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 [email protected] Dae San Kim [email protected]

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

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