On the associated sequences of special polynomials

0.12 MB·English

by  D. S. Kim

ON THE ASSOCIATED SEQUENCES OF SPECIAL
POLYNOMIALS
3
1 D. S. KIM,T. KIM,AND S.-H.RIM
0
2
Abstract. In this paper, we investigate some properties of the asso-

n ciated sequence of Daehee and Changhee polynomials. Finally, we give
a someinterestingidentitiesofassociatedsequenceinvolvingsomespecial
J
polynomials.
6
2

]
T 1. Introduction
N
Inthispaper,weassumethatλ C,withλ = 1. Forα R,theFrobenius
h. -Euler polynomials of order r are∈defined by t6he generati∈ng function to be
t
a 1 λ α ∞ tn
m (1) − ext = H(α)(x λ) , (see[9,11,13,22,27]).
et λ n | n!
[ (cid:18) − (cid:19) n=0
X

1 In the special case, x = 0, H(α)(0λ) = H(α)(λ) are called n-th Frobenius-
n n
v |
Euler numbers of order α, (see[8,9,27]). As is well known, the Bernoulli
2
2 polynomials of order α are defined by the generating function to be:
2
6 (2) t αext = ∞ B(α)(x)tn, (see[10,12,21,28]).
1. et 1 n n!
(cid:18) − (cid:19) n=0
0 X
3 In the special case, x = 0, B(α)(0λ) = B(α) are called n-th Bernoulli num-
n n
1 |
bers of order α. The Stirling numbers of the second kind are defined by
:
v
i ∞ tl
X (3) (et 1)n = n! S (l,n) , (see[5,20,21,26]).
2
− l!
r l=n
a X
and the Stirling numbers of the first kind are given by
n
(4) (x) = x(x 1)...(x n+1) = S (l,n)xl, (see[5,20,21]).
n 1
− −
l=0
X
Let be the set of all formal power series in the variable t over C with
F
∞
a
(5) = f(t) = ktk a C .
k
F { k! | ∈ }
n=0
X
Let us assume that P is the algebra of polynomials in the variable x over C
andP∗ isthevectorspaceofalllinearfunctionalsonP. < L p(x)>denotes
|
the action of the linear functional L on a polynomial p(x) and we remind
that the vector space structure on P∗ is defined by < L+M p(x) >=
|
1
2 D.S.KIM,T.KIM,ANDS.-H.RIM
< L p(x) > + < M p(x)>, and < cL p(x)>= c< L p(x) >,
wher|ecis a complex c|onstant, (see[6,9,20|,21]). For f(t)=| ∞ aktk ,
we define a linear functional on P by setting k=0 k! ∈ F
P
(6) < f(t) xn >= a ,(n 0), (see[20,21]).
n
| ≥
From (6), we note that
(7) < tk xn >= n!δ , (n,k 0),
n,k
| ≥
where δ is the Kronecker’s symbol. Let f (t) = ∞ <L | xk>tk. Then,
n,k L k=0 k!
by(7), weseethat< f (t) xn >=<L xn >. ThemapL f (t)isavec-
L L
tor space isomorphism from| P∗ onto .|Henceforth,P is th7→ought of as both
F F
a formal power series and a linear functional. We call the umbralalgebra.
F
The umbral calculus is the study of umbral algebra (see[7,11,,19,20,21]).
The order O(f(t)) of the non-zero power series f(t) is the smallest inte-
ger k for which the coefficient of tk does not vanish (see[10,20,21]). If
O(f(t)) = 1, then f(t) is called a delta series and a series f(t) having
O(f(t))= 0 is called an invertible series (see[10,20,21]). Let f(t) bea delta
series and g(t) be an invertible series. Then there exists a unique sequence
S (x) of polynomials such that < g(t)f(t)k S (x) >=n!δ where n,k 0.
n n n,k
| ≥
The sequence S (x) is called the Sheffer sequence for (g(t),f(t)) which is
n
denoted by S (x) (g(t),f(t)). If S (x) (1,f(t)), then S (x) is called
n n n
∼ ∼
the associated sequence for f(t). By (7), we see that < eyt p(x) >= p(y).
For f(t) and p(x) P, we have |
∈ F ∈
(8)
∞ < f(t) xk > ∞ < tk p(x) >
f(t)= | tk, p(x) = | xk, (see[11,20,21]).
k! k!
k=0 k=0
X X
and
(9) < f (t)f (t)...f (t) xn >
1 2 m
|
n
= < f (t) xi1 > ... < f (t) xim >,
1 m
i ,...i | |
i1+..X.+im=n(cid:18) 1 m(cid:19)
where f (t),f (t),...,f (t) .
1 2 m
∈ F
By (8), we get
(10) p(k)(0) =<tk p(x)>, < 1 p(k)(x) >=p(k)(0).
| |
Thus, by (10), we see that
dkp(x)
(11) tkp(x) = p(k)(x) = , (k 0), (see[10,20,21]).
dxk ≥
For S (x) (g(t),f(t)), we have
n
∼
∞
(12) 1 eyf¯(t) = Sk(y)tk, for all y C,
g(f¯(t)) k! ∈
k=0
X
ON THE ASSOCIATED SEQUENCES OF SPECIAL POLYNOMIALS 3
where f¯(t) is the compositional inverse of f(t),
n n
n n
(13) S (x+y) = p (y)S (x) = S (y)p (x),
n k n−k n−k k
k k
k=0(cid:18) (cid:19) k=0(cid:18) (cid:19)
X X
where p (y) = g(t)S (y) (1,f(t)), (see[5,11,20,21]).
k k
∼
Let p (x) (1,f(t)) and q (x) (1,g(t)). Then the transfer formula for
n n
∼ ∼
the associated sequence is implies that
f(t) n
(14) q (x) = x x−1p (x), (see[5,11,20,21]).
n n
g(t)
(cid:18) (cid:19)
In this paper, we investigate some properties of the associated sequence
of Daehee and Changhee polynomials arising from umbral calculus. Finally,
we derive some interesting identities of associated sequence of special poly-
nomials from (14).
2. On the associated sequence of special polynomials
For λ C with λ = 1, the Daehee polynomials are defined by the gener-
∈ 6
ating function to be
∞ D (x λ) (1 λ)+t(1+λ) 1+t x
(15) k | tk = − ,
k! (1 λ)(1 t) 1 t
k=0 (cid:18) − − (cid:19)(cid:18) − (cid:19)
X
(see [1,2,14,16,17,18]).
From (15), we note that D (x λ) is the Sheffer sequence for 1−λ,et−1 .
n | et−λ et+1
That is, (cid:16) (cid:17)
1 λ et 1
(16) D (x λ) − , − .
n | ∼ et λ et+1
(cid:18) − (cid:19)
As is known, the Mittag-Leffler sequence is given by
n n et 1
(17) M (x) = (n 1) 2k(x) 1, − .
n k − n−k k ∼ et+1
k=0(cid:18) (cid:19) (cid:18) (cid:19)
X
From (16) and (17), we have
1 λ et 1
(18) M (x) = − D (x λ) 1, − .
n et λ n | ∼ et+1
− (cid:18) (cid:19)
Let us consider the following associated sequence:
1 λ
(19) S (x λ) 1, − t and xn (1,t).
n | ∼ et λ ∼
(cid:18) − (cid:19)
4 D.S.KIM,T.KIM,ANDS.-H.RIM
By (14) and (19), we get
1−λt −n
(20) S (x λ) = x et−λ x−1xn
n
| t
!
= (1 λ)−nx(et λ)nxn−1
− −
n
n
= (1 λ)−nx ( λ)n−leltxn−1
− l −
l=0(cid:18) (cid:19)
X
n
n
= (1 λ)−nx ( λ)n−l(x+l)n−1.
− l −
l=0(cid:18) (cid:19)
X
Therefore by (20), we obtain the following theorem.
Theorem 1 . For r Z = N 0 , let S (x λ) (1, 1−λt).
∈ + ∪{ } n | ∼ et−λ
Then we have
n
x n
S (x λ) = ( λ)n−l(x+l)n−1.
n | (1 λ)n l −
− l=0(cid:18) (cid:19)
X
From (14), (17) and (19), we have
(21)
M (x)
n
1−λt n 1 λ n t(et +1) n
= x et−λ x−1S (x) = x − x−1S (x)
et−1 n et λ et 1 n
!
et+1 (cid:18) − (cid:19) (cid:18) − (cid:19)
1 λ n 2t n
= x − t+ x−1S (x)
et λ et 1 n
(cid:18) − (cid:19) (cid:18) − (cid:19)
1 λ n n n 2t l
= x − tn−l x−1S (x)
et λ l et 1 n
(cid:18) − (cid:19) l=0(cid:18) (cid:19) (cid:18) − (cid:19)
X
1 λ n n n 2t l 1 n n
= x − tn−l ( λ)n−j(x+j)n−1
et λ l et 1 (1 λ)n j −
(cid:18) − (cid:19) l=0(cid:18) (cid:19) (cid:18) − (cid:19) − j=0(cid:18) (cid:19)
X X
1 λ n n n n n 2t l
= x − (1 λ)−n ( λ)n−jtn−l (x+j)n−1.
et λ l − j − et 1
(cid:18) − (cid:19) l=0 j=0(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) − (cid:19)
XX
By (1), (2) and (12), we get
et 1 α et λ α
(22) B(α)(x) − ,t H(α)(x λ) − ,t ,
n ∼ t n | ∼ 1 λ
(cid:18)(cid:18) (cid:19) (cid:19) (cid:18)(cid:18) − (cid:19) (cid:19)
ON THE ASSOCIATED SEQUENCES OF SPECIAL POLYNOMIALS 5
and, by (13), we see that
n n
n n
(23) B(α)(x)= B(α) xk, H(α)(x λ) = H(α)xl.
n k n−k n | l n−l
k=0(cid:18) (cid:19) l=0(cid:18) (cid:19)
X X
Thus, by (21), (22) and (23), we get
(24)
M (x)
n
1 λ n n n n n
=x − (1 λ)−n ( λ)n−j2ltn−lB(l) (x+j)
et λ l − j − n−1
(cid:18) − (cid:19) l=0 j=0(cid:18) (cid:19) (cid:18) (cid:19)
XX
1 λ n n n n n
=x − (1 λ)−n ( λ)n−j2l(n 1) B(l) (x+j)
et λ l − j − − n−l l−1
(cid:18) − (cid:19) l=1 j=0(cid:18) (cid:19) (cid:18) (cid:19)
XX
1 λ n n n n n (n 1)! l−1
=x − (1 λ)−n ( λ)n−j2l − B(l)
et λ l − j − (l 1)! l−1−m
(cid:18) − (cid:19) l=1 j=0(cid:18) (cid:19) (cid:18) (cid:19) − m=0
XX X
l 1
− (x+j)m
× m
(cid:18) (cid:19)
n n l−1
n n (n 1)! l 1
=x (1 λ)−n ( λ)n−j2l − B(l) −
l − j − (l 1)! l−1−m m
l=1 j=0(cid:18) (cid:19) (cid:18) (cid:19) − m=0 (cid:18) (cid:19)
XX X
1 λ n
− (x+j)m
× et λ
(cid:18) − (cid:19)
x(n 1)! n n l−1 n n l−1 ( λ)n−j2l
= − l j m − B(l) H(n)(x+j λ).
(1 λ)n (l 1)! l−1−m m |
(cid:0) (cid:1)(cid:0) (cid:1)(cid:0) (cid:1)
− l=1 j=0m=0 −
XX X
Recall here that, for any g(t) , the Pincherle derivative is given by
∈ F
′
(25) g (t)= g(t)x xg(t)
−
as linear operators on P, (see [22]).
By (18), (24) and (25), we get
et λ
(26) D (x λ) = − M (x)
n n
| 1 λ
−
(n 1)! n n l−1 n n l−1 ( λ)n−j2l
= − l j m − B(l)
(1 λ)n (l 1)! l−1−m
(cid:0) (cid:1)(cid:0) (cid:1)(cid:0) (cid:1)
− l=1 j=0m=0 −
XX X
1
xH(n−1)(x+j λ)+ H(n)(x+j +1λ) .
× m | 1 λ m |
(cid:26) − (cid:27)
Therefore, by (26), we obtain the following theorem.
6 D.S.KIM,T.KIM,ANDS.-H.RIM
Theorem 2 . For n 1, we have
≥
D (x λ)
n
|
(n 1)! n n l−1 n n l−1 ( λ)n−j2l
= − l j m − B(l) x
(1 λ)n (l 1)! l−1−m
(cid:0) (cid:1)(cid:0) (cid:1)(cid:0) (cid:1)
− l=1 j=0m=0 −
XX X
1
xH(n−1)(x+j λ)+ H(n)(x+j+1λ) .
× m | 1 λ m |
(cid:26) − (cid:27)
Remark . (a) Let us consider the following associated sequence for t :
et+1
t
(27) S (x) 1, .
n ∼ et+1
(cid:18) (cid:19)
By (27), we get
n
n
(28) S (x) = x(et+1)nx−1xn = x (x+j)n−1.
n
j
j=0(cid:18) (cid:19)
X
From (14), (17) and (27), we have
t n n n
(29) M (x) = x x−1S (x)= xB(n) (x+j).
n et 1 n j n−1
(cid:18) − (cid:19) j=0(cid:18) (cid:19)
X
Thus, by (18) and (29), we get
(30)
et λ
D (x λ) = − M (x)
n n
| 1 λ
(cid:18) − (cid:19)
n n
1 n λ n
= (x+1)B(n) (x+j +1)+ xB(n) (x+j).
1 λ j n−1 λ 1 j n−1
− j=0(cid:18) (cid:19) − j=0(cid:18) (cid:19)
X X
(b) From (17), we note that we can obtain another expression of D (x λ) as
n
|
follow :
et λ
(31) D (x λ) = − M (x)
n n
| 1 λ
(cid:18) − (cid:19)
n n et λ
= (n 1) 2k − (x)
k − n−k 1 λ k
k=0(cid:18) (cid:19) −
X
n
1 n
= (n 1) 2k (x+1) λ(x) .
1 λ k − n−k { k − k}
− k=0(cid:18) (cid:19)
X
For λ C with λ = 1, Changhee polynomials of order a are defined by
∈ 6
the generating function to be
∞ tk t+1 λ a
(32) C(a)(x λ) = − (1+t)x, (see[15,23,24,25,26]).
k | k! 1 λ
k=0 (cid:18) − (cid:19)
X
ON THE ASSOCIATED SEQUENCES OF SPECIAL POLYNOMIALS 7
(a) (a)
In the specials case, x = 0, C (0λ) = C (λ) are called n-th Changhee
k | k
numbers of order a. From (12) and (21), we note that
1 λ a
(33) C(a)(x λ) − ,et 1 , a = 0.
n | ∼ et λ − 6
(cid:18)(cid:18) − (cid:19) (cid:19)
Thus, by (33), we get
et λ a
(34) − C(a)(x λ) 1,et 1 .
1 λ n | ∼ −
(cid:18) − (cid:19)
From (14), xn (1,t) and (35), we have (cid:0) (cid:1)
∼
et λ a t n
(35) − C(a)(x λ) = x x−1xn
1 λ n | et 1
(cid:18) − (cid:19) (cid:18) − (cid:19)
t n
= x xn−1 = xB(n) (x).
et 1 n−1
(cid:18) − (cid:19)
Thus, by (14), (22), (25) and (35), we get
(36)
1 λ a a 1 λ a+1
C(a)(x λ) = x − et − B(n) (x)
n | et λ − 1 λ et λ n−1
( )
(cid:18) − (cid:19) − (cid:18) − (cid:19)
n−1 n 1 1 λ a a 1 λ a+1
= − B(n) x − et − xn−1−l
l l et λ − 1 λ et λ
( )
l=0 (cid:18) (cid:19) (cid:18) − (cid:19) − (cid:18) − (cid:19)
X
n−1
n 1 a
(n) (a) (a+1)
= − B xH (x λ) H (x+1λ) .
l l n−1−l | − 1 λ n−1−l |
l=0 (cid:18) (cid:19) (cid:26) − (cid:27)
X
Therefore, by (36), we obtain the following theorem.
Theorem 3 . For n N, we have
∈
n−1
n 1 a
C(a)(x λ) = − B(n) xH(a) (x λ) H(a+1)(x+1λ) .
n | l l n−1−l | − 1 λ n−1−l |
l=0 (cid:18) (cid:19) (cid:26) − (cid:27)
X
Let
et λ a
(37) p(x) 1,t − .
∼ 1 λ
(cid:18) (cid:18) − (cid:19) (cid:19)
By xn (1,t) and (14), we get
∼
n
t
(38) p(x)= x x−1xn
a
t et−λ 
1−λ
1(cid:16) λ (cid:17)an
= x − xn−1 = xH(an)(x λ),
et λ n−1 |
(cid:18) − (cid:19)
8 D.S.KIM,T.KIM,ANDS.-H.RIM
where a Z .
+
∈
Therefore, by (38), we obtain the following theorem.
Theorem 4 . For n N and a Z , we have
+
∈ ∈
1 λ a
(an)
xH (x λ) 1,t − .
n−1 | ∼ et λ
(cid:18) (cid:18) − (cid:19) (cid:19)
By the same method of (38), we easily see that
et 1 b
(39) xB(bn)(x) 1,t − , b Z .
n−1 ∼ t ∈ +
!
(cid:18) (cid:19)
For n 1 , by (14), Theorem 4 and (39), we get
≥
b n
t et−1
(40) xH(an)(x λ) = x t x−1xB(an)(x)
n−1 | t(cid:16)et−λ(cid:17)a n−1
1−λ
 
1(cid:16) λ (cid:17)anet 1 bn
(an)
= x − − B (x).
et λ t n−1
(cid:18) − (cid:19) (cid:18) (cid:19)
Thus, from (40), we have
(et λ)an et 1 bn
(an) (bn)
(41) − H (x λ) = − B (x).
(1 λ)an n−1 | t n−1
− (cid:18) (cid:19)
an
1 an
(42) LHS of (40) = ( λ)an−jejtH(an)(x λ)
(1 λ)an j − n−1 |
− j=0(cid:18) (cid:19)
X
an
1 an
= ( λ)an−jH(an)(x+j λ),
(1 λ)an j − n−1 |
− j=0(cid:18) (cid:19)
X
ON THE ASSOCIATED SEQUENCES OF SPECIAL POLYNOMIALS 9
and
et 1 bn
(bn)
(43) RHS of (41) = − B (x)
t n−1
(cid:18) (cid:19)
1 ∞ tj
(bn)
= (bn)! S (j,bn) B (x)
tbn 2 j! n−1
j=bn
X
n−1
S (j +bn,bn)
=(bn)! 2 tjB(bn)(x)
(j +bn)! n−1
j=0
X
n−1
(bn)! (n 1)!
(bn)
= − S (j +bn,bn)B (x)
(j +bn)!(n j 1)! 2 n−1−j
j=0 − −
X
n−1
(bn)! S (j +bn,bn)
=(n 1)! 2 B(bn) (x).
− (j +bn)! (n j 1)! n−1−j
j=0 − −
X
Therefore, by (41), (42) and (43), we obtain the following theorem.
Theorem 5 . For a,b Z and n N, we have
+
∈ ∈
an
an
( λ)an−jH(an)(x+j λ)
j − n−1 |
j=0(cid:18) (cid:19)
X
n−1
(bn)!
= (1 λ)an(n 1)! S (j +bn,bn)B(bn) (x).
− − (j +bn)!(n j 1)! 2 n−1−j
j=0 − −
X
Remark . Let us consider the following Sheffer sequence:
1 λ 1 λ
(44) S (x λ) = − ,t − .
n | et λ et λ
(cid:18) − (cid:18) − (cid:19)(cid:19)
Thus, by (44), we get
1 λ 1 λ
(45) − S (x λ) 1,t − .
et λ n | ∼ et λ
− (cid:18) (cid:18) − (cid:19)(cid:19)
By Theorem 5 and (45), we get
et λ
(n)
S (x λ) = − xH (x λ)
n | 1 λ n−1 |
(cid:18) − (cid:19)
1
(n) (n)
= (x+1)H (x+1λ) λxH (x λ) .
1 λ{ n−1 | − n−1 | }
−
The Daehee polynomials of the second kind are defined by the generating
function to be
∞ tk 1 1 λt x
(46) D∗(x λ) = − ,
k | k! 1 t 1 t
k=0 − (cid:18) − (cid:19)
X
10 D.S.KIM,T.KIM,ANDS.-H.RIM
where λ C with λ = 1, (see[2,14,26,28]).
∈ 6
From (12) and (46), we note that
1 λ et 1
(47) D∗(x λ) − , − .
n | ∼ et λ et λ
(cid:18) − − (cid:19)
Let us consider the λ-analogues of Mittag-Leffler sequence as follows:
et 1
(48) M∗(x λ) 1, − .
n | ∼ et λ
(cid:18) − (cid:19)
From (14), xn (1,t) and (48), we have
∼
et λ n t n
(49) M∗(x λ) = x − t xn−1 = x(et λ)n xn−1
n | et 1 − et 1
(cid:18) − (cid:19) (cid:18) − (cid:19)
n
n
= x ( λ)n−leltB(n) (x)
l − n−1
l=0(cid:18) (cid:19)
X
n
n
= x ( λ)n−lB(n) (x+l).
l − n−1
l=0(cid:18) (cid:19)
X
Therefore, by (49), we obtain the following theorem.
Theorem 6 . For n N, let M∗(x λ) 1, et−1
∈ n | ∼ et−λ
Then we have (cid:16) (cid:17)
n
n
M∗(x λ) = ( λ)n−lxB(n) (x+l).
n | l − n−1
l=0(cid:18) (cid:19)
X
From (13) and Theorem 6, we have
n n−1
n n 1
(50) M∗(x λ) = x − ( λ)n−lB(n) (x+l)j.
n | l j − n−1−j
l=0 j=0(cid:18) (cid:19)(cid:18) (cid:19)
XX
By (47) and (48), we get
(51)
et λ
D∗(x λ) = − M∗(x λ)
n | 1 λ n |
(cid:18) − (cid:19)
n
1 n
= (x+1)B(n) (x+1+j) λxB(n) (x+j) ( λ)n−j.
1 λ j { n−j − n−1 } −
− j=0(cid:18) (cid:19)
X
Let us consider the following associated sequences:
2t
(52) T∗(x) 1, .
n ∼ 1+t2
(cid:18) (cid:19)