RACSAM DOI 10.1007/s13398-015-0270-2 ORIGINAL PAPER
Generalized Boole numbers and polynomials Dae San Kim1 · Taekyun Kim2
Received: 2 September 2015 / Accepted: 16 December 2015 © Springer-Verlag Italia 2015
Abstract In this paper, we consider the generalized Boole numbers and polynomials attached to a Dirichlet character χ and investigate some properties of those numbers and polynomials. Keywords calculus
Generalized Boole polynomials · Generalized Euler polynomials · Umbral
Mathematics Subject Classification
05A40 · 11B83 · 11S80
1 Introduction Let p be a fixed odd prime number and let Z p , Q p and C p denote the ring of p-adic rational integers, the field of p-adic rational numbers and the completion of algebraic closure of Q p . The p-adic absolute value on C p is normalized so that | p| p = 1p . Let C Z p be the space of continuous functions on Z p . For f ∈ C Z p , the fermionic p-adic integral on Z p is defined by Kim to be I−1 ( f ) =
Zp
f (x) dμ−1 (x) = lim
n→∞
n −1 p
f (x) (−1)x ,
(1.1)
x=0
(see [15,20]). By (1.1), we get I−1 ( f 1 ) + I−1 ( f ) = 2 f (0), (see [15,20]), where f 1 (x) = f (x + 1).
B
Taekyun Kim tkkim@kw.ac.kr Dae San Kim dskim@sogang.ac.kr
1
Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea
2
Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea
(1.2)
D. S. Kim, T. Kim
From (1.2), we can derive the following equation : I−1 ( f n ) + (−1)n−1 I−1 ( f ) = 2
n−1
(−1)n−1−l f (l) ,
(1.3)
l=0
where f n (x) = f (x + n), n ∈ N. For d ∈ N with ( p, d) = 1, let X = X d = lim Z/d p N Z, X 1 = Z p , ← N
X∗ =
a + d pZ p ,
0
a + d p N Z p = x ∈ X | x ≡ a mod dp N , where a ∈ Z lies in 0 ≤ a < d p N . Let d ∈ N with d ≡ 1 (mod 2) and let χ be a Dirichlet character with conductor d. Then the generalized Euler polynomials attached to χ are defined by d−1 ∞ χ (a) eat (−1)a tn E n,χ (x) (1.4) e xt . =2 n! edt + 1 n=0
a=0
When x = 0, E n,χ = E n,χ (0) are called the generalized Euler numbers attached to χ (see [1–34]). The Boole polynomials Bln (x|λ) are defined by the generating function to be ∞ n=0
Bln (x|λ)
1 tn (1 + t)x , = n! (1 + t)λ + 1
(1.5)
where λ ∈ Z p with λ = 0. When x = 0, Bln (λ) = Bln (0|λ) are called the Boole numbers (see [15,28]). Thus, by (1.5), we get n n Bln (x|λ) = Blm (λ) (x)n−m m m=0 n n x m! Bln−m (λ) , = m m m=0
where (x)n = x (x − 1) . . . (x − n + 1), (n ≥ 1) and (x)0 = 1. The Boole polynomials and numbers are important and useful in studying polynomial expansion of analytic functions and in the umbral calculus (see [3,4]). Also, the Boole polynomials are Sheffer sequences for the pair g (t) = 1 + eλt , f (t) = et − 1, and a generalization of them are Peters polynomials. In this paper, we will consider the generalized Boole polynomials and numbers attached to a Dirichlet character χ, which are a twisted version of Boole polynomials and numbers studied earlier in [15]. We note here that the generating function of the generalized Boole polynomials are closely related to that of the generalized Euler polynomials. In fact, one can be obtained from the other
Generalized Boole numbers and polynomials
by a simple replacement of variables. Here we investigate their relations with the viewpoint of the Euler polynomials of the second kind. Many properties about the generalized Euler polynomials and numbers attached to χ have been derived from the fermionic p-adic integrals on Z p . Specifically, Kim constructed the Euler p-adic l-functions which interpolate the generalized Euler polynomials and numbers attached to χ at negative integers. After this construction, many researchers like Simsek, Araci, Mansour, Hu, Ozden, Kurt and Cenkci have studied the Kim’s p-adic l-functions and their results have been published in various international journals (see [2,5,18,19,21,25– 27,32,34,37]). In the same spirit as the construction of the generalized Euler polynomials and numbers attached to χ, here we derive the generalized Boole polynomials and numbers attached to χ from the multivariate p-adic fermionic integrals on Z p and study some relations between them in Sect. 2. Also, we study the generalized Boole polynomials in connection with its associated λ-invariant measure in Sect. 3. The generalized Boole polynomials, being Sheffer sequences, can also be studied by using the umbral calculus (see [11,14,24,28,29]). This is done in Sect. 4. Further, in our future work, we would like to construct an analytic function which interpolates the generalized Bool polynomials at negative integers, just as the Euler p-adic l-function interpolating the generalized Euler polynomials at negative integers were constructed. Finally, we highlight important results obtained in this paper. (k)
• We define the Boole polynomials Bln,χ (x|λ) of order k (∈ N) by the generating function as d−1 k ∞ (−1)a χ (a) (1 + t)λa tn x (k) + t) = Bln,χ (1 (x|λ) . λd n! (1 + t) + 1 a=0
n=0
Then we show that the generating function can be expressed as the multivariate p-adic fermionic integrals on X as follows: 2k
∞
(k) Bln,χ (x|λ)
n=0
···
= X
X
tn n!
χ (x1 ) . . . χ (xk ) (1 + t)λx1 +···+λxk +x dμ−1 (x1 ) · · · dμ−1 (xk )
and that Bln(k) (x|λ) =
n 1 l (k) x l) λ S E (n, (n ≥ 0). 1 l 2k λ l=0
d
χ (a)(−1)a (1+t)a
• Let F (t) = 2 a=1 any k ≥ 0, we see that
∈ θ [[t]] . If β is the associated measure, then, for
(1+t)d +1
Zp
x k dβ (x) = E k,χ .
If the -transform of β is defined by k x dβ (x) = β (s) = lim k
Zp
Z× p
x dβ (x) x = s
x × for x ∈ Z p , ω (x)
D. S. Kim, T. Kim
then we show l p (s, χ) = β (−s). Here l p (s, χ) (s ∈ Z p ) is the p-adic l-function due to Kim which is given by x −s χ (x) dμ−1 (x). l p (s, χ) = X∗
• Assume χ is a Dirichlet character modulo d with d ≡ 1 (mod2) (d ∈ Z>0 ) satisfying d−1 a a=0 (−1) χ (a) = 0. Then we see that either χ is the trivial character modulo 1 or χ is an odd character. Then the following holds: (a) If x is an odd character modulo d, then
n λ l Bln+1,χ (x|λ) = x Bln,χ (x − 1|λ) + S1 (n, l) (λd)l−m 2 m l=0
0≤m≤l m≡l−1 (mod 2)
(2)
× El−m (χ) (x − 1)m . (b) If χ is the trivial character modulo 1, then Bln+1 (x|λ) = x Bln (x − 1|λ) −
l n l λ (2) λl−m El−m (1) (x − 1)m . S1 (n, l) m 4 l=0 m=0
Here, for any k ∈ Z≥0 , and any Dirichlet character χ modulo d, (2)
E k (χ) =
d−1
(2)
(−1)a χ (a) a E k
a=0
a d
,
(2)
where E k (x) are the Euler polynomials of order 2.
2 Generalized Boole numbers and polynomials attached to χ For d ∈ N with d ≡ 1(mod 2), let χ be a Dirichlet character with conductor d. Now, we define the generalized Boole polynomial attached to χ as follows: d−1 ∞ (−1)a χ (a) (1 + t)λa tn Bln,χ (x|λ) (2.1) = (1 + t)x , n! (1 + t)λd + 1 n=0
a=0
1 − p−1
where |t| p < p and λ ∈ Z p with λ = 0. When x = 0, Bln,χ (λ) = Bln,χ (0|λ) are called the generalized Boole numbers attached to χ. By replacing t by et − 1 , we get d−1 ∞ (−1)a χ (a) eλat n x 1 t 2 e −1 =2 e( λ )λt Bln,χ (x|λ) λdt n! e +1 n=0
a=0
=
∞ m=0
λm E m,χ
x tm , λ m!
(2.2)
Generalized Boole numbers and polynomials
and 2
∞ n=0
m ∞ n 1 t tm e −1 =2 Bln,χ (x|λ) Bln,χ (x|λ) S2 (m, n) , n! m! m=0
(2.3)
n=0
where S2 (m, n) is the Stirling number of the second kind. Therefore, by (2.2) and (2.3), we obtain the following theorem. Theorem 2.1 For m ∈ Z≥0 , we have m
Bln,χ (x|λ) S2 (m, n) =
n=0
x λm . E m,χ 2 λ
Let us take f (x) = χ (x) et x . Then, by (1.3), we get d−1 χ (x) et x dμ−1 (x) = 2 edt + 1 (−1)l χ (l) elt . X
(2.4)
l=0
From (1.4) and (2.4), we have X
χ (x) e xt dμ−1 (x) = 2 =
d−1
elt edt + 1
(−1)l χ (l)
l=0 ∞
E n,χ
n=0
tn . n!
(2.5)
Therefore, by (2.5), we get χ (x) x n dμ−1 (x) = E n,χ , χ (y) (x + y)n dμ−1 (y) = E n,χ (x) , X
(2.6)
X
where n ∈ Z≥0 . If we take f (x) = χ (x) (1 + t)λx , then we have d−1 χ (x) (1 + t)λx dμ−1 (x) = 2 (−1)a χ (a) (1 + t)λa . (1 + t)λd + 1 X
(2.7)
a=0
Thus, by (2.7), we get X
χ (x) (1 + t)λx dμ−1 (x) = 2 =2
d−1 a=0 ∞
(−1)a χ (a) Bln,χ (λ)
n=0
(1 + t)λa (1 + t)λd + 1
tn , n!
(2.8)
and X
χ (y) (1 + t)λy+x dμ−1 (y) = 2
d−1
(−1)a χ (a)
a=0
=2
∞ n=0
Bln,χ (x|λ)
(1 + t)λa (1 + t)λd + 1 tn . n!
(1 + t)x (2.9)
D. S. Kim, T. Kim
From (2.9), we note that Bln,χ (x|λ) = = = =
1 2 1 2
χ (y) (x + λy)n dμ−1 (y)
X n
1 2
X
l=0
n 1
2
S1 (n, l)
l=0 n
χ (y) (x + λy)l dμ−1 (y)
S1 (n, l) λ
χ (y)
l
x λ
X
S1 (n, l) λl El,χ
l=0
x λ
+y
l
dμ−1 (y)
,
(2.10)
where S1 (n, l) is the Stirling number of the first kind. Therefore, by (2.10), we obtain the following theorem. Theorem 2.2 For n ≥ 0, we have Bln,χ (x|λ) =
n
S1 (n, l)
l=0
x λl . El,χ 2 λ
Now, we observe that
Bln,χ (x|λ) x + yλ 1 χ (y) = dμ−1 (y) n! 2 X n
n x 1 yλ χ (y) dμ−1 (y) = n−m 2 X m m=0 n Blm,χ (λ) x = m! n−m m=0
n x Bln−m,χ (λ) . = m (n − m)!
(2.11)
m=0
Therefore, by (2.11), we obtain the following theorem. Theorem 2.3 For n ≥ 0, we have
n Bln,χ (x|λ) x Bln−m,χ (λ) . = n! m (n − m)! m=0
From (1.5) and (2.1), we have ∞
tn (1 + t)λa+x = (−1)a χ (a) n! (1 + t)λd + 1 d−1
Bln,χ (x|λ)
n=0
a=0
=
d−1 ∞
(−1)a χ (a) Bln (λa + x|λd)
n=0 a=0
Thus, by (2.12), we get Bln,χ (x|λ) =
d−1 a=0
(−1)a χ (a) Bln (λa + x|λd) .
tn . n!
(2.12)
Generalized Boole numbers and polynomials
Let us consider the Boole polynomials of order k (∈ N) as follows : (k) 2k Bln,χ (x|λ) ··· χ (x1 ) . . . χ (xk ) (λx1 + · · · + λxk + x)n dμ−1 (x1 ) . . . dμ−1 (xk ) . (2.13) = X
X
Thus, by (2.13), we get 2k
∞
tn n!
(k) Bln,χ (x|λ)
n=0
χ (x1 ) . . . χ (xk ) (1 + t)λx1 +···+λxk +x dμ−1 (x1 ) . . . dμ−1 (xk )
···
= X
X
= 2
d−1
(1 + t)λa
(−1) χ (a) a
(1 + t)λd + 1
a=0
= 2k
d−1
···
a1 =0
k (1 + t)x
d−1
χ (a1 ) · · · χ (ak ) λa1 +···+λak +x (−1)a1 +···+ak . k (1 + t) λd (1 + t) + 1 ak =0
(2.14)
From (2.13), we have (k)
2k
Bln,χ (x|λ) n!
λx1 + · · · + λxk + x dμ−1 (x1 ) . . . dμ−1 (xk ) = ··· χ (x1 ) . . . χ (xk ) n X X n n−l k λx1 1 λx2 = ··· χ (xi ) l1 l2 X X l1 =0
i=1
n−l1 −···−lk−2
× ··· ×
λxk−1 lk−1
lk−1 =0
= 2k
n n−l 1 l1 =0 l2 =0
n−l1 −···−lk−2
···
lk−1 =0
l2 =0
λxk + x dμ−1 (x1 ) . . . dμ−1 (xk ) n − l1 − · · · − lk−1
Bll1 ,χ (λ) Bll2 ,χ (λ) l1 !l2 !
Bllk−1 ,χ (λ) Bln−l1 −···−lk−1 ,χ (x|λ) × ··· × . lk−1 ! (n − l1 − · · · − lk−1 )! Therefore, by (2.15), we obtain the following theorem. Theorem 2.4 For n ≥ 0, we have (k) Bln,χ (x|λ) =
n n−l 1 l1 =0 l2 =0
n−l1 −···−lk−2
···
lk−1 =0
n!Bll1 ,χ (λ) Bll2 ,χ (λ) l1 !l2 !
Bllk−1 ,χ (λ) Bln−l1 −···−lk−1 ,χ (x|λ) × ··· × lk−1 ! (n − l1 − · · · − lk−1 )! n−l1 −···−lk−2 n−l n 1 n = ··· l1 , l2 , . . . , lk−1 , n − l1 − · · · − lk−1 l1 =0 l2 =0
lk−1 =0
× Bll1 ,χ (λ) Bll2 ,χ (λ) · · · Bllk−1 ,χ (λ) Bln−l1 −···−lk−1 ,χ (x|λ) .
(2.15)
D. S. Kim, T. Kim
The Boole polynomials of order k (∈ N) are defined by the generating function to be ∞
Bln(k) (x|λ)
n=0
tn = n!
1
k
(1 + t)λ + 1
(1 + t)x .
From (2.14), we can derive the following Eq. (2.16): (k) Bln,χ (x|λ)
=
d−1
···
a1 =0
d−1
(−1)a1 +···+ak χ (a1 ) . . . χ (ak ) Bln(k) (λa1 + · · · + λak + x|λd) . (2.16)
ak =0
Therefore, by (2.16), we obtain the following theorem. Theorem 2.5 For n ≥ 0, we have (k) Bln,χ (x|λ)
=
d−1
d−1
···
a1 =0
(−1)a1 +···+ak χ (a1 ) . . . χ (ak ) Bln(k) (λa1 + · · · + λak + x|λd) .
ak =0
By (2.15), we easily get 2k
(k)
Bln (x|λ) n! ··· =
λx1 + · · · + λxk + x dμ−1 (x1 ) . . . dμ−1 (xk ) n Zp Zp n−l1 −···−lk−2 n n−l 1 λx1 λx2 ··· ··· = l1 l2 Zp Zp l1 =0 l2 =0 lk−1 =0
λxk + x λxk−1 dμ−1 (x1 ) . . . dμ−1 (xk ) × ··· × lk−1 n − l1 − · · · − lk−1 = 2k
n n−l 1
n−l1 −···−lk−2
l1 =0 l2 =0
× ··· ×
···
Bll1 (λ) Bll2 (λ) l1 !l2 !
lk−1 =0
Bllk−1 (λ) Bln−l1 −···−lk−1 (x|λ) . lk−1 ! (n − l1 − · · · − lk−1 )!
(2.17)
Therefore, by (2.17), we obtain the following corollary. Corollary 2.6 For n ≥ 0, we have Bln(k) (x|λ) =
n n−l 1 l1 =0 l2 =0
n−l1 −···−lk−2
···
lk−1 =0
n l1 , l2 , . . . , lk−1 , n − l1 − · · · − lk−1
× Bll1 (λ) Bll2 (λ) · · · Bllk−1 (λ) Bln−l1 −···−lk−1 (x|λ) .
Generalized Boole numbers and polynomials
From (2.15), we have 2k Bln(k) (x|λ) = ··· Zp
=
n
Zp
= =
l=0 n
S1 (n, l)
l=0 n
(λx1 + · · · + λxk + x)n dμ−1 (x1 ) . . . dμ−1 (xk )
Zp
···
S1 (n, l) λl
(λx1 + · · · + λxk + x)l dμ−1 (x1 ) . . . dμ−1 (xk )
Zp
Zp
(k)
S1 (n, l) λl El
···
Zp
x
l=0
λ
x1 + · · · + xk +
x l dμ−1 (x1 ) . . . dμ−1 (xk ) λ
,
(2.18)
(k)
where E n (x) are the Euler polynomials of order k (∈ N). Therefore, by (2.18), we obtain the following theorem. Theorem 2.7 For n ≥ 0, we have Bln(k) (x|λ) =
n 1 l (k) x l) λ . S E (n, 1 l 2k λ l=0
By replacing t by et − 1, we get ∞
Bln(k) (x|λ)
n=0
et − 1 n!
n
k 1 e xt eλt + 1 k
x 2 1 = k e( λ )λt 2 eλt + 1 ∞ 1 m (k) x t m , = k λ Em 2 λ m! =
(2.19)
m=0
and ∞
Bln(k) (x|λ)
n=0
∞ ∞ n 1 t 1 tm e −1 = Bln(k) (x|λ) n! S2 (m, n) n! n! m=n m! n=0 m ∞ tm Bln(k) (x|λ) S2 (m, n) = . m! m=0
(2.20)
n=0
Therefore, by (2.19) and (2.20), we obtain the following theorem. Theorem 2.8 For m ≥ 0, we have m 1 m (k) x (k) = λ E Bln (x|λ) S2 (m, n) . m 2k λ n=0
3 Associated measure x Let ω denote the Teichmüller character mod p. For x ∈ X ∗ , we set x = ω(x) . 1 − p−1 s , x is defined by exp s log p x , for s ∈ Z p . Note that since | x − 1| p < p
D. S. Kim, T. Kim
For s ∈ Z p , Kim defined p-adic l-function which is given by x −s χ (x) dμ−1 (x) , l p (s, χ) = X∗
(see [17]). Thus, we note that
l p −n, χw k = E n,χ − p n χ ( p) E n,χ ,
where 2
d−1 (−1)a χ (a) eat
edt + 1
a=0
=
∞
E n,χ
n=0
tn , n!
(see [17,27]). Let θ be the ring of integers of the finite extension over Q p in C p generated by the values of χ and let be the ring of θ -valued measure on Z p . Then is isomorphic to the ring θ [[t]]; explicitly, if α ∈ , then the power series associated with α is defined by F (t) = (1 + t)x dα (x) Zp
=
∞
x dα (x) t n . Zp n
n=0
Thus, by (3.1), we get
x dα (x) = k
Zp
d (t + 1) dt
k
F (t)
(3.1)
.
(3.2)
t=0
Let Fλ (t) be defined by Fλ (t) = 2
d χ (a) (−1)a (1 + t)λa a=1
(1 + t)λd + 1
.
(3.3)
Then Fλ (t) ∈ θ [[t]]. Hence if βλ is the associated measure, then by (3.2), for any k ≥ 0, we have
d k k x dβλ (x) = (t + 1) Fλ (t) . (3.4) dt Zp t=0
For λ = 1, let us assume that F (t) = 2 =2
d χ (a) (−1)a (1 + t)a a=1 ∞ n=0
(1 + t)d + 1 Bln,χ (1)
tn . n!
(3.5)
Then, from (3.5), we note that F (t) ∈ θ [[t]]. Hence if β = β1 is the associated measure, then by (3.2), for any k ≥ 0, we have
Generalized Boole numbers and polynomials
k d x k dβ (x) = (1 + t) F (t) dt t=0
k d = F ez − 1 dz
Zp
z=0
= E k,χ .
(3.6)
For s ∈ Z p , let k vary through a sequence of positive integers with (i) k → s p-adically (ii) k ≡ 0 mod p − 1 (iii) k → ∞ in the usual sequence. Then the -transform of β is defined by
β (s) = lim k
=
Z× p
Zp
x k dβ (x)
x s dβ (x) ,
(3.7)
x for x ∈ Z× where x = ω(x) p. By (3.7), we get
l p (s, χ) = β (−s) .
4 Umbral calculus approach For the background on umbral calculus, one is referred to [13,28]. Lemma 4.1 (a) For each d ∈ Z>0 , there is a Dirichlet character χ modulo d such that d−1
(−1)a χ (a) = 0.
a=0
d−1 (b) Let d ∈ Z>0 with d ≡ 1 (mod 2). If a=1 (−1)a χ (a) = 0, then either χ is the trivial character modulo 1 or χ is an odd character. d−1 Proof (a) For d = 1, the trivial character χ is the only character and a=0 (−1)a χ (a) = 1. Let d ≥ 2. Then d−1 d−1 a χ (a) = −φ (d) , (−1) χ (a) = (−1)a χ
mod d
a=0
as χ
If
d−1 a=0
mod
a=0
χ
mod d
0, if a = 1 with 0 ≤ a ≤ d − 1, χ (a) = φ , if a = 1. (d) d
(−1)a χ (a) = 0, for all χ, we would have φ (d) = 0, a contradiction.
D. S. Kim, T. Kim
(b) Let d ≥ 3. If χ is the trivial character modulo d, then d−1
(−1)a χ (a) =
a=0
(−1)a =
1≤a≤d−1 (a,d)=1
=
1≤a≤d−1 (a,d)=1
(−1)a + (−1)d
1≤a≤ d−1 2
(−1)a +
(−1)d−a
1≤a≤ d−1 2 (d−a,d)=1
(−1)a = 0.
1≤a≤ d−1 2
(a,d)=1
(a,d)=1
Assume now that χ is a nontrivial even Dirichlet character modulo d. Let A = χ Then A = − (a). 1≤a≤d−1 1≤a≤d−1 χ (a). Also, aeven
A=
aodd
χ (a) =
1≤a≤d−1 aeven
So A = 0 and
d−1
1≤a≤d−1 aeven
a=0
χ (a) = −A.
1≤a≤d−1 aodd
(−1)a χ (a) =
χ (d − a) =
χ (a) −
1≤a≤d−1 aeven
χ (a) = 2 A = 0.
1≤a≤d−1 aodd
Assume from now on till the end of this paper that χ is a Dirichlet character with modulo d−1 d with d ≡ 1 (mod2) (d ∈ Z>0 ) satisfying a=0 (−1)a χ (a) = 0. Then, in view of Lemma 4.1, either χ is the trivial character modulo 1 or χ is an odd character. Under this d−1 (−1)a χ (a)eλt d−1 assumption, the constant term 21 a=0 is nonzero. Thus, (−1)a χ (a) of a=0 eλdt +1 −1 d−1 (−1)a χ (a)eλt g (t) = is as invertible series. We now easily see that Bln,χ ( x| λ) is a=0 eλdt +1
−1 d−1 (−1)a χ (a)eλt t the Sheffer sequence for the pair g (t) = , f (t) = e − 1 . a=0 eλdt +1 Thus,
⎛ ⎞ −1 d−1 a λt (−1) χ (a) e , e t − 1⎠ . Bln,χ ( x| λ) ∼ ⎝ eλdt + 1
(4.1)
a=0
From (1.4) , we observe that g (t)−1 =
∞
tn 1 E n,χ λn . 2 n!
(4.2)
n=0
Theorem 4.2
n n 1 l l−m S1 (n, l) El−m,χ x m Bln,χ ( x| λ) = λ 2 m m=0 l=m ⎞ ⎛ n n n ⎝ = S1 (l, j) Bln−l,χ (λ)⎠ x j l j=0
=
(4.4)
l= j
n n m=0
(4.3)
m
Bn−m,χ (λ) (x)m
(4.5)
Generalized Boole numbers and polynomials
Proof d−1 −1 (−1)a χ (a) eλat eλdt
a=0
Bln,χ (x|λ) ∼ 1, et − 1 ,
+1 (x)n =
n
S1 (n, l) x l ∼ 1, et − 1 .
l=0
Thus, from (4.2), we have Bln,χ (x|λ) =
n
S1 (n, l)
l=0
=
n l=0
=
a=0 ∞
1 S1 (n, l) 2
n 1
2
d−1 (−1)a χ (a) eλat
S1 (n, l)
m=0
l
eλdt + 1 tm E m,χ λ m!
E m,χ λm
m=0
l=0
m
xl
xl
(l)m l−m x m!
n n 1 l l−m λ = S1 (n, l) El−m, χ x m . 2 m m=0
l=m
So we get (4.3). For sn (x) ∼ (g (t) , f (t)), the conjugation formula says sn (x) =
n −1 1 g f (t) f (t) j x n x j , j! j=0
where f (t) is the compositional inverse of f (t) with f f (t) = f ( f (t)) = t. From (4.1), and this formula, we have d−1 n 1 (−1)a χ (a) (1 + t)λa j n Bln,χ (x|λ) = + t)) (log (1 x xj λd j! + t) + 1 (1 a=0 j=0 ∞ n tm 1 j n = Blm,χ (λ) (log (1 + t)) x x j . j! m! m=0
j=0
Here
tm Blm,χ (λ) (log (1 + t)) j x n m! m=0 ∞ t m j n Blm,χ (λ) (log (1 + t)) x = m! m=0 ∞ ∞ t m tl n Blm,χ (λ) j! S1 (l, j) x = m! l! ∞
m=0
= j!
n l= j
l= j
n S1 (l, j) Bln−l, χ (λ) . l
(4.6)
(4.7)
D. S. Kim, T. Kim
Thus, we obtain (4.4).
t i n Bli,χ (y|λ) x Bln,χ (y|λ) = i! i=0 ∞ ti y n = Bli,χ (λ) (1 + t) x i! i=0 ∞ t i y n = Bli,χ (λ) (1 + t) x i! i=0 ∞ ∞ t i y m n Bli,χ (λ) = t x i! m m=0 i=0
n y = . (n)m Bln−m,χ (λ) m
∞
m=0
From this, we get (4.5). For sn (x) ∼ (g (t) , f (t)) and with pn (x) = g (t) sn (x), we have the sheffer identity n n sn (x + y) = s j (x) pn− j (y) . j j=0
Thus, from (4.1), (4.6) and (4.7), we get Bln,χ ( x + y| λ) =
n n
j
j=0
Bl j,χ (x|λ) (y)n− j .
For sn (x) ∼ (g (t) , f (t)), f (t) sn (x) = nsn−1 (x). So Bln,χ ( x + 1| λ) − Bln,χ (x|λ) = et − 1 Bln,χ (x|λ) = n Bln−1,χ (x|λ) . For sn (x) ∼ (g (t) , f (t)) , we have the recursive formula
1 g (t) sn+1 (x) = x − sn (x) . g (t) f (t) Thus, from (4.1), we obtain Bln+1,χ (x|λ) = x Bln,χ (x − 1|λ) − e−t where g (t) =
g (t) Bln,χ (x) , g (t)
d−1 −1 (−1)a χ (a) eλat a=0
eλdt + 1
.
(4.8)
Here d−1 g (t) (−1)a χ (a) eλat g (t) . = (log g (t)) = − g (t) eλdt + 1 a=0
(4.9)
Generalized Boole numbers and polynomials
From (4.6) and (4.7), we get Bln+1,χ (x|λ) = x Bn,χ (x − 1|λ) d−1 (−1)a χ (a) eλat g (t) Bln,χ (x|λ) + e−t eλdt + 1 a=0
= x Bn,χ (x − 1|λ) +
n
S1 (n, l) e−t
d−1 (−1)a χ (a) eλat eλdt + 1
a=0
l=0
xl .
(4.10)
It is easy to see that d−1 (−1)a χ (a) eλat eλdt + 1
a=0
=
χ (−1) e−λdt + 1
+
2
1 eλdt
+1
d
(−1)a χ (a) λae−λat
a=1
2
d−1
(−1)a χ (a) λaeλat .
(4.11)
a=0
Assume now that χ is an odd character with modulo d with d ≡ 1 (mod 2). Then χ (d) = 0. Under this assumption, (4.11) becomes d−1 (−1)a χ (a) eλat eλdt + 1 a=0
=− +
a (−λdt)m λ E m(2) (−1)a χ (a) a 4 d m! λ 4
d−1
∞
a=0
m=0
d−1
(−1)a χ (a) a
a=0
∞
E m(2)
a (λdt)m d
m=0
m!
.
(2)
Here E m (x) is the Euler polynomial of order 2 whose generating function is given by
2 ∞ 2 tn xt e = E n(2) (x) . (4.12) t e +1 n! n=0
Thus, we have d−1 (−1)a χ (a) eλat a=0
eλdt + 1
xl = −
l λ l (2) (−λd)l−m El−m (χ) x m m 4 m=0
+
l λ l (2) (λd)l−m El−m (χ) x m , 4 m m=0
where, for any k ∈ Z≥0 , and any Dirichlet character χ modulo d, we define (2)
E k (χ) =
d−1 a=0
(2)
(−1)a χ (a) a E k
a d
.
(4.13)
D. S. Kim, T. Kim
Finally, from (4.10) and (4.13), we obtain Bln+1,χ (x|λ) = x Bln,χ ( x − 1| λ) +
λ S1 (n, l) 2 n
l=0
0≤m≤l m≡l−1 (mod2)
l (λd)l−m m
(2)
× El−m (χ) (x − 1)m .
(4.14)
If χ is the trivial character modulo 1, then Bln (x|λ) ∼ g (t) = eλt + 1, f (t) = et − 1 . Proceeding similarly to the above, one easily gets Bln+1 (x|λ)
l n l λ λl−m S1 (n, l) m 4
= x Bln ( x − 1| λ) −
l=0 m=0
×
(2) El−m
(1) (x − 1) . m
(4.15)
Thus, from (4.14) and (4.15), we obtain the following theorem. Theorem 4.3 For any k ∈ Z≥0 , and any Dirichlet character χ modulo d, we let (2)
E k (χ) =
d−1
(2)
(−1)a χ (a) a E k
a=0
a d
.
Assume d−1 thata χ is a Dirichlet character with modulo d with d ≡ 1 (mod2) satisfying a=0 (−1) χ (a) = 0. Then we have the following. (a) If x is an odd character modulo d, then
n λ l S1 (n, l) Bln+1,χ (x|λ) = x Bln,χ (x − 1|λ) + (λd)l−m 2 m l=0
0≤m≤l m≡l−1(mod2)
(2)
× El−m (χ) (x − 1)m . (b) If χ is the trivial character modulo 1, then Bln+1 (x|λ) = x Bln (x − 1|λ) −
l n λ l (2) S1 (n, l) λl−m El−m (1) (x − 1)m . 4 m l=0 m=0
Acknowledgments University in 2014.
The work reported in this paper was conducted during the sabbatical year of Kwangwoon
References 1. Araci, S., Acikgoz, M.: A note on the Frobenius-Euler numbers and polynomials associated with Bernstein polynomials. Adv. Stud. Contemp. Math. 22(3), 399–406 (2012)
Generalized Boole numbers and polynomials 2. Bayad, A., Chikhi, J.: Apostol-Euler polynomials and asymptotics for negative binomial reciprocals. Adv. Stud. Contemp. Math. 24(1), 33–37 (2014) 3. Boas, R.P., Buck, R.C.: Polynomial expansions of analytic functions, Springer (1958) 4. Boole, G.: Calculus of Finite Differences, Boole Press (2007) 5. Can, M., Cenkci, M., Kurt, V., Simsek, Y.: Twisted Dedekind type sums associated with Barnes’ type multiple Frobenius-Euler l -functions. Adv. Stud. Contemp. Math. 18(2), 135–160 (2009) 6. Comtet, L.: Advanced combinatorics. D. Reidel Publishing Co., Dordrecht (1974) 7. Dere, R., Simsek, Y.: Applications of umbral algebra to some special polynomials. Adv. Stud. Contemp. Math. 22(3), 433–438 (2012) 8. Dolgy, D.V., Kim, T., Lee, B., Lee, S.-H.: Some new identities on the twisted Bernoulli and Euler polynomials. J. Comput. Anal. Appl. 15(3), 441–451 (2013) 9. Hwang, K.-W., Dolgy, D.V., Kim, D.S., Kim, T., Kee, S.H.: Some theorems on Bernoulli and Euler numbers. Ars Combin. 109, 285–297 (2013) 10. Jeong, J.-H., Park, J.-W., Rim, S.-H.: New approach to the analogue of Lebesgue-Radon-Nikodym theorem with respect to weighted p-adic q-measure on Z p . J. Comput. Anal. Appl. 15, 1310–1316 (2013) 11. Jordan, C.: Calculus of finite differences (AMS Chelsea Publishing), 3 edn., American Mathematical Society, 1 (1965) 12. Kim, D.S., Kim, T.: Daehee numbers and polynomials. Appl. Math. Sci. 7(120), 5969–5976 (2013) 13. Kim, D.S., Kim, T.: Higher-order Cauchy of the first kind and poly-Cauchy of the first kind mixed type polynomials. Adv. Stud. Contemp. Math. 33, 621–636 (2013) 14. Kim, D.S., Kim, T., Rim, S.H., Dolgy, D.V.: Some identities of higher-order Bernoulli, Euler, and Hermite polynomials arising from umbral calculus. J. Inequal. Appl. 2013(211), 10 (2013) 15. Kim T.: A note on Boole polynomials (Communicated) 16. Kim, T.: q-Volkenborn integration. Russ. J. Math. Phys. 9(3), 288–299 (2002) 17. Kim, T.: On p-adic q-l-functions and sums of powers. J. Math. Anal. Appl. 329(2), 1472–1481 (2007) 18. Kim, T.: On p-adic interpolating function for q-Euler numbers and its derivatives. J. Math. Anal. Appl. 339(1), 598–608 (2008) 19. Kim, T.: New approach to q -Euler polynomials of higher order. Russ. J. Math. Phys. 17(2), 218–225 (2010) 20. Kim, T.: An identity of the symmetry for the Frobenius-Euler polynomials associated with the fermionic p-adic invariant q-integrals on Z p . Rocky Mt. J. Math. 41(1), 239–247 (2011) 21. Kim, T.: On a p -adic interpolation function for the multiple generalized Euler numbers and its derivatives. J. Comput. Anal. Appl. 14(3), 410–416 (2012) 22. Kim, T., Adiga, C.: Sums of products of generalized Bernoulli numbers. Int. Math. J. 5(5), 1–7 (2004) 23. Kim, T., Kim, D.S., Dolgy, D.V., Rim, S.H.: Some identities on the Euler numbers arising from Euler basis polynomials. Ars Combin. 109, 433–446 (2013) 24. Kim, T., Kim, D.S., Toufik, M., Rim, S.-H., Schork, M.: Umbral calculus and Sheffer sequences of polynomials. J. Math. Phys. 54(8), 083504 (2015) 25. Kwon, J., Park, J.-W.: A note on (h, q)-Boole polynomials. Adv. Differ. Equ. 2015(198), 11 (2015) 26. Mansour, T., Schork, M., Shattuck, M.: On the Stirling numbers associated with the meromorphic Weyl algebra. Appl. Math. Lett. 25(11), 1767–1771 (2012) 27. Rim, S.-H., Kim, T.: Explicit p-adic expansion for alternating sums of powers. Adv. Stud. Contemp. Math. 14(2), 241–250 (2007) 28. Roman, S.: The umbral calculus. Dover Publ. Inc, New York (2005) 29. Rotam, G.-C., Taylor, B.D.: The classical umbral calculus. SIAM J. Math. Anal. 25, 694–711 (1994) 30. Shiratani, K.: On Euler numbers. Mem. Fac. Sci. Kyushu Univ. Ser. A 27, 1–5 (1973) 31. Shiratani, K., Yamamoto, S.: On a p-adic interpolation function for the Euler numbers and its derivatives. Mem. Fac. Sci. Kyushu Univ. Ser. A 39(1), 113–125 (1985) 32. Simsek, Y.: On twisted generalized Euler numbers. Bull. Korean Math. Soc. 41(2), 299–306 (2004) 33. Simsek, Y.: Generating functions of the twisted Bernoulli numbers and polynomials associated with their interpolation functions. Adv. Stud. Contemp. Math. 16(2), 251–278 (2008) 34. Simsek, Y.: Complete sum of products of (h, q) -extension of Euler polynomials and numbers. J. Differ. Equ. Appl. 16(11), 1331–1348 (2010) 35. Sinnott, W.: On the μ-invariant of the -transform of a rational function. Invent. Math. 75(2), 273–282 (1984) 36. Washington, L.C.: Introduction to cyclotomic fields, Graduate Texts in Mathematics, vol. 83. SpringerVerlag, New York (1982) 37. Wei, C.-F., Qi, F.: Several closed expressions for the Euler numbers. J. Inequal. Appl. 2015(219), 8 (2015) 38. Zhang, Z., Yang, H.: Some closed formulas for generalized Bernoulli-Euler numbers and polynomials. Proc. Jangjeon Math. Soc. 11(2), 191–198 (2008)
