + b + ) (c + b + ) 1 B (a + d + ) a + d + 7 f ()! 2 (c a)+ ((c a)+)((d b)+) B ((d b) + ) A i! (ai + bj) ci+bj i (ci + bj) (ai + dj) B j (ai + dj) (a + b + ) c+b+! (c + b + ) (a + d + ) B (a + d + ) Rear For 0 idetity (17) ad Theore 10 give Propositio 1 give i [3] ad idetity (20) ad Theore 12 give Propositio 3 give i [3] Acowledgets The authors wish to warly tha their referee ad the editor for their valuable advice ad coets which helped to greatly iprove the quality of this paper Refereces [1] Charalabos A Charalabides Euerative Cobiatorics Chapa & Hall/CRC A CRC Press Copay (2001) [2] L Cotet Advaced Cobiatorics D Reidel Publishig Copay Dordrecht-Hollad / Bosto-USA (1974) [3] M Mihoubi Bell polyoials ad bioial type sequeces Discrete Math 308 (2008)
17 INTEGERS: 11 (2011) 17 [4] M Mihoubi The role of bioial type sequeces i deteriatio idetities for Bell polyoials To appear Ars Cobi Preprit available at [5] M Mihoubi Soe cogrueces for the partial Bell polyoials J Iteger Seq 12 (2009) Article 0941 [6] J Riorda Cobiatorial Idetities Hutigto New Yor (1979) [7] S Roa The Ubral Calculus Acadeic Press (1984) [8] J Zeg Multioial Covolutio Polyoials Discrete Matheatics 160 (1996)
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