Partial Bell Polynomials and Inverse Relations

Transcription

1 Joural of Iteger Seueces, Vol. 13 (2010, Article Partial Bell Polyomials ad Iverse Relatios Miloud Mihoubi 1 USTHB Faculty of Mathematics P.B. 32 El Alia Algiers Algeria miloudmihoubi@hotmail.com Abstract Chou, Hsu ad Shiue gave some applicatios of Faà di Bruo s formula for the characterizatio of iverse relatios. I this paper, we use partial Bell polyomials ad biomial-type seuece of polyomials to develop complemetary iverse relatios. 1 Itroductio Recall that the (expoetial partial Bell polyomials B,k are defied by their geeratig fuctio B,k (x 1,x 2, t = 1 ( t m k x m (1 k! m! =k ad are give explicitly by the formula B,k (x 1,x 2,... = k 1!k 2! π(,k m=1 (x 1 1! k1 (x 2 k2, 2! where π(,k is the set of all oegative itegers (k 1,k 2,... such that k 1 + k 2 + k 3 + = k ad k 1 + 2k 2 + 3k 3 + =, 1 Research supported by LAID3 Laboratory of USTHB Uiversity. 1

2 Comtet [3] has studied the partial ad complete Bell polyomials ad has give their basic properties. Riorda [6] has show the applicatios of the Bell polyomials i combiatorial aalysis ad Roma [7] i umbral calculus. Chou, Hsu ad Shiue [2] have used these polyomials to characterize some iverse relatios. They have proved that, for ay fuctio F havig power formal series with compositioal iverse F 1, the followig iverse relatios hold Dj x=af(xb,j (x 1,x 2,..., x = Dj x=f(a F 1 (xb,j (y 1,y 2,... I this paper, we lik their results to those of Mihoubi [5, 6, 7] o partial Bell polyomials ad biomial-type seuece of polyomials. 2 Bell polyomials ad iverse relatios Usig the compositioal iverse fuctio with biomial-type seuece of polyomials, we determie some iverse relatios ad the coectios with the partial Bell polyomials. Theorem 1. Let {f (x} be a biomial-type seuece of polyomials with expoetial geeratig fuctio (f(t x. The the compositioal iverse fuctio of h(t = t(f(t x = 1 f 1 (x t is give by h 1 (t = 1 f 1 ( x t. Proof. To obtai the compositioal iverse fuctio of h it suffices to solve the euatio z = tf(z x. The Lagrage iversio formula esures that the last euatio has a uiue solutio defied aroud zero by z = h 1 (t = 1 Dz=0(f(z 1 x t = f 1 ( x t. 1 Corollary 2. Let {f (x} be a biomial-type seuece of polyomials ad let a be a real umber. The the compositioal iverse fuctio of h(t;a = 1 x a( 1 + x f 1(a( 1 + x t is give by h 1 (t;a = 1 x a( 1 x f 1(a( 1 x t. 2

3 Proof. This result follows by replacig {f (x} i Theorem 1, by the biomial-type seuece of polyomials {f (x;a}, where see Mihoubi [5, 6, 7]. f (x;a = x a + x f (a + x, (2 Theorem 3. Let {f (x} be a biomial-type seuece of polyomials ad a be a real umber. The the followig iverse relatios hold xj f a(j 1+x j 1(a(j 1 + xb,j (x 1,x 2,... x = xj f a(j 1 jx j 1(a(j 1 jxb,j (y 1,y 2,... (3 Proof. For ay fuctio F havig power formal series with compositioal iverse F 1, Chou, Hsu ad Shiue [2, Remark 1] have proved that To prove (3, it suffices to take ad the use Corollary 2. Dj x=af(xb,j (x 1,x 2,... x = Dj x=f(a F 1 (xb,j (y 1,y 2,... F(t := =1 x a( 1 + x f 1(a( 1 + x t Now, let f (x i Theorem 3 be oe of the ext biomial-type seuece of polyomials f (x = x, f (x = (x ( := x(x 1 (x + 1, 1, with (x (o = 1, f (x = (x ( ( := x(x + 1 (x + 1, 1, with (x (o = 1, x f (x = := j=o B,j( ( 1 1,...,i!( 1,...(x i (j, f (x = B (x := j=o S(,kxk, ( k where B (., S(,k ad are, respectively, the sigle variable Bell polyomials, the Stirlig umbers of secod kid ad the coefficiets defied by (1 + x + x x k = 0 ( k x, see Belbachir, Bouroubi ad Khelladi [4]. We deduce the followig results: 3

4 Corollary 4. Let a ad x be real umbers. The the followig iverse relatios hold: For f (x = x, we get For f (x = (x (, we get For f (x = (x (, we get ( x For f (x = xj(a(j 1 + xj 2 B,j (x 1,x 2,..., x = xj(a(j 1 jxj 2 B,j (y 1,y 2,... xj (a(j 1 + x a(j 1+x (j 1B,j (x 1,x 2,..., x = xj (a(j 1 jx a(j 1 jx (j 1B,j (y 1,y 2,... xj (a(j 1 + a(j 1+x x(j 1 B,j (x 1,x 2,..., x = xj (a(j 1 a(j 1 jx jx(j 1 B,j (y 1,y 2,..., we get For f (x = B (x, we get Example 5. x = x(j 1! a(j 1+x xj! a(j 1 jx ( a(j 1+x j 1 ( a(j 1 jx j 1 B,j(x 1,x 2,..., B,j(y 1,y 2,... xj B a(j 1+x j 1(a(j 1 + xb,j (x 1,x 2,..., x = xj B a(j 1 jx j 1(a(j 1 jxb,j (y 1,y 2,... For a = 0, x = 1 i (5, we obtai j=0 ( j xj x j with x 0 = 1 2, x = 1 j=0 ( 1j (2j! (j! B,j+1(y 1,y 2,... (4 (5 (6 (7 (8 (9 For x = 2 ( 2/2, x = 1 2 or x = i (9, we get 1 j=0 (2j! (j! ( 4 j S(,j + 1 = ( For x 1 = 1, x = 0, 2, i (9, we get +1 ( ( + 1 2j ( 1 j = 0, 0. j j= [/2] 4

5 Take x = a = x 1 = 1, x 2 = 2 ad x = 0, 3, i (5 ad from the idetity of Ceralosi [1] j!b,j (1!, 2!, 0, 0,... = F, 1, we obtai ( 1 j j!b,j (1!F 1, 2!F 2,... = 0, 3, where F, = 0, 1, 2,, are the Fiboacci umbers. Take x = 1, a = x 1 = 0, x 2 = 2 ad x =, 3, i (5, from the idetity of Ceralosi [1] j!b,j (0, 2!, 3!,... = F 2, 2, we obtai ( 1 j 1 j!b,j (0, 2!F 0, 3!F 1,... =, 2. Theorem 6. Let r,s be oegative itegers, rs 0, ad let {u } be a seuece of real umbers with u 1 = 1. The y = s ( j Uj +j 1 1BUj U j U j +j 1,U j (1,u 2,u 3,...B,j (x 1,x 2,..., x 1 = y 1 ad for 2 we have (10 x = y s ( j Vj +j 1 1BVj j=2 V j V j +j 1,V j (1,u 2,u 3,...B,j (y 1,y 2,..., where U j = (r + 2s(j 1 + s ad V j = (r + s(j 1 s. Proof. Let,r,s be oegative itegers, r(r + s 1, z (r := B (r+1,r(1,u 2,u 3,..., ad cosider the biomial-type seuece of polyomials {f (x} defied by f (x := B,j (z 1 (r,z 2 (r,...x j with f o (x = 1, r( (r+1 r see Roma [8]. The from the idetity B,j(z 1 (r,z 2 (r,...s j = s ( 1 (r s B (r+1+s,r+s(1,u 2,u 3,..., r + s r + s see Mihoubi [5, 6, 7], we get f (s = s ( 1 (r s B (r+1+s,r+s(1,u 2,u 3,... (11 r + s r + s To obtai (10, we set a = 0, x = s i (3 ad use the expressio of f (s give by (11, with r + 2s istead of r. 5

6 Example 7. From the well-kow idetity B,k (1!, 2!,...,i!,... = ( 1 k 1 k!, we get y = s j!((r+2s+1(j 1+s 1! ((r+2s(j 1+s! B,j (x 1,x 2,..., x 1 = y 1 ad for 2 we have x = y s j=2 j!((r+s+1(j 1+s 1! ((r+s(j 1+s! B,j (y 1,y 2,... Similar relatios ca be obtaied for the Stirlig umbers of the first kid, the usiged Stirlig umbers of the first kid ad the Stirlig umbers of the secod kid by settig u = ( 1 1 ( 1!, u = ( 1! ad u = 1 for all 1, respectively. Corollary 8. Let u,r,s be oegative itegers, a,α be real umbers, αurs 0, ad {f (x} be a biomial-type seuece of polyomials. The y = s j D T j α T j u(j 1 T j z=0(e αz f j 1 (T j x + z;ab,j (x 1,x 2,..., x 1 = y 1 ad for 2 we have x = y s j j=2 D R j α R j u(j 1 R j z=0(e αz f j 1 (R j x + z;ab,j (y 1,y 2,..., where T j = (u + r + 2s(j 1 + s ad R j = (u + r + s(j 1 s. Proof. Set i Theorem 6 u = (u( 1 + 1α Du( 1+1 z=0 (e αz f 1 ((u( 1 + 1x + z;a ad use the first idetity of Mihoubi [6, Theorem 2]. Corollary 9. Let u,r,s be oegative itegers, urs 0, a be real umber ad let {f (x} be a biomial-type seuece of polyomials. The y = s j! D T j α T j u(j 1 (T j +j 1!T j z=0f Tj +j 1(T j x + z;ab,j (x 1,x 2,..., x 1 = y 1 ad for 2 we have x = y s j! j=2 D R j α R j u(j 1 (R j +j 1!R j z=0f Rj +j 1(R j x + z;ab,j (y 1,y 2,..., where T j = (u + r + 2s(j 1 + s ad R j = (u + r + s(j 1 s. Proof. Set i Theorem 6 u = Du( 1+1 z=0 f ((u+1( 1+1 ((u( 1 + 1x + z;a ((u + 1( 1 + 1!(u( 1 + 1α ad use the secod idetity of Mihoubi [6, Theorem 2]. 6

7 Theorem 10. Let d be a iteger 1. The iverse relatios hold y = ( 1 j (d + j (j 1 B,j (x 1,x 2,..., x = ( 1 j (d + j (j 1 B,j (y 1,y 2,... (12 Proof. Let f(t = t ( x t d ad f 1 (t = t ( 1 + t d. 1y The proof ow follows from Comtet [3, Theorem F, p. 151]. Example 11. Take d = 1 ad x =, 1, i Theorem 10, we get f(t = t 1 t ad f 1 (t = t 1 + t, i.e. y = ( 1, ad the relatios (12 give ( ( + j ( 1 j =. j + 1 Take d = 2 ad x =, 1, we get f(t = t ad f 1 (t = 1 ( t 2, 1 t 2 2t i.e. y = ( 1 (2!, 1, ad the relatios (12 give ( + 1! ( ( 2 + j ( 1 j = ( 2 = C, j where C, = 0, 1, 2,, are the Catala umbers. Theorem 12. The followig iverse relatios hold ( y = 1 (r+1 1B(r+1,r (1,x r r 1,x 2,..., r 1, x = ( + 1 B, j(y 1,y 2,...( 1 j 1 (r 1 j 1. Proof. From Mihoubi [7, Theorem 1] we have x k 1 B,j (y 1,y 2,...(k r j 1 = xr 1 k ( with y = 1 (r+1 1B(r+1,r (1,x r r 1,x 2,..., rk 1. It just suffices to set k = 1, x 1 = 1, ad replace x by x 1. 7 ( 1 + k B +k,k(x 1,x 2,x 3,..., k

8 3 Ackowledgmets The author thaks the aoymous referee for his/her careful readig ad valuable suggestios. He also thaks Professor Beaissa Larbi for his Eglish correctios. Refereces [1] M. Cerasoli, Two idetities betwee Bell polyomials ad Fiboacci umbers. Boll. U. Mat. Ital. A, (5 18 (1981, [2] W. S. Chou, L. C. Hsu, P. J. S. Shiue, Applicatio of Faà di Bruo s formula i characterizatio of iverse relatios. J. Comput. Appl. Math 190 (2006, [3] L. Comtet, Advaced Combiatorics. D. Reidel Publishig Compay, [4] H. Belbachir, S. Bouroubi, A. Khelladi, Coectio betwee ordiary multiomials, geeralized Fiboacci umbers, partial Bell partitio polyomials ad covolutio powers of discrete uiform distributio. A. Math. Iform. 35 (2008, [5] M. Mihoubi, Bell polyomials ad biomial type seueces. Discrete Math. 308 (2008, [6] M. Mihoubi, The role of biomial type seueces i determiatio idetities for Bell polyomials. To appear, Ars Combi. Preprit available at [7] M. Mihoubi, Some cogrueces for the partial Bell polyomials. J. Iteger Se. 12 (2009, Article [8] S. Roma, The Umbral Calculus, Mathematics Subject Classificatio: Primary 05A10, 05A99; Secodary 11B73, 11B75. Keywords: iverse relatios, partial Bell polyomials, biomial-type seuece of polyomials. Received December ; revised versio received April Published i Joural of Iteger Seueces, April Retur to Joural of Iteger Seueces home page. 8