On central complete Bell polynomials
aa r X i v : . [ m a t h . N T ] N ov ON CENTRAL COMPLETE BELL POLYNOMIALS
TAEKYUN KIM, DAE SAN KIM, AND GWAN-WOO JANG
Abstract.
In this paper, we consider central complete and incomplete Bellpolynomials which are generalizations of the recently introduced centralBell polynomials and ’central’ analogues for the complete and incompleteBell polynomials. We investigate some properties and identities for thesepolynomials. Especially, we give explicit formulas for the central completeand incomplete Bell polynomials related to central factorial numbers of thesecond kind.
1. Introduction
The Stirling numbers of the second kind are given by1 k ! ( e t − k = ∞ X n = k S ( n, k ) t n n ! , (see [4 , , , , , . (1.1)It is well known that the Bell polynomials (also called Tochard polynomialsor exponential polynomials) are defined by e x ( e t − = ∞ X n =0 B n ( x ) t n n ! , (see [1 , , , , , . (1.2)From (1.1) and (1.2), we note that B n ( x ) = e − x ∞ X k =0 k n k ! x k = n X k =0 x k S ( n, k ) , ( n ≥ , (see [2 , , . (1.3)When x = 1, B n = B n (1) are called Bell numbers. Mathematics Subject Classification.
Key words and phrases. central incomplete Bell polynomials, central complete Bell poly-nomials, central complete Bell numbers.
The (exponential) incomplete Bell polynomials (also called (exponential) par-tial Bell polynomials) are defined by the generating function1 k ! (cid:16) ∞ X m =1 x m t m m ! (cid:17) k = ∞ X n = k B n,k ( x , · · · , x n − k +1 ) t n n ! , ( k ≥ , (see [12 , . (1.4)Thus, by (1.4), we get B n,k ( x , · · · , x n − k +1 ) = X n ! i ! i ! · · · i n − k +1 ! (cid:16) x (cid:17) i (cid:16) x (cid:17) i × · · ·× (cid:16) x n − k +1 ( n − k + 1)! (cid:17) i n − k +1 , (1.5)where the summation is over all integers i , · · · , i n − k +1 ≥ i + i + · · · + i n − k +1 = k and i + 2 i + · · · + ( n − k + 1) i n − k +1 = n .From (1.1) and (1.4), we note that B n,k (1 , , · · · , | {z } n − k +1 − times = S ( n, k ) , ( n, k ≥ . (1.6)By (1.5), we easily get B n,k ( αx , αx , · · · , αx n − k +1 ) = α k B n,k ( x , x , · · · , x n − k +1 ) (1.7)and B n,k ( αx , α x , · · · , α n − k +1 x n − k +1 ) = α n B n,k ( x , x , · · · , x n − k +1 ) , (1.8)where α ∈ R (see [12 , ∞ X n = k B n,k ( x, , , , · · · , t n n ! = 1 k ! (cid:0) xt + t (cid:1) k = t k k ! k X n =0 (cid:18) kn (cid:19)(cid:16) t (cid:17) n x k − n = k X n =0 ( n + k )! k ! (cid:18) kn (cid:19) n x k − n t n + k ( n + k )! , (1.9)and ∞ X n = k B n,k ( x, , , , · · · , t n n ! = ∞ X n =0 B n + k,k ( x, , , · · · , t n + k ( n + k )! . (1.10) . Kim, D. S. Kim and G.-W. Jang 3 By comparing the coefficients on both sides of (1.9) and (1.10), we get B n + k,k ( x, , , · · · ,
0) = ( n + k )! k ! (cid:18) kn (cid:19) n x k − n , (0 ≤ n ≤ k ) . (1.11)By replacing n by n − k in (1.11), we get B n,k ( x, , , · · · ,
0) = n ! k ! (cid:18) kn − k (cid:19) x k − n (cid:16) (cid:17) n − k , ( k ≤ n ≤ k ) . (1.12)The (exponential) complete Bell polynomials are defined byexp (cid:16) ∞ X i =1 x i t i i ! (cid:17) = ∞ X n =0 B n ( x , x , · · · , x n ) t n n ! . (1.13)Then, by (1.4) and (1.13), we get B n ( x , x , · · · , x n ) = n X k =0 B n,k ( x , x , · · · , x n − k +1 ) (1.14)From (1.3), (1.6), (1.7) and (1.14), we have B n ( x, x, · · · , x ) = n X k =0 x k B n,k (1 , , · · · , n X k =0 x k S ( n, k ) = B n ( x ) , ( n ≥ . (1.15)It is known that the central factorial numbers of the second kind are given by1 k ! (cid:0) e t − e − t (cid:1) k = ∞ X n = k T ( n, k ) t n n ! , (see [3 , , , (1.16)where k ≥ T ( n, k ) = 1 k ! k X j =0 (cid:18) kj (cid:19) ( − k − j (cid:0) j − k (cid:1) n , (1.17)where n, k ∈ Z with n ≥ k ≥
0, (see [8 , , B ( c ) n ( x ) are defined by B ( c ) n ( x ) = n X k =0 T ( n, k ) x k , ( n ≥ . (1.18)When x = 1, B ( c ) n = B ( c ) n (1) are called the central Bell numbers. On central complete Bell polynomials
From (1.18), we can derive the generating function for the central Bell poly-nomials as follows: e x (cid:0) e t − e − t (cid:1) = ∞ X n =0 B ( c ) n ( x ) t n n ! , (see [9]) . (1.19)Thus, by (1.19), we have the following Dobinski-like formula B ( c ) n ( x ) = ∞ X l =0 ∞ X j =0 (cid:18) l + jj (cid:19) ( − j l + j )! (cid:16) l − j (cid:17) n x l + j , (1.20)where n ≥
2. On central complete and incomplete Bell polynomials
In view of (1.13), we consider the central incomplete Bell polynomials whichare given by1 k ! (cid:16) ∞ X m =1 m ( x m − ( − m x m ) t m m ! (cid:17) k = ∞ X n = k T n,k ( x , x , · · · , x n − k +1 ) t n n ! , (2.1)where k = 0 , , , , · · · .For n, k ≥ n − k ≡ T n,k ( x , x , · · · , x n − k +1 ) = X n ! i ! i ! · · · i n − k +1 ! (cid:16) x (cid:17) i (cid:16) · (cid:17) i × ( x · (cid:17) i · · · (cid:16) x n − k +1 n − k ( n − k + 1)! (cid:17) i n − k +1 , (2.2)where the summation is over all integers i , i , · · · , i n − k +1 ≥ i + · · · + i n − k +1 = k and i + 2 i + · · · + ( n − k + 1) i n − k +1 = n .From (1.5) and (2.2), we note that T n,k ( x , x , · · · , x n − k +1 ) = B n,k (cid:0) x , , x , , · · · , x n − k +1 n − k (cid:1) , (2.3)where n, k ≥ n − k ≡ n ≥ k .Therefore, we obtain the following lemma. . Kim, D. S. Kim and G.-W. Jang 5 Lemma 2.1.
For n, k ≥ with n ≥ k and n − k ≡ (mod ), we have T n,k ( x , x , · · · , x n − k +1 ) = B n,k (cid:0) x , , x , , · · · , x n − k +1 n − k (cid:1) . For n, k ≥ n ≥ k and n − k ≡ ∞ X n = k T n,k ( x, x , x , · · · , x n − k +1 ) t n n ! = 1 k ! (cid:16) xt + x t
3! + x t
5! + · · · (cid:17) k = 1 k ! (cid:16) e x t − e − x t (cid:17) k = 1 k ! e − kx t (cid:16) e xt − (cid:17) k = 1 k ! k X l =0 (cid:18) kl (cid:19) ( − k − l e ( l − k ) xt = 1 k ! k X l =0 (cid:18) kl (cid:19) ( − k − l ∞ X n =0 (cid:0) l − k (cid:1) n x n t n n != ∞ X n =0 (cid:16) x n k ! k X l =0 (cid:18) kl (cid:19) ( − k − l (cid:0) l − k (cid:1) n (cid:17) t n n ! . (2.4)Therefore, by comparing the coefficients on both sides of (2.4), we obtain thefollowing theorem. Theorem 2.2.
For n, k ≥ with n − k ≡ (mod ), we have x n k ! k X l =0 (cid:18) kl (cid:19) ( − k − l (cid:0) l − k (cid:1) n = (cid:26) T n,k ( x, x , · · · , x n − k +1 ) , if n ≥ k, , if n < k. (2.5) In particular k ! k X l =0 (cid:18) kl (cid:19) ( − k − l (cid:0) l − k (cid:1) n = (cid:26) T n,k (1 , , · · · , , if n ≥ k, , if n < k. (2.6)For n, k ≥ n − k ≡ n ≥ k , by (1.17) and (2.6), we get T n,k (1 , , · · · ,
1) = T ( n, k ) . (2.7)Therefore, by (2.5) and (2.6), we obtain the following corollary Corollary 2.3.
For n, k ≥ with n − k ≡ (mod ), n ≥ k , we have T n,k ( x, x , · · · , x n − k +1 ) = x n T n,k (1 , , · · · , On central complete Bell polynomials and T n,k (1 , , · · · ,
1) = T ( n, k ) = B n,k (cid:0) , , , · · · , n − k (cid:1) = X n ! i ! i ! · · · i n − k +1 ! (cid:16) (cid:17) i (cid:16) (cid:17) i · · · (cid:16) n − k ( n − k + 1)! (cid:17) i n − k +1 , where i + i + · · · + i n − k +1 = k and i + 3 i + · · · + ( n − k + 1) i n − k +1 = n . For n, k ≥ n ≥ k and n − k ≡ ∞ X n = k T n,k ( x, , , , · · · , t n n ! = 1 k ! ( xt ) k . (2.8)Thus we have T n,k ( x, , , · · · ,
0) = x k (cid:18) n − k (cid:19) . For n, k ≥ n − k ≡ n ≥ k , by (2.2), we get T n,k ( x , x , · · · , x n − k +1 ) = X n ! i ! i ! · · · i n − k +1 ! (cid:16) x (cid:17) i ( x · (cid:17) i × · · · × (cid:16) x n − k +1 n − k ( n − k + 1)! (cid:17) i n − k +1 , (2.9)where the summation is over all integers i , i , · · · , i n − k +1 ≥ i + i + · · · + i n − k +1 = k and i + 3 i + · · · + ( n − k + 1) i n − k +1 = n .By (2.9), we easily get T n,k ( x, x, · · · , x ) = x k T n,k (1 , , · · · ,
1) (2.10)and T n,k ( αx , αx , · · · , αx n − k +1 ) = α k T n,k ( x , x , · · · , x n − k +1 ) , where n, k ≥ n − k ≡ n ≥ k . . Kim, D. S. Kim and G.-W. Jang 7 Now, we observe thatexp (cid:16) x ∞ X i =1 (cid:0) (cid:1) i ( x i − ( − i x i ) t i i ! (cid:17) = ∞ X k =0 x k k ! (cid:16) ∞ X i =1 (cid:0) (cid:1) i ( x i − ( − i x i ) t i i ! (cid:17) k = 1 + ∞ X k =1 x k k ! (cid:16) ∞ X i =1 (cid:0) (cid:1) i ( x i − ( − i x i ) t i i ! (cid:17) k = 1 + ∞ X k =1 x k ∞ X n = k T n,k ( x , x , · · · , x n − k +1 ) t n n != 1 + ∞ X n =1 (cid:16) n X k =1 x k T n,k ( x , x , · · · , x n − k +1 ) (cid:17) t n n ! . (2.11)In view of (1.13), we define the central complete Bell polynomials byexp (cid:16) x ∞ X i =1 (cid:0) (cid:1) i ( x i − ( − i x i ) t i i ! (cid:17) = ∞ X n =0 B ( c ) n ( x | x , x , · · · , x n ) t n n ! . (2.12)Thus, by (2.11) and (2.12), we get B ( c ) n ( x | x , x , · · · , x n ) = n X k =0 x k T n,k ( x , x , · · · , x n − k +1 ) . (2.13)When x = 1, B ( c ) n (1 | x , x , · · · , x n ) = B ( c ) n ( x , x , · · · , x n ) are called the centralcomplete Bell numbers .For n ≥
0, we have B ( c ) n ( x , x , · · · , x n ) = n X k =0 T n,k ( x , x , · · · , x n − k +1 ) (2.14)and B ( c )0 ( x , x , · · · , x n ) = 1 . By (1.18) and (2.13), we get B ( c ) n (1 , , · · · ,
1) = n X k =0 T n,k (1 , , · · · ,
1) = n X k =0 T ( n, k ) = B ( c ) n , (2.15)and B ( c ) n ( x | , , · · · ,
1) = n X k =0 x k T n,k (1 , , · · · ,
1) = n X k =0 x k T ( n, k ) = B ( c ) n ( x ) . (2.16) On central complete Bell polynomials
From (2.11), we note thatexp (cid:16) ∞ X i =1 (cid:0) (cid:1) i ( x i − ( − i x i ) t i i ! (cid:17) = 1 + ∞ X n =1 n ! (cid:16) ∞ X i =1 (cid:0) (cid:1) i ( x i − ( − i x i ) t i i ! (cid:17) n = 1 + 11! ∞ X i =1 (cid:0) (cid:1) i ( x i − ( − i x i ) t i i ! + 12! (cid:16) ∞ X i =1 (cid:0) (cid:1) i ( x i − ( − i × x i ) t i i ! (cid:17) + 13! (cid:16) ∞ X i =1 (cid:0) (cid:1) i ( x i − ( − i x i ) t i i ! (cid:17) + · · · = 1 + 11! x t + 12! x t + (cid:16) x + x (cid:17) t + · · · = ∞ X n =0 (cid:16) X m +2 m + ··· + nm n = n n ! m ! m ! · · · m n ! (cid:16) x (cid:17) m (cid:16) (cid:17) m × (cid:16) x (cid:17) m · · · (cid:16) x n (cid:0) − ( − n (cid:1) n !2 n (cid:17) m n (cid:17) t n n ! . (2.17)Now, for n ∈ N with n ≡ B ( c ) n ( x , x , · · · , x n ) = n X k =0 T n,k ( x , x , · · · , x n − k +1 )= X m +3 m + ··· + nm n = n n ! m ! m ! · · · m n ! (cid:16) x (cid:17) m (cid:16) x (cid:17) m · · · (cid:16) x n n !2 n − (cid:17) m n . (2.18)Therefore, by (2.18), we obtain the following theorem Theorem 2.4.
For n ∈ N with n ≡ (mod ), we have B ( c ) n ( x , x , · · · , x n ) = = X m +3 m + ··· + nm n = n n ! m ! m ! · · · m n ! (cid:16) x (cid:17) m × (cid:16) x (cid:17) m · · · (cid:16) x n n !2 n − (cid:17) m n . . Kim, D. S. Kim and G.-W. Jang 9 We note thatexp (cid:16) x ∞ X i =1 (cid:0) (cid:1) i (cid:0) − ( − i (cid:1) t i i ! (cid:17) = 1 + ∞ X k =1 x k k ! (cid:16) ∞ X n = k (cid:0) (cid:1) i (cid:0) − ( − i (cid:1) t i i ! (cid:17) k = 1 + ∞ X k =1 x k ∞ X n = k T n,k (1 , , · · · , t n n != 1 + ∞ X n =1 (cid:16) n X k =1 x k T n,k (1 , , · · · , (cid:17) t n n ! . (2.19)On the other hand, from (1.19) we haveexp (cid:16) x ∞ X i =1 (cid:0) (cid:1) i (cid:0) − ( − i (cid:1) t i i ! (cid:17) = exp (cid:16) x (cid:0) t + 12 t + 12 t + · · · (cid:1)(cid:17) = exp (cid:16) x (cid:0) e t − e − t (cid:1)(cid:17) = ∞ X n =0 B ( c ) n ( x ) t n n ! . (2.20)Therefore, by (2.19) and (2.20), we obtain the following theorem. Theorem 2.5.
For n, k ≥ with n ≥ k , we have n X k =0 x k T n,k (1 , , · · · ,
1) = B ( c ) n ( x ) . From Theorem 2.5, we note that n X k =0 x k T n,k (1 , , · · · ,
1) = n X k =0 T n,k ( x, x, · · · , x ) = B ( c ) n ( x, x, · · · , x ) . (2.21)Therefore, by Theorem 2.5 and (2.21), we obtain the following corollary. Corollary 2.6.
For n ≥ , we have B ( c ) n ( x, x, · · · , x ) = B ( c ) n ( x ) . It is known that the Stirling numbers of the first kind are given by the gen-erating function1 k ! (cid:0) log(1 + t ) (cid:1) k = ∞ X n = k S ( n, k ) t n n ! , ( k ≥ , (see [4 , . (2.22) By (2.22), we easily get1 k ! log (cid:16) x − x (cid:17) = ∞ X l = k S ( l, k ) 1 l ! (cid:16) x − x (cid:17) l = ∞ X l = k S ( l, k ) x l l ! (cid:0) − x (cid:1) − l = ∞ X l = k l ! S ( l, k ) ∞ X n = l (cid:18) n − l − (cid:19)(cid:0) (cid:1) n − l x n = ∞ X n = k (cid:16) n X l = k l ! S ( l, k ) (cid:18) n − l − (cid:19)(cid:0) (cid:1) n − l (cid:17) x n . (2.23)From (2.1) and (2.23), we can derive the following equation. ∞ X n = k T n,k (0! , , , · · · , ( n − k )! (cid:17) t n n != 1 k ! (cid:16) t + (cid:0) (cid:1) t (cid:0) (cid:1) t (cid:0) (cid:1) t · · · (cid:17) k = 1 k ! (cid:16) log (cid:0) t (cid:1) − log (cid:0) − t (cid:1)(cid:17) k = 1 k ! (cid:18) log (cid:16) t − t (cid:17)(cid:19) k = 1 k ! (cid:16) log (cid:0) t − t (cid:1)(cid:17) k = ∞ X n = k (cid:16) n X l = k S ( l, k ) l ! (cid:18) n − l − (cid:19)(cid:0) (cid:1) n − l (cid:17) t n . (2.24)By comparing the coefficients on both sides of (2.24), we obtain the followingtheorem. Theorem 2.7.
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Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic ofKorea
E-mail address : [email protected] Department of Mathematics, Sogang University, Seoul 121-742, Republic of Ko-rea
E-mail address : [email protected] Department of Mathematics, Kwnagwoon University, Seoul 139-701, Republic ofKorea
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