A note on Boole polynomials

A NOTE ON BOOLE POLYNOMIALS

arXiv:1310.8025v1 [math.NT] 30 Oct 2013

DAE SAN KIM AND TAEKYUN KIM Abstract. Boole polynomials play an important role in the area of number theory, algebra and umbral calculus. In this paper, we investigate some properties of Boole polynomials and consider Witt-type formulas for the Boole numbers and polynomials. Finally, we derive some new identities of those polynomials from the Witt-type formulas which are related to Euler polynomials.

1. Introduction Let p be a fixed prime number. Throughout this paper, Zp , Qp and Cp will denote the ring of p-adic integers, the field of p-adic numbers and the completion of algebraic closure of Qp . The p-adic norm |·|p is normalized as |p|p = 1p . Let C (Zp ) be the space of continuous functions on Zp . For f ∈ C (Zp ), the fermionic p-adic integral on Zp is defined by Kim to be (1)

I (f ) =

ˆ

f (x) dµ (x) = lim

N →∞

Zp

N pX −1

f (x) (−1)x ,

x=0

(see [9]). From (1), we can derive (2)

I (f1 ) = −I (f ) + 2f (0) ,

(see [1,2,4-8,12,14-16]). Here f1 (x) = f (x + 1). For k ∈ N, the Euler polynomials of order k are defined by the generating function to be k ∞ X tn 2 xt (k) e = E (x) (3) , n et + 1 n! n=0 (see [4, 6, 9, 16]). (k) (k) When x = 0, En = En (0) are called the Euler numbers of order k. (1) In particular, for k = 1, En (x) = En (x) are called the Euler polynomials. The Stirling numbers of the first kind are defined by (4)

(x)n = x (x − 1) · · · (x − n + 1) =

n X

S1 (n, l) xl ,

l=0

(see [13]). As is well known, the Stirling numbers of the second kind are given by the generating function to be (5)

et − 1

n

= n!

∞ X l=n

(see [13]). 1

S2 (l, n)

tl , l!

2

DAE SAN KIM AND TAEKYUN KIM

The Boole polynomials are defined by the generating function to be ∞ X

(6)

Bln (x|λ)

n=0

1 tn (1 + t)x , = λ n! 1 + (1 + t)

(see [3, 10, 11, 13]). When λ = 1, 2Bln (x|1) = Chn (x) are Changhee polynomials which are defined by ∞ X tn 2 x Chn (x) = (1 + t) , n! t + 2 n=0

(see [7]). In this paper, we investigate some properties of Boole polynomials and consider Witt-type formulas for the Boole numbers and polynomials. Finally, we derive some new identities of those polynomials from the Witt-type formulas which are related to Euler polynomials. 2. A Note on Boole Polynomials 1

In this section, we assume that t ∈ Cp with |t|p < p− p−1 and λ ∈ Zp . Let us λx

take f (x) = (1 + t) . From (2), we have ˆ x+λy (1 + t) dµ−1 (y) = (7)

2

Zp

= It is easy to show that ˆ ∞ ˆ X x+λy (1 + t) dµ−1 (y) = (8) Zp

(1 + t)

1 + (1 + t) ∞ X tn 2Bln (x|λ) . n! n=0

Zp

n=0

x

λ

(x + yλ)n dµ−1 (y)

tn . n!

Therefore, by (7) and (8), we obtain the following theorem.

Theorem 1. For n ≥ 0, we have ˆ (x + yλ)n dµ−1 (y) = 2Bln (x|λ) . Zp

By (6), we get ∞ X

(9)

2Bln (x|λ)

n=0

and (10)

∞ X

n=0

2Bln (x|λ)

∞ n x X (et − 1) tm 2 λm , = λt ext = Em n! e +1 λ m! m=0

n 1 t e −1 n!

= =

∞ X

n=0 ∞ X

∞ X 1 tm n! S2 (m, n) n! m=n m! ! ∞ X tm 2 Bln (x|λ) S2 (m, n) . m! n=0

2Bln (x|λ)

m=0

Therefore, by (9) and (10), we obtain the following theorem. Theorem 2. For m ≥ 0, m X

n=0

Bln (λ|x) S2 (m, n) =

x 1 λm . Em 2 λ


Remark. From Theorem 1, we note that ˆ n X 2Bln (x|λ) = S1 (n, l) =

l

(x + yλ) dµ−1 (y)

Zp

l=0

∞ X

3

S1 (n, l) λl El

l=0

x λ

.

Let us consider the Boole polynomials of order k(∈ N) as follows : ˆ ˆ (λx1 + · · · + λxk + x)n dµ−1 (x1 ) · · · dµ−1 (xk ) . ··· (11) 2k Bln(k) (x|λ) = Zp

Zp

Thus, by (11), we get 2k Bln(k) (x|λ) =

(12)

n X

(k)

S1 (n, l) λl El

l=0

x λ

.

(k)

From (11), we can derive the generating function of Bln (x) as follows : ∞ X tn 2k Bln(k) (x|λ) (13) n! n=0 ˆ ˆ λx +···+λxk +x (1 + t) 1 dµ−1 (x1 ) · · · dµ−1 (xk ) ··· = Zp

=

Zp

2 λ

1 + (1 + t)

!k

x

(1 + t) .

By replacing t by et − 1, we get ∞ X n 1 t (14) e −1 2k Bln(k) (x|λ) n! n=0

= =

2 λt e +1

∞ X

k

(k) λm Em

m=0

and (15)

∞ X

2

k

Bln(k)

n=0

ext x tm λ

m!

,

! m ∞ n X X (et − 1) tm k (k) (x|λ) 2 Bln (x|λ) S2 (m, n) = . n! m! m=0 n=0

Therefore, by (14) and (15), we obtain the following theorem. Theorem 3. For m ≥ 0, we have m X λm (k) x Bln(k) (x|λ) S2 (m, n) = k Em . 2 λ n=0 For n ≥ 0, the rising factorial sequence is defined by

(16)

x(n)

n

= x (x + 1) · · · (x + n − 1) = (−1) (−x)n n X n l = x, l l=0

n n−l where = (−1) S1 (n, l) . l Now, we define the Boole polynomials of the second kind as follows: ˆ ˆ n (x|λ) = 1 (−λy + x)n dµ−1 (y) , (n ≥ 0) . (17) Bl 2 Zp

4


Then, by (17), we get n

(18)

ˆ n (x|λ) Bl

=

1X S1 (n, l) (−1)l λl 2

l x − + y dµ−1 (y) λ Zp

ˆ

l=0

=

n 1X

2

l=0

x l . S1 (n, l) (−1) λl El − λ

ˆ n (λ) = Bl ˆ n (0|λ) are called the Boole numbers of the second When x = 0, Bl kind. ˆ n (x|λ) is given by The generating function of Bl ∞ X

n

ˆ n (x|λ) t Bl n! n=0

(19)

1 2

=

ˆ

−λy+x

(1 + t)

dµ−1 (y)

Zp λ

(1 + t)

=

x

λ

1 + (1 + t)

(1 + t) .

By replacing t by et − 1, we get ∞ X n λm 1 t λ + x tm ˆ Bln (x|λ) e −1 = Em , n! 2 λ m! m=0 n=0 ∞ X

(20) and (21)

! m ∞ X X n tm 1 t ˆ ˆ Bln (x|λ) S2 (m, n) Bln (x|λ) e −1 = . n! m! m=0 n=0 n=0 ∞ X

Therefore, by (20) and (21), we obtain the following theorem. Theorem 4. For m ≥ 0, we have λm Em 2

λ+x λ

=

m X

ˆ n (x|λ) S2 (m, n) , Bl

n=0

and ˆ m (x|λ) = Bl

m X

l

S1 (m, l) (−1)

l=0

λl x . El − 2 λ

For k ∈ N, let us consider the Boole polynomials of the second kind with order k as follows : (22)

ˆ (k) (x|λ) Bl n ˆ ˆ 1 = k (− (λx1 + · · · + λxk ) + x)n dµ−1 (x1 ) · · · dµ−1 (xk ) . ··· 2 Zp Zp

Thus, by (22), we get

(23)

ˆ n(k) (x|λ) = 2k Bl

n X l=0

l

(k)

S1 (n, l) λl (−1) El

x . − λ


5

ˆ n(k) (x|λ) is given by The generating function of Bl (24)

∞ X

n

ˆ n(k) (x|λ) t Bl n! n=0 ˆ ˆ 1 −(λx1 +···+λxk )+x (1 + t) dµ−1 (x1 ) · · · dµ−1 (xk ) ··· = k 2 Zp Zp !k (1 + t)λ x = (1 + t) . 1 + (1 + t)λ

By replacing t by et − 1, we get (25)

∞ X

ˆ (k) (x|λ) 1 et − 1 n Bl n n! n=0

= =

and (26)

1 2k

2 eλt + 1

k

e(λk+x)t

∞ X x tm λm (k) E k + m 2k λ m! m=0

! m ∞ X X n 1 t tm (k) (k) ˆ ˆ Bln (x|λ) Bln (x|λ) S2 (m, n) e −1 = . n! m! n=0 m=0 n=0 ∞ X

Therefore, by (25) and (26), we obtain the following theorem. Theorem 5. For m ≥ 0, we have m x l X (k) l λ ˆ (k) (x|λ) = Bl S (m, l) (−1) E − 1 m 2k l λ l=0

and

m x X ˆ (k) λm (k) Bln (x|λ) S2 (m, n) . E k + = 2k m λ n=0

Now, we observe that (27)

Bln (x|λ) (−1) n! n

= = = = =

and (28)

n

(−1)

ˆ n (x|λ) Bl n!

x + yλ dµ−1 (y) (−1) n Zp ˆ −yλ − x + n − 1 dµ−1 (y) n Zp ˆ n X n−1 −yλ − x dµ−1 (y) n − m Zp m m=0 ˆ n−1 n X −yλ − x m−1 dµ−1 (y) m! m! m Zp m=1 ˆ n X n − 1 Bl m (−x|λ) , m! m − 1 m=1 n

ˆ

ˆ n X n−1 −x + yλ dµ−1 (y) m − 1 Zp m m=0 ˆ n X n − 1 Bl m (−x|λ) = . m − 1 m! m=1

=

Therefore, by (27) and (28), we obtain the following theorem.

6


Theorem 6. For n ≥ 1, we have n

(−1) and

ˆ n X Bln (x|λ) n − 1 Bl m (−x|λ) = m−1 n! m! m=1

n X ˆ n (x|λ) n − 1 Blm (−x|λ) Bl = . (−1) m−1 n! n! m=1 n

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Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea E-mail address : [email protected] Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea E-mail address : [email protected]