On the q-Extensions of the Bernoulli and Euler Numbers, Related Identities and Lerch Zeta Function

0.16 MB·English

by  Taekyun Kim

ON THE q-EXTENSIONS OF THE BERNOULLI AND EULER
NUMBERS, RELATED IDENTITIES AND LERCH ZETA
FUNCTION
9
0
0 TAEKYUNKIM,YOUNG-HEEKIM,ANDKYUNG-WONHWANG
2

n
a Abstract Recently, λ-Bernoulli and λ-Euler numbers are studied in [5, 10]. The purpose
J of this paper is to present a systematic study of some families of the q-extensions of the λ-

2 Bernoulliandtheλ-Eulernumbersbyusingthebosonicp-adicq-integralandthefermionicp-adic

q-integral. The investigation of these λ-q-Bernoulli and λ-q-Euler numbers leads to interesting
]
T identitiesrelatedtotheseobjects. Theresultsofthepresentpapercoverearlierresultsconcerning
N q-Bernoulliandq-Euler numbers. Byusing derivative operator to the generating functions of λ-
q-Bernoulliandλ-q-Eulernumbers,wegivetheq-extensionsofLerchzetafunction.
.
h
t 2000 Mathematics Subject Classification : 11S80, 11B68?
a
m Key words and phrases : λ-Bernoulli numbers, λ-Euler numbers, p-adic q-integral, Lerch
[ zetafunction

1
1. Introduction, Definitions and Notations
v
9
Throughout this paper, the symbols Z ,Q ,C and C denote the ring of p-adic
4 p p p
rationalintegers,thefieldofp-adicrationalnumbers,thecomplexnumberfieldand
2
0 the completion of algebraic closure of Qp, respectively. Let N be the set of natural
. numbers.
1
0 Thesymbolq canbetreatedasacomplexnumber,q ∈C,orasap-adicnumber,
9 q ∈ Cp. If q ∈ C, then we always assume that |q| < 1. If q ∈ Cp, then we
0 usually assume that |1−q| < 1. Here |·| stands for the p-adic absolute value
p p
v: in Cp with |p|p = 1p. The q-basic natural numbers are defined by [n]q = 11−−qqn =
Xi 1+q+q2 +···+qn−1 (n ∈ N) and [n]−q = 1−1(+−qq)n. In this paper, we use the
r notation
a 1−qx 1−(−q)x
[x] = and [x] = , see [1-19].
q 1−q −q 1+q
Hence lim[x] =x for any x with |x| ≤1 in the present p-adic case.
q p
q→1
Forx∈Z ,wesaythatg isauniformlydifferentiablefunctionatapointa∈Z ,
p p
andwrite g ∈UD(Z ),the setofuniformlydifferentiable function, ifthe difference
p
quotients
g(y)−g(x)
F (x,y)=
g y−x
have a limit l=g′(a) as (x,y)→(a,a). For f ∈UD(Z ), the q- deformed bosonic
p
p-adic integral is defined as
pN−1 qx
(1) I (f)= f(x)dµ (x)= lim f(x) , see [1-19],
q Z q N→∞ [pN]q
Z p x=0
X
1
2 TAEKYUNKIM,YOUNG-HEEKIM,ANDKYUNG-WONHWANG
and the q-deformed fermonic p-adic integral is defined by
pN−1 (−q)x
I (f)= f(x)dµ (x)= lim f(x) , (see [1-19]).
−q ZZp −q N→∞x=0 [pN]−q
X
For n∈N, let f (x)=f(x+n). Then
n
n−1
(2) qnI (f )=(−1)nI (f)+[2] (−1)n−1−lqlf(l).
−q n −q q
l=0
X
The classicalBernoullipolynomials B (x) and the Euler polynomialsE (x) are
n n
defined as
t ∞ tn 2ext ∞ tn
(3) ext = B (x) and = E (x) .
et−1 n n! et+1 n n!
x=0 x=0
X X
The Bernoulli numbers B and the Euler numbers E are defined as B = B (0)
n n n n
and E =E (0), (see [1-19]).
n n
From (1), we note that
q−1
(4) qI (f )=I (f)+(q−1)f(0)+ f′(0),
q 1 q logq
for f (x)=f(x+1). By (4), we see that I (f )=I (f)+f′(0), (see [7]).
1 1 1 1
Let u be algebraic in C (or C). Then the Frobenius-Euler polynomials are
p
defined as
1−u ∞ tn
(5) ext = H (u,x) , (see [5]).
et−u n n!
n=0
X
In case x=0, H (u,0)=H (u), which are called the Frobenius Euler numbers.
n n
Let Cpn be the cyclic group consisting of all pn-th roots of unity in Cp for any
n ≥ 0 and Tp be the direct limit of Cpn with respect to the natural morphisms,
hence Tp is the union of all Cpn with discrete topology.
For λ∈T with λ6=1, if we use (4), then we have
p
t
(6) etxλxdµ (x)= .
Z 1 λet−1
Z p
From (6), the λ−Bernoulli numbers are defined as
t ∞ tn
(7) =eB(λ)t = B (λ) , (see [5])
λet−1 n n!
n=0
X
with the usual convention of replacing Bi(λ) by B (λ). Thus, B (λ) can be deter-
i k
mined inductively by
1, if k =1,
(8) λ(B(λ)+1)k−B (λ)=
k
(0, if k >1, (see [5]).
By the definition of the Frobenius-Euler numbers, we see that
t ∞ 1 (m+1)H (λ−1)
(9) = · m tm+1, ( see [7]).
λet−1 (m+1)! λ−1
m=0
X
3
For m≥1 and λ6=1, we have
m
(10) B (λ)= xmλxdµ (x)= H (λ−1), (see [5]).
m Z 1 λ−1 m−1
Z p
We can also easily see that λxdµ (x)=0 and
Z 1
p
tλx R ∞ tn
etx= lim = lim xmλxdµ (x)λx.
m→∞ λet−1 n!m→∞ Z 1
λ∈XCpm nX=0 λ∈XCpmZ p
Consequently, we have
1
xn = B (1)+ H (λ−1)λx
n λ−1 n−1
λX∈Tp
λ6=1
B (λ)
= B (1)+ n λx, ( see [5]).
n n
λX∈Tp
λ6=1
From (6) and (8), we note that
1 2λ
B (λ)=0, B (λ)= , B (λ)=− , ··· .
0 1 λ−1 2 (λ−1)2
The Genocchi numbers are defined by the generating function
2t ∞ tn
= G .
et+1 nn!
n=0
X
These numbers satisfy the relation G = 0,G = 1,G = G = ··· = G = 0,
0 1 3 5 2k+1
and the even coefficients are G =2(1−2n)B .
n n
For λ∈C with |λ|<1, by (2), we have
p
2
(11) λxextdµ (x)= .
Z −1 λet+1
Z p
By (11), we define the λ-Euler numbers as follows :
∞
2 E (λ)
(12) = n tn, (see [7,9,10]).
λet+1 n!
n=0
X
Note that E (λ)= 2 H (−λ−1).
n λ+1 n
From (12), we can easily derive
2
(13) xnλxdµ (x)=E (λ)= H (−λ−1).
Z −1 n λ+1 n
Z p
The λ-Genocchi numbers are also defined as
2t ∞ tn
t xnλxdµ (x)= = G (x) .
Z −1 λet+1 n n!
Z p n=0
X
Thus, we have G (λ)=0, G (λ)= 2 , ··· , E (λ)= Gn+1(λ).
0 1 λ+1 n n+1
Inthispaper,westudy theq-extensionofλ-Bernoullinumberandλ-Eulernum-
bers related to Lerch zeta function. The purpose of this paper is to present a
systematicstudy ofsomefamilies ofthe q-extensionofthe λ-Bernoulliandλ-Euler
numbersbyusingthebosonicp-adicq-integralandtheferminionicp-adicq-integral.
4 TAEKYUNKIM,YOUNG-HEEKIM,ANDKYUNG-WONHWANG
Theinvestigationoftheseλ-q-Bernoulliandλ-q-Eulernumbersleadstointeresting
identities related to these objects. The results of the present paper cover earlier
results concerning q-Bernoulli and q-Euler numbers. By using derivative operator
to the generating functions of λ-q-Bernoulli and λ-q-Euler numbers, we can give
the q-extension of Lerch zeta function.
2. q-extension of λ-Bernoulli numbers and polynomials
For λ∈T , let us consider the q-extension of λ-Bernoulli numbers as follows.
p
(14) β (λ)= λx[x]kdµ (x).
k,q q q
Z
Z p
From (14), we note that
pN−1
1
β (λ) = lim λx[x]kqx
k,q N→∞[pN]q q
x=0
X
pN−1 k
1 k 1
= lim (λq)x( (−1)lqlx)
N→∞[pN]q x=0 l=0(cid:18)l(cid:19) (1−q)k
X X
1−q k k 1−(λql+1)pN
= (−1)l
(1−q)k l 1−λql+1
l=0(cid:18) (cid:19)
X
k
1 k l+1
= (−1)l .
(1−q)k−1 l 1−λql+1
l=0(cid:18) (cid:19)
X
Therefore, we obtain the following theorem.
Theorem 1. For k ∈N∪{0} and λ∈T , we have
p
k
1 k l+1
β (λ)= (−1)l .
k,q (1−q)k−1 l 1−λql+1
l=0(cid:18) (cid:19)
X
Let F(t,λ:q) be the generating functions of β (λ) with
n,q
∞ tn
F(t,λ:q)= β (λ) .
n,q n!
n=0
X
5
Then we have
∞ tn
F(t,λ:q) = β (λ) = λxe[x]qtdµ (x)
n,q n! Z q
n=0 Z p
X
∞ tn
= λx[x]ndµ (x)
Z q q n!
n=0Z p
X
∞ 1 k k ∞ tk
(15) = { (−1)l(l+1) λmq(l+1)m}
(1−q)k−1 l k!
k=0 l=0(cid:18) (cid:19) m=0
X X X
∞ 1 ∞ k k tk
= λm (−1)l(l+1)q(l+1)m
(1−q)k−1 l k!
k=0 m=0 l=0(cid:18) (cid:19)
X X X
∞ 1 ∞ k k tk
= qmλm l (−1)lqlm
(1−q)k−1 l k!
k=0 m=0 l=0 (cid:18) (cid:19)
X X X
∞ 1 ∞ k k tk
+ qmλm (−1)lqlm .
(1−q)k−1 l k!
k=0 m=0 l=0(cid:18) (cid:19)
X X X
Since l k =k k−1 , the first term of the last equation in (15) equals
l l−1
(cid:0) (cid:1) (cid:0) (cid:1)
∞ ∞ 1 k k−1 tk
qmλm{ (−1)lqlm}
(1−q)k−1 l−1 (k−1)!
m=0 k=0 l=1(cid:18) (cid:19)
X X X
∞ ∞ 1 k−1 k−1 tk
(16) =− q2mλm{ (−1)lqlm}
(1−q)k−1 l (k−1)!
m=0 k=1 l=0(cid:18) (cid:19)
X X X
∞ ∞ tk ∞
=−t q2mλm [m]k =−t q2mλme[m]qt.
qk!
m=0 k=0 m=0
X X X
The second term of the last equation in (15) equals
∞ 1 ∞ tk
qmλm(1−qm)k
(1−q)k−1 k!
k=0 m=0
X X
∞ ∞ tk ∞
(17) =(1−q) qmλm [m]k =(1−q) qmλme[m]qt.
qk!
m=0 k=0 m=0
X X X
From (15), (16) and (17), we obtain the following proposition.
Proposition 2. Let F(t,λ:q)= ∞ β (λ)tn . Then we have
n=0 n,q n!
P
∞ ∞
F(t,λ:q)=−t q2mλme[m]qt+(1−q) qmλme[m]qt.
m=0 m=0
X X
6 TAEKYUNKIM,YOUNG-HEEKIM,ANDKYUNG-WONHWANG
Since q2m =qm{[m] (q−1)+1}, it follows that
q
dkF (t,λ:q)
β (λ) = q |
k,q (dt)k t=0
∞ ∞
= −k q2mλm[m]k−1+(1−q) qmλm[m]k
q q
m=0 m=0
X X
∞ ∞ ∞
= −k(q−1) qmλm[m]k−k qmλm[m]k−1+(1−q) qmλm[m]k
q q q
m=0 m=0 m=0
X X X
∞ ∞
= (1−q)(k+1) qmλm[m]k−k qmλm[m]k−1.
q q
m=0 m=0
X X
Therefore, we obtain the following theorem.
Theorem 3. For k ∈N∪{0} and λ∈T , we have
p
∞ ∞
β (λ)=(1−q)(k+1) qmλm[m]k−k qmλm[m]k−1.
k,q q q
m=0 m=0
X X
Now we consider another q-extension of λ-Bernoulli numbers as follows.
(18) B (λ)= q−xλx[x]ndµ (x).
n,q q q
Z
Z p
From (18), we can derive
B (λ) = q−xλx[x]ndµ (x)
n,q q q
Z
Z p
n
1 n
= q−x(−1)lλxqlxdµ (x)
(1−q)n l Z q
l=0(cid:18) (cid:19)Z p
X
n
1 n l
= (−1)l .
(1−q)n−1 l 1−λql
l=0(cid:18) (cid:19)
X
Thus, we obtain the following theorem.
Theorem 4. For n∈N∪{0} and λ∈T , we have
p
n
1 n l
B (λ)= (−1)l .
n,q (1−q)n−1 l 1−λql
l=0(cid:18) (cid:19)
X
Let F∗(t,λ:q) be the generating functions of B (λ) with
n,q
∞ tn
F∗(t,λ:q)= B (λ) .
n,q n!
n=0
X
7
Then we have
∞ tn
F∗(t,λ:q) = B (λ) = q−xλxe[x]qtdµ (x)
n,q n! Z q
n=0 Z p
X
∞ tn
= { q−xλx[x]ndµ (x)}
Z q q n!
n=0 Z p
X
∞ 1 n n l tn
= { (−1)l }
(1−q)n−1 l 1−λql n!
n=0 l=0(cid:18) (cid:19)
X X
∞ 1 n n ∞ tn
= { (−1)ll λmqlm}
(1−q)n−1 l n!
n=0 l=0(cid:18) (cid:19) m=0
X X X
∞ ∞ n n n−1 tn
= λm{ (−1)lqlm}
(1−q)n−1 l−1 n!
m=0 n=1 l=1(cid:18) (cid:19)
X X X
∞ ∞ n tn
= − λmqm (1−qm)n−1
(1−q)n−1 n!
m=0 n=1
X X
∞ ∞ (1−qm)ntn+1
= − λmqm
(1−q)n n!
m=0 n=0
X X
∞
= −t λmqme[m]qt.
m=0
X
Therefore we obtain the following lemma.
Lemma 5. Let F∗(t,λ:q)= ∞ B (λ)tn . Then we have
n=0 n,q n!
∞
P
F∗(t,λ:q)=−t λmqme[m]qt.
m=0
X
We also have
dkF (t,λ:q) ∞
B (λ)= q | =−k qmλm[m]k−1.
k,q (dt)k t=0 q
m=0
X
Therefore we obtain the following theorem.
Theorem 6. For k ∈N∪{0} and λ∈T , we have
p
∞
B (λ)=−k qmλm[m]k−1.
k,q q
m=0
X
3. q-extension of λ-Euler numbers and polynomials
In this section, we assume that p is an odd prime number and λ ∈ C with
p
|1 − λ| < 1. By using the fermionic p-adic q-integral on Z , we consider the
p p
q-extensions of λ-Euler numbers as follows.
For n∈N∪{0}, we define the q−extension of λ-Euler numbers as
(19) E (λ)= q−xλx[x]ndµ (x).
n,q q −q
Z
Z p
8 TAEKYUNKIM,YOUNG-HEEKIM,ANDKYUNG-WONHWANG
From (19), we note that
E (λ) = q−xλx[x]ndµ (x)
n,q q −q
Z
Z p
pN−1
1+q
= lim (−1)x[x]nλx
N→∞ 1+qpN q
x=0
X
[2] 1 n n 1+qpNλpN
= q (−1)l lim
2 (1−q)n l N→∞ 1+qlλ
l=0(cid:18) (cid:19)
X
n
[2] 1 n 2
= q (−1)l
2 (1−q)n l 1+qlλ
l=0(cid:18) (cid:19)
X
n
[2] n 1
= q (−1)l .
(1−q)n l 1+qlλ
l=0(cid:18) (cid:19)
X
Therefore we obtain the following theorem.
Theorem 7. For n∈N∪{0}, we have
n
[2] n 1
E (λ)= q (−1)l .
n,q (1−q)n l 1+qlλ
l=0(cid:18) (cid:19)
X
Let g(t,λ:q) be the generating function of E (λ) with
n,q
∞ tn
g(t,λ:q)= E (λ) .
n,q n!
n=0
X
Then we have
∞ tn
g(t,λ:q) = E (λ) = q−xλxe[x]qtdµ (x)
n,q n! Z −q
n=0 Z p
X
∞ tn
= { q−xλx[x]ndµ (x)}
Z q −q n!
n=0 Z p
X
∞ 1 n n 1 tn
= [2] { (−1)l }
q (1−q)n l 1+λql n!
n=0 l=0(cid:18) (cid:19)
X X
∞ 1 n n ∞ tn
= [2] (−1)l{ (−1)mλmqlm}
q (1−q)n l n!
n=0 l=0(cid:18) (cid:19) m=0
X X X
∞ ∞ 1 n n tn
= [2] (−1)mλm (−1)lqlm
q (1−q)n l n!
m=0 n=0 l=0(cid:18) (cid:19)
X X X
∞ ∞ tn
= [2] (−1)mλm [m]n
q q n!
m=0 n=0
X X
∞
= [2] (−1)mλme[m]qt.
q
m=0
X
Thus, we have the following lemma.
9
∞
Lemma 8. Let g(t,λ:q)= E (λ)tn. Then we have
n,q n!
n=0
P ∞
(20) g(t,λ:q)=[2] (−1)mλme[m]qt.
q
m=0
X
By (20), we can also consider the λ-q-Genocchi numbers as follows.
∞ ∞ tn
(21) t q−xλxe[x]qtdµ (x)=[2] t (−1)mλme[m]qt = G (λ) .
Z −q q n,q n!
Z p m=0 n=0
X X
From (21), we note that G (λ)=0 and
0,q
G (λ)
q−xλx[x]ndµ (x)= n+1,q .
Z q −q n+1
Z p
Thus, we see that
G (λ)
E (λ)= q−xλx[x]ndµ (x)= n+1,q .
n,q Z q −q n+1
Z p
Hence
n−1
n n−1 1
G (λ)=[2] (−1)l ,
n,q q(1−q)n−1 l 1+qlλ
l=0(cid:18) (cid:19)
X
where n=1,2,3, ···. Indeed,
[2]
G (λ) = q ,
1,q 1+λ
1
2[2] 1 1 [2] 2 2
G (λ) = q (−1)l = q ( − )
2,q 1−q l 1+qlλ 1−q 1+λ 1+qλ
l=0(cid:18) (cid:19)
X
λ
= −2[2] ( ).
q (1+λ)(1+qλ)
Now, we consider the q-extension of λ-Euler polynomials as follows.
(22) E (λ,x)= q−yλy[x+y]ndµ (y).
n,q q −q
Z
Z p
From (22), we can easily derive
n
[2] n 1
E (λ,x)= q (−1)lqlx .
n,q (1−q)n l 1+qlλ
l=0(cid:18) (cid:19)
X
∞
Let g(x,λ:q)= E (λ,x)tn. Then we have
n,q n!
n=0
P ∞ tn
g(x,λ:q) = En,q(λ,x)n! = Z q−yλye[x+y]tqdµ−q(y)
n=0 Z p
X
∞ [2] n n ∞ tn
= q (−1)lqlx( (−1)mqmlλm)
(1−q)n l n!
n=0 l=0(cid:18) (cid:19) m=0
X X X
∞
= [2] (−1)mλme[m+x]qt.
q
m=0
X
10 TAEKYUNKIM,YOUNG-HEEKIM,ANDKYUNG-WONHWANG
It follows that
dn(g(x,λ:g)) ∞
E (λ,x)= | =[2] (−1)mλm[m+x]n.
n,q (dt)n t=0 q q
m=0
X
Then we obtain the following theorem.
Theorem 9. For n∈N∪{0}, we have
∞
E (λ,x)=[2] (−1)mλm[m+x]n.
n,q q q
m=0
X
By the same method, we consider the λ-q-Genocchi polynomials as follows.
∞
(23) t q−xλxe[x+y]qtdµ (x) = [2] t (−1)mλme[m+x]qt
−q q
Z
Z p m=0
X
∞ tn
= G (λ,x) .
n,q n!
n=0
X
By (23), we see
G (λ,x)
(24) E (λ,x)= q−yλy[x+y]ndµ (y)= n+1,q
n,q Z q −q n+1
Z p
and G (λ,x)=0. From the definition of λ-q-Euler polynomials, we derive
0,q
E (λ,x) = q−yλy[x+y]ndµ (y)
n,q q −q
Z
Z p
n
n
(25) = [x]n−lqlx q−yλy[y]ldµ (y)
l q q −q
Z
l=0(cid:18) (cid:19) Z p
X
d−1
[2] x+a
= q [d]n (−1)aλa q−dyλdy[ +y]n dµ (y,)
[2]qd q a=0 ZZp d qd −q
X
for d∈N with d≡1( mod 2). By (25), we see that
n
n
E (λ,x)= = [x]n−lqlxE (λ,x)
n,q l q l,q
l=0(cid:18) (cid:19)
X
d−1
[2] x+a
= q [d]n (−1)aλaE (λd, ).
[2] q n,qd d
qd a=0
X
It is easy to see that
qI (f )+I (f)=[2] f(0),
−q 1 −q q
where f (x)=f(x+1). Thus, we have
1
q q−y−1λy+1[x+1+y]ndµ (y)+ q−yλy[x+y]ndµ (y)=[2] [x]n.
q −q q −q q q
Z Z
Z p Z p
Therefore, we obtain the following theorem.
Theorem 10. For n∈N∪{0}, we have
λE (λ,x+1)+E (λ,x)=[2] [x]n.
n,q n,q q q
By Theorem 10 and (24), we have the following result.