Some identities arising from Sheffer sequences for the powers of Sheffer pairs under umbral composition
aa r X i v : . [ m a t h . N T ] M a r Some identities arising from Sheffer sequencesfor the powers of Sheffer pairs under umbralcomposition. byDae San Kim and Taekyun Kim
Abstract
In this paper, we study some properties of Sheffer sequences for thepowers of Sheffer pairs under umbral composition. From our propertieswe derive new and interesting identities of Sheffer sequences of specialpolynomials for the powers of Sheffer pairs under umbral composition.
For α ∈ R , the Bernoulli polynomials of order α are defined by the generatingfunction to be (cid:18) te t − (cid:19) α e xt = ∞ X n =0 B ( α ) n ( x ) t n n ! , (see [1 , , . (1)In the special case, x = 0, B ( α ) n (0) = B ( α ) n are called the n -th Bernoulli numbersof order α .The Stirling number of the first kind is defined by( x ) n = n X k =0 S ( n, k ) x k , (2)where ( x ) n = x ( x − · · · ( x − n + 1).From (2), we note that x ( n ) = x ( x + 1) · · · ( x + n −
1) = ( − n ( − x ) n = n X k =0 | S ( n, k ) | x k . (3)1et F be the set of all formal power series in the variable t over C with F = ( f ( t ) = ∞ X k =0 a k k ! t k | a k ∈ C ) . (4)Suppose that P is the algebra of polynomials in the variable x over C and P ∗ is the vector space of all linear functionals on P . The action of the linear func-tional L on a polynomial p ( x ) is denoted by h L | p ( x ) i . For f ( t ) = P ∞ k =0 a k k ! t k ∈F , let us define a linear functional on P by setting h f ( t ) | x n i = a n , ( n ≥ , (see [2 , . (5)By (4) and (5), we easily get h t k | x n i = n ! δ n,k ( n, k ≥ , (see [2 , , (6)where δ n,k is the Kronecker’s symbol.For f L ( t ) = P ∞ k =0 h L | x k i k ! t k , we have h f L ( t ) | x n i = h L | x n i .Thus, we note that the map L f L ( t ) is a vector space isomorphism from P ∗ onto F . Henceforth, F is thought of as both a formal power series and alinear functional. We call F the umbral algebra. The umbral calculus is thestudy of umbral algebra (see [4]).The order O ( f ( t )) of the nonzero power series f ( t ) is the smallest integer k forwhich the coefficient of t k does not vanish (see [2 , O ( f ( t )) = 0, then f ( t ) is called an invertible series. If O ( f ( t )) = 1, then f ( t ) is called a delta series. For O ( f ( t )) = 1 and O ( g ( t )) = 0, there exists aunique sequence s n ( x ) of polynomials such that h g ( t ) f ( t ) k | s n ( x ) i = n ! δ n,k for n, k ≥ s n ( x ) is called the Sheffer sequence for ( g ( t ) , f ( t )) which is de-noted by s n ( x ) ∼ ( g ( t ) , f ( t )).Let f ( t ) ∈ F and p ( x ) ∈ P . Then we see that f ( t ) = ∞ X k =0 h f ( t ) | x k i k ! t k , p ( x ) = ∞ X k =0 h t k | p ( x ) i k ! x k , (see [4]) . (7)By (7), we easily see that t k p ( x ) = p ( k ) ( x ) = d k p ( x ) dx k , (see [2 , . (8)2et s n ( x ) ∼ ( g ( t ) , f ( t )). Then the generating function of Sheffer sequence s n ( x ) is given by 1 g ( ¯ f ( t )) e x ¯ f ( t ) = ∞ X k =0 s k ( x ) t k k ! , (see [2 , , (9)where ¯ f ( t ) is the compositional inverse of f ( t ).For p n ( x ) ∼ (1 , f ( t )), q n ( x ) ∼ (1 , g ( t )), we note that q n ( x ) = x (cid:18) f ( t ) g ( t ) (cid:19) n x − p n ( x ) , (see [2 , . (10)The pair ( g ( t ) , f ( t )) will be called a Sheffer pair where O ( g ( t )) = 0 and O ( f ( t )) = 1 (see [2, 4]). Let m be nonnegative integer. The m -th powerof an invertible series is denoted by ( g ( t )) m , while the compositional powerof a delta series f ( t ) is denoted by f m ( t ) = f ◦ f ◦ · · · ◦ f | {z } m − times ( t ). Let p n ( x ) and q n ( x ) = P nk =0 q n,k x k be sequences of polynomials. Then the umbral composi-tion of q n ( x ) with p n ( x ) is defined by( q n ◦ p ) ( x ) = n X k =0 q n,k p k ( x ) , (see [2 , . (11)Suppose that s n ( x ) ∼ ( g ( t ) , f ( t )) and r n ( x ) ∼ ( h ( t ) , l ( t ))Then we note that( r n ◦ s )( x ) = r n ( s ( x )) ∼ ( g ( t ) h ( f ( t )) , l ( f ( t ))) . (12)The identity under umbral composition is the sequence x n and the inverse ofsequence s n ( x ) is the Sheffer sequence for (cid:0) g ( ¯ f ( t )) − , ¯ f ( t ) (cid:1) (see [2, 4]).By (12), we easily see that the m -th power under umbral composition of s n ( x ) ∼ ( g ( t ) , f ( t )) is given by s ( m ) n ( x ) ∼ m − Y i =0 g ( f i ( t )) , f m ( t ) ! , where m ∈ N . (13)For n ≥
0, let us assume that s n ( x ) = n X k =0 s n,k x k = ∞ X k =0 s n,k x k , (14)3here we agree that s i,j = 0 if i < j .If we define s ( m ) n ( x ) by s ( m ) n ( x ) = n X k =0 s ( m ) n,k x k = ∞ X k =0 s ( m ) n,k x k , (15)then, by (11),(14) and (15), we easily get s ( m ) n,k = n X l , ··· ,l m − =0 s n,l s l ,l · · · s l m − ,l m − s l m − ,k , (see [2]) . (16)From (9) and (13), we can derive the generating function of s ( m ) n ( x ) as follows: ∞ X k =0 s ( m ) k ( x ) k ! t k = Q m − i =0 g ( f i ( ¯ f m ( t ))) ! e x ¯ f m ( t ) (17)= m − Y i =0 g ( ¯ f ( m − i ) ( t )) ! − e x ¯ f m ( t ) . In this paper, we study some properties of Sheffer sequences for the powers ofSheffer pairs under umbral composition. From our properties, we derive newand interesting identities of Sheffer sequences of special polynomials for thepowers of Sheffer pairs under umbral composition.
Let us take the sequence s n ( x ) of special polynomial as follows: s n ( x ) = x ( n ) = n X k =0 | S ( n, k ) | x k ∼ (1 , f ( t ) = 1 − e − t ) . (18)For m ∈ N , let us assume that the m -th power under umbral composition of s n ( x ) is given by s ( m ) n ( x ) = n X k =0 s ( m ) n,k x k . (19)4y (16), (18) and (19), we get s ( m ) n,k = n X l , ··· ,l m − =0 | S ( n, l ) || S ( l , l ) | · · · | S ( l m − , k ) | (20)= n X l , ··· ,l m − =0 | S ( n, l ) S ( l , l ) · · · S ( l m − , k ) | . It is known that x n ∼ (1 , t ) , s n ( x ) = x ( n ) ∼ (1 , f ( t ) = 1 − e − t ) . (21)By (10) and (21), we get s n ( x ) = x (cid:18) tf ( t ) (cid:19) n x − x n = x (cid:18) tf ( t ) (cid:19) n x n − . (22)From (22), we note that f ( t ) m x − s n ( x ) = f ( t ) m (cid:18) tf ( t ) (cid:19) n x n − = (cid:18) tf ( t ) (cid:19) n − m t m x n − (23)= (cid:18) t − e − t (cid:19) n − m t m x n − = ∞ X l =0 ( − l B ( n − m ) l l ! t l + m x n − = n − − m X l =0 ( − l B ( n − m ) l l ! ( n − l + m x n − − l − m , where n ≥
1, 0 ≤ m ≤ n − n ≥
1, by (13), (18), we get s (2) n ( x ) = x (cid:18) f ( t ) f ( t ) (cid:19) n x − s n ( x ) = x (cid:18) f ( t )1 − e − f ( t ) (cid:19) n x − s n ( x ) (24)= x n − X k =0 B ( n ) k k ! ( − k f ( t ) k x − s n ( x ) . s (2) n ( x ) = x n − X k =0 B ( n ) k k ! ( − k n − − k X k =0 ( − k B ( n − k ) k k ! ( n − k + k x n − − k − k (25)= n − X k =0 n − − k X k =0 ( n − − k + k k ! k !( n − k − k − B ( n ) k B ( n − k ) k x n − k − k = X k + k + l = n − (cid:18) n − k , k , l (cid:19) ( − k + k B ( n ) k B ( n − k ) k x l +1 = n X k =1 ( X k + k = n − k (cid:18) n − k , k , k − (cid:19) ( − k + k B ( n ) k B ( n − k ) k ) x k . From s (3) n ( x ) ∼ (1 , f ( t )) and s (2) ( x ) ∼ (1 , f ( t )), we get s (3) n ( x ) = x (cid:18) f ( t ) f ( t ) (cid:19) n x − s (2) n ( x ) = x (cid:18) f ( t )1 − e − f ( t ) (cid:19) n x − s (2) n ( x ) (26)= x n − X k =0 B ( n ) k k ! ( − k (cid:18) f ( t ) f ( t ) (cid:19) n − k ( f ( t )) k x − s n ( x )= x n − X k =0 B ( n ) k k ! ( − k (cid:18) f ( t )1 − e − f ( t ) (cid:19) n − k ( f ( t )) k x − s n ( x ) . From (48), (23) and (26), we have s (3) n ( x ) = x n − X k =0 n − − k X k =0 ( − k + k B ( n ) k B ( n − k ) k k ! k ! (27) × n − − k − k X k =0 ( − k B ( n − k − k ) k k ! ( n − k + k + k x n − − k − k − k = X k + k + k + l = n − ( − k + k + k (cid:18) n − k , k , k , l (cid:19) B ( n ) k B ( n − k ) k B ( n − k − k ) k x l +1 = n X k =1 ( X k + k + k = n − k ( − k + k + k (cid:18) n − k , k , k , k − (cid:19) × B ( n ) k B ( n − k ) k B ( n − k − k ) k ) x k . s ( m ) n ( x ) = n X k =1 ( X k + ··· + k m = n − k ( − k + ··· + k m (cid:18) n − k , · · · , k m , k − (cid:19) B ( n ) k m (28) × B ( n − k m ) k m − · · · B ( n − k m −···− k ) k ) x k . Therefore, by (19), (20) and (28), we obtain the following theorem.
Theorem 1.
For m, n ≥ , we have n X l , ··· ,l m − =0 | S ( n, l ) S ( l , l ) · · · S ( l m − , k ) | = X k + ··· + k m = n − k ( − k + ··· + k m (cid:18) n − k , · · · , k m , k − (cid:19) × B ( n ) k m B ( n − k m ) k m − · · · B ( n − k m − k m − −···− k ) k . Let us consider the following Sheffer sequence: s n ( x ) = L n ( x ) = n X k =0 L ( n, k )( − x ) k ∼ (cid:18) , f ( t ) = tt − (cid:19) , (29)where L ( n, k ) are the Lah numbers with L ( n, k ) = (cid:18) n − k − (cid:19) n ! k ! , for 1 ≤ k ≤ n, (30) L ( n, k ) = 0 , for k > n ≥ ,L ( n,
0) = 0 , for n ≥ ,L n (0 ,
0) = 1 . n ≥
1, 0 ≤ m ≤ n −
1, we have f ( t ) m x − s n ( x ) = f ( t ) m (cid:18) tf ( t ) (cid:19) n x n − = (cid:18) tf ( t ) (cid:19) n − m t m x n − (31)= ( t − n − m t m x n − = n − m − X l =0 (cid:18) n − ml (cid:19) ( − n − m − l ( n − m t l x n − − m = n − m − X l =0 (cid:18) n − ml (cid:19) ( − n − m − l ( n − m ( n − − m ) l x n − − m − l = n − m − X l =0 (cid:18) n − ml (cid:19) ( − n − m − l ( n − l + m x n − − m − l . For n ≥
1, from s (2) n ( x ) ∼ (1 , f ( t )) and s n ( x ) ∼ (cid:0) , f ( t ) = tt − (cid:1) , we get s (2) n ( x ) = x (cid:18) f ( t ) f ( t ) (cid:19) n x − s n ( x ) = x ( f ( t ) − n x − s n ( x ) (32)= x n − X k =0 (cid:18) nk (cid:19) ( − n − k f ( t ) k x − s n ( x ) . From (31) and (32), we can derive the following equation: s (2) n ( x ) = x n − X k =0 (cid:18) nk (cid:19) ( − n − k n − − k X k =0 (cid:18) n − k k (cid:19) ( − n − k − k (33) × ( n − k + k x n − − k − k = n − X k =0 n − − k X k =0 ( − n − k +( n − k − k ) n !( n − k − k )! × (cid:18) n − k , k , n − − k − k (cid:19) x n − k − k = X k + k + l = n − ( − n − k + l +1 n !( l + 1)! (cid:18) n − k , k , l (cid:19) x l +1 = n X k =1 ( X k + k = n − k ( − ( n − k )+ k n ! k ! (cid:18) n − k , k , k − (cid:19)) x k s (3) n ( x ) ∼ (1 , f ( t )) and s (2) n ( x ) ∼ (1 , f ( t )), we get s (3) n ( x ) = x (cid:18) f ( t ) f ( t ) (cid:19) n x − s (2) n ( x ) = x f ( t ) f ( t ) f ( t ) − ! n x − s (2) n ( x ) (34)= x (cid:0) f ( t ) − (cid:1) n x − s (2) n ( x ) = x n − X k =0 (cid:18) nk (cid:19) ( − n − k (cid:0) f ( t ) (cid:1) k x − s (2) n ( x )= x n − X k =0 ( − n − k (cid:18) nk (cid:19) (cid:18) f ( t ) f ( t ) (cid:19) n − k f ( t ) k x − s n ( x )= x n − X k =0 (cid:18) nk (cid:19) ( − n − k n − − k X k =0 (cid:18) n − k k (cid:19) ( − n − k − k ( f ( t )) k + k x − s n ( x ) . From (31) and (34), we have s (3) n ( x ) = X k + k + k + l = n − ( − ( n − k )+( n − k − k )+( l +1) n !( l + 1)! (cid:18) n − k , k , k , l (cid:19) x l +1 (35)= n X k =1 ( X k + k + k = n − k ( − ( n − k )+( n − k − k )+ k n ! k ! (cid:18) n − k , k , k , k − (cid:19)) x k . Continuing this process, we get s ( m ) n ( x ) = n X k =1 ( X k + ··· + k m = n − k ( − ( n − k m )+ ··· +( n − k m − k m − −···− k )+ k n ! k ! × (cid:18) n − k , k , · · · , k m , k − (cid:19) ) x k (36)= n X k =1 s ( m ) n,k x k , where m ≥ . By (14), (15), (16), (29) and (36), we easily get s ( m ) n,k = n X l , ··· ,l m − =0 s n,l s l ,l · · · s l m − ,k (37)= n X l , ··· ,l m − =0 ( − l + l + ··· + l m − + k L ( n, l ) L ( l , l ) · · · L ( l m − , k ) . Therefore, by (36) and (37), we obtain the following theorem.9 heorem 2.
For m, n ≥ , ≤ k ≤ n , we have n X l ,l ··· ,l m − =0 ( − l + l + ··· + l m − + k L ( n, l ) L ( l , l ) · · · L ( l m − , k )= X k + ··· ,k m = n − k ( − ( n − k m )+( n − k m − k m − )+ ··· +( n − k m −···− k )+ k n ! k ! (cid:18) n − k , k , · · · , k m , k − (cid:19) Let us take Abel sequence as follows: s n ( x ) = A n ( x : a ) = x ( x − an ) n − = n X k =1 (cid:18) n − k − (cid:19) ( − an ) n − k x k (38) ∼ (cid:0) , f ( t ) = te at (cid:1) , where a = 0 . Thus by (38), we get s n,k = (cid:18) n − k − (cid:19) ( − an ) n − k , ( n, k ≥
0) (39)From (16) and (39), we note that s ( m ) n,k = n X l , ··· ,l m − =0 s n,l s l ,l · · · s l m − ,l m − s l m − ,k (40)= n X l , ··· ,l m − =0 (cid:18) n − l − (cid:19) (cid:18) l − l − (cid:19) · · · (cid:18) l m − − l m − − (cid:19) (cid:18) l m − − k − (cid:19) × ( − a ) n − k n n − l l l − l · · · l l m − − l m − m − l l m − − km − . From s n ( x ) = A n ( x : a ) ∼ (1 , f ( t ) = te at ) and x n ∼ (1 , t ), we note that f ( t ) m x − s n ( x ) = f ( t ) m (cid:18) tf ( t ) (cid:19) n x n − = (cid:18) tf ( t ) (cid:19) n − m t m x n − (41)= (cid:18) tte at (cid:19) n − m t m x n − = e − a ( n − m ) t t m x n − = n − − m X l =0 ( − a ( n − m )) l ( n − l + m l ! x n − − l − m . n ≥
1, from s (2) n ( x ) ∼ (1 , f ( t )) and s n ( x ) ∼ (1 , f ( t ) = te at ), we get s (2) n ( x ) = x (cid:18) f ( t ) f ( t ) (cid:19) n x − s n ( x ) = x (cid:18) f ( t ) f ( t ) e af ( t ) (cid:19) n x − s n ( x ) (42)= xe − anf ( t ) x − s n ( x ) = x n − X k =0 ( − an ) k k ! ( f ( t )) k x − s n ( x ) . From (41) and (42), we can derive the following equation (43): s (2) n ( x ) = n − X k =0 n − − k X k =0 (cid:18) n − k , k , n − − k − k (cid:19) ( − an ) k ( − a ( n − k )) k x n − k − k (43)= X k + k + l = n − (cid:18) n − k , k , l (cid:19) ( − an ) k ( − a ( n − k )) k x l +1 = n X k =1 ( X k + k = n − k (cid:18) n − k , k , k − (cid:19) ( − an ) k ( − a ( n − k )) k ) x k . From s (3) n ( x ) ∼ (1 , f ( t )) and s (2) n ( x ) ∼ (1 , f ( t )), we get s (3) n ( x ) = x (cid:18) f ( t ) f ( t ) (cid:19) x − s (2) n ( x ) = xe − anf ( t ) x − s (2) n ( x ) (44)= x n − X k =0 ( − an ) k k ! (cid:0) f ( t ) (cid:1) k x − s (2) n ( x )= x n − X k =0 ( − an ) k k ! (cid:18) f ( t ) f ( t ) (cid:19) n − k f ( t ) k x − s n ( x )= x n − X k =0 ( − an ) k k ! e − a ( n − k ) f ( t ) ( f ( t )) k x − s n ( x )= x n − X k =0 ( − an ) k k ! n − − k X k =0 ( − a ( n − k )) k k ! ( f ( t )) k + k x − s n ( x ) . s (3) n ( x ) = X k + k + k + l = n − (cid:18) n − k , k , k , l (cid:19) ( − an ) k ( − a ( n − k )) k ( − a ( n − k − k )) k x l +1 (45)= n X k =1 ( X k + k + k = n − k (cid:18) n − k , k , k , k − (cid:19) ( − an ) k ( − a ( n − k )) k × ( − a ( n − k − k )) k ) x k . Continuing this process, we get s ( m ) n ( x ) = n X k =1 ( X k + ··· + k m = n − k (cid:18) n − k , k , · · · , k m , k − (cid:19) × m Y i =1 ( − a ( n − k m − · · · − k i +1 )) k i ! ) x k . (46)Therefore, by (40) and (46), we obtain the following theorem. Theorem 3.
For n, m ≥ , ≤ k ≤ n , we have n X l , ··· ,l m − =0 (cid:18) n − l − (cid:19) (cid:18) l − l − (cid:19) · · · (cid:18) l m − − l m − − (cid:19) (cid:18) l m − − k − (cid:19) ( − an ) n − l × ( − al ) l − l · · · ( − al m − ) l m − − l m − ( − al m − ) l m − − k = X k + k + ··· + k m = n − k (cid:18) n − k , k , · · · , k m , k − (cid:19) m Y i =1 ( − a ( n − k m − · · · − k i +1 )) k i ! . Remark . Let us consider the Mittag-Leffler sequences as follows: s n ( x ) = M n ( x ) = n X r =0 (cid:18) nr (cid:19) ( n − r − r ( x ) r (47)= n X k =0 ( n X r = k (cid:18) nr (cid:19) ( n − r − r S ( r, k ) ) x k ∼ (cid:18) , e t − e t + 1 = f ( t ) (cid:19) .
12y the same method, we get, for m, n ≥
1, 1 ≤ k ≤ n , n X l , ··· ,l m − =0 n X r = l · · · l m − X r m − = l m − l m − X r m = k (cid:18) nr (cid:19) (cid:18) l r (cid:19) · · · (cid:18) l m − r m − (cid:19) (cid:18) l m − r m (cid:19) × ( n − l − · · · ( l m − − l m − − r − r − · · · ( r m − − r m − × r + r + ··· + r m × S ( r , l ) S ( r , l ) · · · S ( r m − , l m − ) S ( r m , k )= X k + ··· + k m = n − k (cid:18) n − k , · · · , k m , k − (cid:19) m − Y i =0 E ( k + ··· + k i − n )2 i +1 B ( n − k −···− k i )2 i +2 ! × m − Y i =0 n − ( k + k + ··· + k i ) ! . Here, for α ∈ R , the Euler polynomials of order α are defined by the generatingfunction to be (cid:18) e t + 1 (cid:19) α e xt = ∞ X n =0 E ( α ) n ( x ) t n n ! , (see [1 , , . (48)In the special case, x = 0, E ( α ) n (0) = E ( α ) n are called the n -th Euler numbersof order α . References [1] M. Acikgoz, D. Erdal, S. Araci,
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