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Some results for the q-Bernoulli, q-Euler numbers and polynomials Advances in Difference Equations 2011, 2011:68 doi:10.1186/1687-1847-2011-68 Daeyeoul Kim daeyeoul@nims.re.kr Min-Soo Ki

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Some results for the q-Bernoulli, q-Euler numbers and polynomials

Advances in Difference Equations 2011, 2011:68 doi:10.1186/1687-1847-2011-68

Daeyeoul Kim (daeyeoul@nims.re.kr) Min-Soo Kim (minsookim@kaist.ac.kr)

ISSN 1687-1847

Article type Research

Submission date 2 September 2011

Acceptance date 23 December 2011

Publication date 23 December 2011

Article URL http://www.advancesindifferenceequations.com/content/2011/1/68

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which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Daeyeoul Kim 1 and Min-Soo Kim∗2

1 National Institute for Mathematical Sciences, Doryong-dong, Yuseong-gu

Daejeon 305-340, South Korea

2 Department of Mathematics, KAIST, 373-1 Guseong-dong, Yuseong-gu,

Daejeon 305-701, South Korea

∗Corresponding author: minsookim@kaist.ac.kr

Email address:

DK: daeyeoul@nims.re.kr

Abstract The q-analogues of many well known formulas are derived by

us-ing several results of Bernoulli, Euler numbers and polynomials The

analogues of ζ-type functions are given by using generating functions of

q-Bernoulli, q-Euler numbers and polynomials Finally, their values at

non-positive integers are also been computed.

2010 Mathematics Subject Classification: 11B68; 11S40; 11S80.

Keywords: Bosonic p-adic integrals; Fermionic p-adic integrals; q-Bernoulli

polynomials; q-Euler polynomials; generating functions; q-analogues of ζ-type

functions; q-analogues of the Dirichlet’s L-functions.

1 Introduction

Carlitz [1,2] introduced q-analogues of the Bernoulli numbers and polynomials.

From that time on these and other related subjects have been studied by various

authors (see, e.g., [3–10]) Many recent studies on q-analogue of the Bernoulli,

Euler numbers, and polynomials can be found in Choi et al [11], Kamano [3], Kim [5,6,12], Luo [7], Satoh [9], Simsek [13,14] and Tsumura [10]

For a fixed prime p, Z p , Q p, and Cp denote the ring of p-adic integers, the field

of p-adic numbers, and the completion of the algebraic closure of Q p , respectively Let | · | p be the p-adic norm on Q with |p| p = p −1 For convenience, | · | p will also

be used to denote the extended valuation on Cp

The Bernoulli polynomials, denoted by B n (x), are defined as

(1.1) B n (x) =

n

X

k=0

µ

n k

¶

B k x n−k , n ≥ 0, where B k are the Bernoulli numbers given by the coefficients in the power series

e t − 1 =

∞

X

k=0

B k

t k

k! . From the above definition, we see B k’s are all rational numbers Since t

e t −1 − 1 + t

2

is an even function (i.e., invariant under x 7→ −x), we see that B k = 0 for any odd

integer k not smaller than 3 It is well known that the Bernoulli numbers can also

1

be expressed as follows

N →∞

1

p N

pXN −1 a=0

a k

(see [15,16]) Notice that, from the definition B k ∈ Q, and these integrals are independent of the prime p which used to compute them The examples of (1.3)

are:

(1.4)

lim

N →∞

1

p N

pXN −1 a=0

a = lim

N →∞

1

p N

p N (p N − 1)

1

2 = B1,

lim

N →∞

1

p N

pXN −1 a=0

a2= lim

N →∞

1

p N

p N (p N − 1)(2p N − 1)

1

6 = B2.

Euler numbers E k , k ≥ 0 are integers given by (cf [17–19])

(1.5) E0= 1, E k = −

k−1

X

i=0

2|k−i

µ

k i

¶

E i for k = 1, 2,

The Euler polynomial E k (x) is defined by (see [20, p 25]):

(1.6) E k (x) =

k

X

i=0

µ

k i

¶

E i

2i

µ

x −1

2

¶k−i

,

which holds for all nonnegative integers k and all real x, and which was obtained by Raabe [21] in 1851 Setting x = 1/2 and normalizing by 2 k gives the Euler numbers

µ 1 2

¶

, where E0= 1, E2= −1, E4= 5, E6= −61, Therefore, E k 6= E k (0), in fact ([19,

p 374 (2.1)])

k + 1 (1 − 2

k+1 )B k+1 , where B kare Bernoulli numbers The Euler numbers and polynomials (so-named by Scherk in 1825) appear in Euler’s famous book, Institutiones Calculi Differentialis (1755, pp 487–491 and p 522)

In this article, we derive q-analogues of many well known formulas by using sev-eral results of q-Bernoulli, q-Euler numbers, and polynomials By using generating functions of q-Bernoulli, q-Euler numbers, and polynomials, we also present the q-analogues of ζ-type functions Finally, we compute their values at non-positive

integers

This article is organized as follows

In Section 2, we recall definitions and some properties for the q-Bernoulli, Euler numbers, and polynomials related to the bosonic and the fermionic p-adic integral

on Zp

In Section 3, we obtain the generating functions of the q-Bernoulli, q-Euler num-bers, and polynomials We shall provide some basic formulas for the q-Bernoulli and q-Euler polynomials which will be used to prove the main results of this article.

In Section 4, we construct the q-analogue of the Riemann’s ζ-functions, the Hurwitz ζ-functions, and the Dirichlet’s L-functions We prove that the value of their functions at non-positive integers can be represented by the Bernoulli,

q-Euler numbers, and polynomials

2 q-Bernoulli, q-Euler numbers and polynomials related to the Bosonic

and the Fermionic p-adic integral on Z p

In this section, we provide some basic formulas for p-adic q-Bernoulli, p-adic q-Euler numbers and polynomials which will be used to prove the main results of

this article

Let U D(Z p , C p) denote the space of all uniformly (or strictly) differentiable Cp -valued functions on Zp The p-adic q-integral of a function f ∈ U D(Z p) on Zp is defined by

(2.1) I q (f ) = lim

N →∞

1

[p N]q

pXN −1 a=0

f (a)q a =

Z

Zp

f (z)dµ q (z), where [x] q = (1 − q x )/(1 − q), and the limit taken in the p-adic sense Note that

q→1 [x] q = x for x ∈ Z p , where q tends to 1 in the region 0 < |q − 1| p < 1 (cf [22,5,12]) The bosonic p-adic integral on Z p is considered as the limit q → 1, i.e.,

(2.3) I1(f ) = lim

N →∞

1

p N

pXN −1 a=0

f (a) =

Z

Zp

f (z)dµ1(z).

From (2.1), we have the fermionic p-adic integral on Z p as follows:

(2.4) I −1 (f ) = lim

q→−1 I q (f ) = lim

N →∞

pXN −1 a=0

f (a)(−1) a =

Z

Zp

f (z)dµ −1 (z).

In particular, setting f (z) = [z] k

q in (2.3) and f (z) =£z +1

2

¤k

q in (2.4), respectively,

we get the following formulas for the p-adic q-Bernoulli and p-adic q-Euler numbers, respectively, if q ∈ C p with 0 < |q − 1| p < 1 as follows

(2.5) B k (q) =

Z

Zp

[z] k

q dµ1(z) = lim

N →∞

1

p N

pXN −1 a=0

[a] k

q ,

(2.6) E k (q) = 2 k

Z

Zp

·

z +1

2

¸k

q

dµ −1 (z) = 2 k lim

N →∞

pXN −1 a=0

·

a +1

2

¸k

q

(−1) a

Remark 2.1 The q-Bernoulli numbers (2.5) are first defined by Kamano [3] In (2.5) and (2.6), take q → 1 Form (2.2), it is easy to that (see [17, Theorem 2.5])

B k (q) → B k=

Z

Z

z k dµ1(z), E k (q) → E k=

Z

Z

(2z + 1) k dµ −1 (z).

For |q − 1| p < 1 and z ∈ Z p , we have

(2.7) q iz=

∞

X

n=0

(q i − 1) n

µ

z n

¶

and |q i − 1| p ≤ |q − 1| p < 1, where i ∈ Z We easily see that if |q − 1| p < 1, then q x = 1 for x 6= 0 if and only if

q is a root of unity of order p N and x ∈ p NZp (see [16])

By (2.3) and (2.7), we obtain

(2.8)

I1(q iz) = 1

q i − 1 N →∞lim

(q i)p N

− 1

p N

= 1

q i − 1 N →∞lim

1

p N

( ∞ X

m=0

µ

p N

m

¶

(q i − 1) m − 1

)

= 1

q i − 1 N →∞lim

1

p N

∞

X

m=1

µ

p N

m

¶

(q i − 1) m

= 1

q i − 1 N →∞lim

∞

X

m=1

1

m

µ

p N − 1

m − 1

¶

(q i − 1) m

= 1

q i − 1

∞

X

m=1

1

m

µ

−1

m − 1

¶

(q i − 1) m

= 1

q i − 1

∞

X

m=1

(−1) m−1 (q i − 1) m

m

= i log q

q i − 1 since the series log(1 + x) =P∞ m=1 (−1) m−1 x m /m converges at |x| p < 1 Similarly,

by (2.4), we obtain (see [4, p 4, (2.10)])

(2.9) I −1 (q iz) = lim

N →∞

pXN −1 a=0

(q i)a (−1) a= 2

q i+ 1.

From (2.5), (2.6), (2.8) and (2.9), we obtain the following explicit formulas of B k (q) and E k (q):

(2.10) B k (q) = log q

(1 − q) k

k

X

i=0

µ

k i

¶

(−1) i i

q i − 1 ,

(2.11) E k (q) = 2k+1

(1 − q) k

k

X

i=0

µ

k i

¶

(−1) i q1i 1

q i+ 1,

where k ≥ 0 and log is the p-adic logarithm Note that in (2.10), the term with

i = 0 is understood to be 1/log q (the limiting value of the summand in the limit

i → 0).

We now move on to the p-adic q-Bernoulli and p-adic q-Euler polynomials The p-adic q-Bernoulli and p-adic q-Euler polynomials in q xare defined by means of the

bosonic and the fermionic p-adic integral on Z p:

(2.12) B k (x, q) =

Z

Z

[x + z] k q dµ1(z) and E k (x, q) =

Z

Z

[x + z] k q dµ −1 (z),

where q ∈ C p with 0 < |q − 1| p < 1 and x ∈ Z p , respectively We will rewrite the

above equations in a slightly different way By (2.5), (2.6), and (2.12), after some elementary calculations, we get

(2.13) B k (x, q) =

k

X

i=0

µ

k i

¶

[x] k−i

q q ix B i (q) = (q x B(q) + [x] q)k

and

(2.14)

E k (x, q) =

k

X

i=0

µ

k i

¶

E i (q)

2i

·

x − 1

2

¸k−i

q

q i(x−1 )=

Ã

q x−1

2 E(q) +

·

x −1

2

¸

q

!k

,

where the symbol B k (q) and E k (q) are interpreted to mean that (B(q)) k and

(E(q)) k must be replaced by B k (q) and E k (q) when we expanded the one on the right, respectively, since [x + y] k

q = ([x] q + q x [y] q)k and

(2.15)

[x + z] k q =

· 1 2

¸k

q

Ã

[2x − 1] q1 + q x−1

· 1 2

¸−1

q

·

z +1

2

¸

q

!k

=

· 1 2

¸k

q

k

X

i=0

µ

k i

¶

[2x − 1] k−i q q (x−1)i

· 1 2

¸−i

q

·

z +1

2

¸i

q

(cf [4,5]) The above formulas can be found in [7] which are the q-analogues of the corresponding classical formulas in [17, (1.2)] and [23], etc Obviously, put x = 1

2

in (2.14) Then

(2.16) E k (q) = 2 k E k

µ 1

2, q

¶

6= E k (0, q) and lim

q→1 E k (q) = E k ,

where E k are Euler numbers (see (1.5) above)

Lemma 2.2 (Addition theorem)

B k (x + y, q) =

k

X

i=0

µ

k i

¶

q iy B i (x, q)[y] k−i q (k ≥ 0),

E k (x + y, q) =

k

X

i=0

µ

k i

¶

q iy E i (x, q)[y] k−i

q (k ≥ 0).

Proof Applying the relationship [x + y − 1

2]q = [y] q + q y [x − 1

2]q to (2.14) for

x 7→ x + y, we have

E k (x + y, q) =

Ã

q x+y−1

2 E(q) +

·

x + y −1

2

¸

q

!k

=

Ã

q y

Ã

q x−1

2 E(q) +

·

x −1

2

¸

q

!

+ [y] q

!k

=

k

X

i=0

µ

k i

¶

q iy

Ã

q x−1

2 E(q) +

·

x − 1

2

¸

q

!i

[y] k−i q

=

k

X

i=0

µ

k i

¶

q iy E i (x, q)[y] k−i

q

Similarly, the first identity follows ¤

Remark 2.3. From (2.12), we obtain the not completely trivial identities

lim

q→1 B k (x + y, q) =

k

X

i=0

µ

k i

¶

B i (x)y k−i = (B(x) + y) k ,

lim

q→1 E k (x + y, q) =

k

X

i=0

µ

k i

¶

E i (x)y k−i = (E(x) + y) k ,

where q ∈ C p tends to 1 in |q − 1| p < 1 Here B i (x) and E i (x) denote the classical

Bernoulli and Euler polynomials, see [17,15] and see also the references cited in each of these earlier works

Lemma 2.4 Let n be any positive integer Then

k

X

i=0

µ

k i

¶

q i [n] i

q B i (x, q n ) = [n] k

q B k

µ

x + 1

n , q

n

¶

,

k

X

i=0

µ

k i

¶

q i [n] i q E i (x, q n ) = [n] k q E k

µ

x + 1

n , q

n

¶

.

Proof. Use Lemma 2.2, the proof can be obtained by the similar way to [7, Lemma

We note here that similar expressions to those of Lemma 2.4 are given by Luo

[7, Lemma 2.3] Obviously, Lemma 2.4 are the q-analogues of

k

X

i=0

µ

k

i

¶

n i B i (x) = n k B k

µ

x + 1 n

¶

,

k

X

i=0

µ

k i

¶

n i E i (x) = n k E k

µ

x + 1 n

¶

,

respectively

We can now obtain the multiplication formulas by using p-adic integrals.

From (2.3), we see that

(2.17)

B k (nx, q) =

Z

Zp

[nx + z] k

q dµ1(z)

= lim

N →∞

1

np N

npXN −1 a=0

[nx + a] k

q

= 1

n N →∞lim

1

p N

n−1

X

i=0

pXN −1 a=0

[nx + na + i] k

q

=[n]

k q

n

n−1X

i=0

Z

Zp

·

x + i

n + z

¸k

q n

dµ1(z)

is equivalent to

(2.18) B k (x, q) = [n]

k q

n

n−1X

i=0

B k

µ

x + i

n , q

n

¶

.

If we put x = 0 in (2.18) and use (2.13), we find easily that

(2.19)

B k (q) = [n]

k q

n

n−1

X

i=0

B k

µ

i

n , q

n

¶

= [n]

k q

n

n−1

X

i=0

k

X

j=0

µ

k j

¶ ·

i n

¸k−j

q n

q ij B j (q n)

= 1

n

k

X

j=0

[n] j q

µ

k j

¶

B j (q n)

n−1

X

i=0

q ij [i] k−j

q Obviously, Equation (2.19) is the q-analogue of

B k = 1

n(1 − n k)

k−1

X

j=0

n j

µ

k j

¶

B j n−1X

i=1

i k−j ,

which is true for any positive integer k and any positive integer n > 1 (see [24,

(2)])

From (2.4), we see that

(2.20)

E k (nx, q) =

Z

Zp

[nx + z] k

q dµ −1 (z)

= lim

N →∞

n−1X

i=0

pXN −1 a=0

[nx + na + i] k

q (−1) na+i

= [n] k q n−1X

i=0

(−1) i

Z

Zp

·

x + i

n + z

¸k

q n

dµ (−1) n (z).

By (2.12) and (2.20), we find easily that

(2.21) E k (x, q) = [n] k

q n−1X

i=0

(−1) i E k

µ

x + i

n , q

n

¶

if n odd.

From (2.18) and (2.21), we can obtain Proposition 2.5 below

Proposition 2.5 (Multiplication formulas) Let n be any positive integer Then

B k (x, q) = [n]

k q

n

n−1X

i=0

B k

µ

x + i

n , q

n

¶

,

E k (x, q) = [n] k

q n−1X

i=0

(−1) i E k

µ

x + i

n , q

n

¶

if n odd.

3 Construction generating functions of q-Bernoulli, q-Euler numbers,

and polynomials

In the complex case, we shall explicitly determine the generating function F q (t)

of q-Bernoulli numbers and the generating function G q (t) of q-Euler numbers:

(3.1) F q (t) =

∞

X

k=0

B k (q) t

k

k! = e

B(q)t and G q (t) =

∞

X

k=0

E k (q) t

k

k! = e

E(q)t ,

where the symbol B k (q) and E k (q) are interpreted to mean that (B(q)) k and

(E(q)) k must be replaced by B k (q) and E k (q) when we expanded the one on the

right, respectively

Lemma 3.1

F q (t) = e 1−q t +t log q

1 − q

∞

X

m=0

q m e [m] q t ,

G q (t) = 2

∞

X

m=0

(−1) m e 2[m+1]q t

Proof Combining (2.10) and (3.1), F q (t) may be written as

F q (t) =

∞

X

k=0

log q (1 − q) k

k

X

i=0

µ

k i

¶

(−1) i i

q i − 1

t k

k!

= 1 + log q

∞

X

k=1

1

(1 − q) k

t k

k!

à 1

log q +

k

X

i=1

µ

k i

¶

(−1) i i

q i − 1

!

.

Here, the term with i = 0 is understood to be 1/log q (the limiting value of the summand in the limit i → 0) Specifically, by making use of the following

well-known binomial identity

k

µ

k − 1

i − 1

¶

= i

µ

k i

¶

(k ≥ i ≥ 1).

Thus, we find that

F q (t) = 1 + log q

∞

X

k=1

1

(1 − q) k

t k

k!

à 1

log q + k

k

X

i=1

µ

k − 1

i − 1

¶

(−1) i 1

q i − 1

!

=

∞

X

k=0

1

(1 − q) k

t k

k! + log q

∞

X

k=1

k (1 − q) k

t k

k!

∞

X

m=0

q m

k−1

X

i=0

µ

k − 1 i

¶

(−1) i q mi

= e 1−q t + log q

1 − q

∞

X

k=1

k (1 − q) k−1

t k

k!

∞

X

m=0

q m (1 − q m)k−1

= e 1−q t +t log q

1 − q

∞

X

m=0

q m

∞

X

k=0

µ

1 − q m

1 − q

¶k

t k

k! .

Next, by (2.11) and (3.1), we obtain the result

G q (t) =

∞

X

k=0

2k+1

(1 − q) k

k

X

i=0

µ

k i

¶

(−1) i q1i 1

q i+ 1

t k

k!

= 2

∞

X

k=0

2k

∞

X

m=0

(−1) m

Ã

1 − q m+1

1 − q

!k

t k

k!

= 2

∞

X

m=0

(−1) m

∞

X

k=0

·

m +1

2

¸k

q

(2t) k

k!

= 2

∞

X

m=0

(−1) m e 2[m+1 ]q t

Remark 3.2. The remarkable point is that the series on the right-hand side of Lemma 3.1 is uniformly convergent in the wider sense

From (2.13)and (2.14), we define the q-Bernoulli and q-Euler polynomials by

(3.2) F q (t, x) =

∞

X

k=0

B k (x, q) t k

k! =

∞

X

k=0

(q x B(q) + [x] q)k t k

k! ,

(3.3) G q (t, x) =

∞

X

k=0

E k (x, q) t k

k! =

∞

X

k=0

Ã

q x−1E(q)

2 +

·

x − 1

2

¸

q

!k

t k

k! .

Hence, we have

Lemma 3.3

F q (t, x) = e [x] q t F q (q x t) = e 1−q t +t log q

1 − q

∞

X

m=0

q m+x e [m+x] q t