New identities for Bell s polynomials New approaches

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1 Rostoc. Math. Kolloq. 6, ) Subject Classification AMS) 05A6 Sade Bouroubi, Moncef Abbas New identities for Bell s polynomials New approaches ABSTRACT. In this wor we suggest a new approach to the determination of new identities for Bell s polynomials, based on the Lagrange inversion formula, and the binomial sequences. This approach allows the easy recovery of nown identities and deduction of some new identities including these polynomials. KEY WORDS. Bell s polynomials, Bell s numbers, Lagrange inversion formula, binomial sequences. Introduction Using a proof by recurrence, Salim KHELIFA and Yves CHERRUAULT gave the following identity on Bell s polynomials [3]. Theorem For n and N, n, it holds B n, 0, 2, 3 2,... ) ) n n n ) This identity allowed the authors to demonstrate a new theorem of convergence for the Adomian decomposition method [4], but the proof is excessively long 7 pages) and consequently it requires another proof. Here we propose two new identities, with shorter proofs. The first uses the Lagrange inversion formula, having as an immediate consequence the identity ), and the second uses binomial sequences, begetting new identities, and the nown identities in the literature. Definition 2 The Bell polynomials are the polynomials B n, x, x 2,...) in an infinite number of variables x, x 2,..., defined by see [2], p. 33) ) t m x m t n B n,, 0,, 2,... 2)! m! m n

2 50 S. Bouroubi, M. Abbas Their exact expression is see [2], p. 34) B n, x, x 2,...) πn)! 2!!) 2!) 2 x x , where π n) denotes a partition of n, with n; i is of course, the number of parts of size i. Also is the number of parts in the partition. 2 Main results 2. Method based on the Lagrange inversion formula Let f be an analytic function about the origin such that f 0) 0 and for n and m N let where D is the differential operator Theorem 3 D n [f w)] m if n f n m) f 0)) m if n 0 d d w. For n and N, n, it holds B n, f 0 ), f 2), f 2 3),...) ) n f n n). Proof: For z C, let us consider the equation of the unnown w C, w zf w) 0. This equation admits a unique solution w g z) around the origin see [], p. 234) and for any analytic function F around the origin we have by Lagrange inversion formula F g z)) F 0) + n If we choose F w) w, we get from 3) g z) n n D n F w) [f w)] n} D n [f w)] n f n n) zn 3)

3 New identities for Bell s polynomials New approaches 5 Thus from 2) we have )! g z)) f n n) zn! n n B n, f 0 ), f 2), f 2 3),...) zn On the other hand, if we choose F w) w!, we get by 3)! g z)) )! )! D n w [f w)] n} n D n w } D j [f w)] n w j j! n j 0 D n )! n j 0 )! D n n j n )! )! n )! f n n) zn n n )f n n) zn n } f j n) w j+ j! } f j + n) j + )! wj Corollary 4 Let a R. We have for all n and N, n, B n, a) 0, 2a), 3a) 2,... ) ) n an) n Proof: We have just to apply Theorem 3 by putting f w) e aw, that gives am) n if n f n m) if n 0 Remar 5 ) If we choose a we find the identity ).

4 52 S. Bouroubi, M. Abbas 2) It is obvious that the identity of Corollary 4 is not the only consequence of Theorem 3, because it depends on the choice of f. If, for instance, we choose the function f w) + aw, we get f n m) a n [m] n if n if n 0 where [m] n m m ) m n + ). Thus we have B n,!a 0, 2!a, 3!a 2,... ) ) n a n! If we choose : a, we recover the nown identity a 0, we get B n,!, 2!, 3!,...) ) n! B n,, 0, 0,...) 0, except B n,n. 2.2 Method based on binomial sequences A sequence of definite functions ϕ n x)) n on a subset I of R is called binomial if, n ) n ϕ n x + y) ϕ x) ϕ n y), x, y I. 0 Theorem 6 Let ϕ n x)) n be a binomial sequence defined on I, N I R, with ϕ 0 0. Then for all n and N, n, we have Proof: ϕ n x)) n, i.e. B n, ϕ 0 ), 2ϕ ), 3ϕ 2 ),...) ) n ϕ n ). Let by Φ x denote the exponential generating function associated to the sequence Φ x t) n 0 ϕ n x) tn We suppose, of course, that the radius of convergence satisfies R > 0.) The sequence ϕ n x)) n is binomial, then we have, from Cauchy product Φ x+y t) Φ x t) Φ y t), x, y I. Hence Φ t) Φ t)), N. 4)

5 New identities for Bell s polynomials New approaches 53 It comes then, on the one hand! tφ t))! n 0! n On the other hand by 4), we have n ) ϕ n ) tn+ ) nϕ n ) tn B n, ϕ 0 ), 2ϕ ), 3ϕ 2 ),...) tn! tφ t))! t Φ t) ϕ n ) tn+! n 0 n )ϕ n ) tn n Application Let S n, ) denote the Stirling number of the second ind, and put B n x) n S n, ) x. 0 The sequence B n x)) n is defined in R, where B 0 x) and B n ) B n, the Bell numbers. Corollary 7 We have ) n n B n, B 0, 2B, 3B 2,...) S n, j) j Proof: It is well nown and easily verified that In fact, from 5) it follows that n0 + n0 j0 B n x) zn exp x ez )) 5) + B n x + y) zn exp x + y) ez )) B n x) zn n0 + n0 B n y) zn

6 54 S. Bouroubi, M. Abbas Therefore B n x + y) n 0 ) n B x) B n y) Thus the sequence B n x)) n is binomial and the result is proved by means of Theorem 6. Remar 8 Corollary 7 is not the only consequence of Theorem 6. It all depends on the choice of binomial sequence. If we choose for example the binomial sequence defined on R by ϕ n x) x n, we recover the nown identity B n,, 2, 3,...) ) n n. References [] Evgrafov, M., and Coll : Recueil de problèmes sur la théorie des fonctions analytiques. Traduction française, Editions Mir. 974 [2] Comtet, L. : Advanced combinatorics. D. Reidel, 974 [3] Khelifa, S., and Cherruault, Y. : Nouvelle identité pour les polynômes de Bell. Maghreb Mathematical Review. Vol.9, n and 2, June and December 2000, 5-23 [4] Khelifa, S., and Cherruault, Y. : New results on Adomian method. Kybernetes. Vol. 29; n 3, 2000) [5] Khelifa, S. : Equations aux dérivées partielles et méthode décompositionnelle d Adomian. Thèse de Doctorat Es Sciences en Mathématiques, U.S.T.H.B., Alger 2002

7 New identities for Bell s polynomials New approaches 55 received: March 26, 2003 Authors: Sade Bouroubi U.S.T.H.B. Faculty of Mathematics Department of Operational Research B.P El-Alia Bab-Ezzouar Algiers Algeria Moncef Abbas U.S.T.H.B. Faculty of Mathematics Department of Operational Research B.P El-Alia Bab-Ezzouar Algiers Algeria bouroubis@yahoo.fr