The p-adic Generalized Twisted (h, q)-Euler-l-Function and
Its Applications
Mehmet Cenkci
Department of Mathematics, Akdeniz University, 07058-Antalya, Turkey
cenkci@akdeniz.edu.tr
Abstract : The main purpose of this paper is to construct the p-adic twisted (h, q)-Euler-l-
function, which interpolates generalized twisted (h, q)-Euler numbers associated with a primitive
Dirichlet character χ. This is a partial answer for the open question which was remained in [13].
An application of this function leads general congruences systems for generalized twisted (h, q)-
Euler numbers associated with χ, in particular, Kummer-type congruences for these numbers are
obtained.
Keywords : p-adic q-Volkenborn integration, Euler numbers and polynomials, Kummer con-
gruences.
MSC 2000 : 11B68, 43A05, 11S80, 11A07.
1. Introduction
Let N, Z, Q, R and C denote, respectively, sets of positive integer, integer, rational, real and
complex numbers as usual. Let p be a fixed odd prime number and x ∈ Q. Then there exists
integers m, n and νp (x) such that x = p
νp(x)m/n and p does not divide either m or n. Let |·|p
be defined such that |x|p = p
−νp(x) and |0|p = 0. Then |·|p is a valuation on Q which satisfies the
non-Archimedean property
|x+ y|p 6 max
|x|p , |y|p
Completion ofQ with respect to |·|p is denoted by Qp and called the field of p-adic rational numbers.
But Qp itself is not complete with respect to |·|p. Cp is the completion of the algebraic closure of
Qp and Zp =
x ∈ Qp : |x|p 6 1
is called the ring of p-adic rational integers (see [14], [17]).
Let d be a fixed positive odd integer and let
X = Xd = lim←−
Z/dpNZ
, X1 = Zp,
0<a<dp
(a,p)=1
a+ dpNZp
a+ dpNZp =
x ∈ X : x ≡ a
moddpN
where N ∈ N and a ∈ Z with 0 6 a < dpN ([1], [5], [12], [18]).
When talking about q-extensions, q can variously be considered as an indeterminate, a complex
number q ∈ C or a p-adic number q ∈ Cp. If q ∈ C, we normally assume that |q| < 1. If q ∈ Cp,
we assume that |1− q|p < p
−1/(p−1) so that for |x|p 6 1, we have q
x =exp(xlogq) ([1], [2], [5], [6],
http://arxiv.org/abs/0704.1397v1
[7]). We use the notations
[x]q =
1− qx
and [x]
1− (−q)
1 + q
We say that f is uniformly differentiable function at a point a ∈ X, and denote this property by
f ∈ UD (X), if the quotient of the differences
f (x)− f (y)
has a limit l = f ′ (a) as (x, y)→ (a, a). For f ∈ UD (X), the p-adic invariant q-integral on X was
defined by
Iq (f) =
f (t) dµq (t) = lim
[dpN ]q
dpN−1
f (a) qa
(cf. [5], [7]), where for any positive integer N
a+ dpNZp
[dpN ]q
(cf. [5], [6], [7]).
The concept of twisted has been applied by many authors to certain functions which interpolate
certain number sequences. In [15], Koblitz defined twisted Dirichlet L-function which interpolates
twisted Bernoulli numbers in the field of complex numbers. In [20], Simsek constructed a q-
analogue of the twisted L-function interpolating q-twisted Bernoulli numbers. Kim et.al. [12]
derived a p-adic analogue of the twisted L-function by using p-adic invariant integrals. By using
the definition of h-extension of p-adic q-L-function which is constructed by Kim [11], Simsek
[22, 23] and Jang [4] defined twisted p-adic generalized (h, q)-L-function. In [18], Satoh derived
p-adic interpolation function for q-Frobenius-Euler numbers. Simsek [21] gave twisted extensions
of q-Frobenius-Euler numbers and their interpolating function q-twisted l-series. In [1], Cenkci
et.al. constructed generalized p-adic twisted l-function in p-adic number field. Recently, Kim and
Rim [13] defined twisted q-Euler numbers by using p-adic invariant integral on Zp in the fermionic
sense. In that paper, they raised the following question: Find a p-adic analogue of the q-twisted l-
function which interpolates E
(h,1)
n,ξ,q,χ, the generalized twisted q-Euler numbers attached to χ [8], [10].
In a forthcoming paper, Rim et.al. [16] answered this question by constructing partial (h, q)-zeta
function motivating from a method of Washington [24, 25].
In this paper, we construct p-adic generalized twisted (h, q)-Euler-l-function by employing p-adic
invariant measure on p-adic number field. This is the answer of the part of the question posed in
[13]. This way of derivation of p-adic generalized twisted (h, q)-Euler-l-function is different from
that of [16], and leads an explicit integral representation for this function. As an application of
the derived integral representation, we obtain general congruences systems for generalized twisted
q-Euler numbers associated with χ, containing Kummer-type congruences.
2. Generalized Twisted q-Euler Numbers
In this section, we give a brief summary of the concepts p-adic q-integrals and Euler numbers
and polynomials. Let UD (X) be the set of all uniformly differentiable functions on X. For any
f ∈ UD (X), Kim defined a q-analogue of an integral with respect to a p-adic invariant measure in
[5, 7] which was called p-adic q-integral. The p-adic q-integral was defined as follows:
Iq (f) =
f (t) dµq (t) = lim
[dpN ]q
dpN−1
f (a) qa.
Note that
I1 (f) = lim
Iq (f) =
f (t) dµ1 (t) = lim
dpN−1
f (a)
is the Volkenborn integral (see [17]).
The Euler zeta function ζE (s) is defined by means of
ζE (s) = 2
for s ∈ C with Re(s) > 1 (cf. [8]). For a Dirichlet character χ with conductor d, d ∈ N, d is odd,
the l-function associated with χ is defined as ([8])
l (s, χ) = 2
χ (k) (−1)
for s ∈ C with Re(s) > 1. This function can be analytically continued to whole complex plane,
except s = 1 when χ = 1; and when χ = 1, it reduces to Euler zeta function ζE (s). In [9],
(h, q)-extension of Euler zeta function is defined by
E,q (s, x) = [2]q
[k + x]
with s, h ∈ C, Re(s) > 1 and x 6=negative integer or zero. (h, q)-Euler polynomials are defined by
the p-adic q-integral as
E(h)n,q (x) =
q(h−1)t [t+ x]
q dµ−q (t) ,
for h ∈ Z. E
n,q (0) = E
n,q are called (h, q)-Euler numbers. In [9], it has been shown that for
n ∈ Z, n > 0
E,q (−n, x) = E
n,q (x) ,
thus we have
E(h)n,q (x) = [2]q
qhk [k + x]
from which the following entails:
E(h)n,q (x) =
(1− q)
1 + qh+j
In [8, 9], (h, q)-extension of the l-function associated with χ is defined by
l(h)q (s, χ) = [2]q
χ (k) (−1)
for h, s ∈ C with Re(s) > 1. The negative integer values of s are determined explicitly by
l(h)q (−n, χ) = E
n,q,χ,
for n ∈ Z, n > 0 where E
n,q,χ are the generalized (h, q)-Euler numbers associated with χ defined
E(h)n,q,χ =
χ (t) q(h−1)t [t]
q dµ−q (t)
= [2]q
χ (k) (−1)
qhk [k]
Now assume that q ∈ Cp with |1− q|p < 1. From the definition of p-adic invariant q integral on
X, Kim [8] defined the integral
I−1 (f) = lim
Iq (f) =
f (t) dµ−1 (t) (2.1)
for f ∈ UD (X). Note that
I−1 (f1) + I−1 (f) = 2f (0) , (2.2)
where f1 (t) = f (t+ 1). Repeated application of last formula yields
I−1 (fn) = (−1)
I−1 (f) + 2
n−1−j
f (j) , (2.3)
with fn (t) = f (t+ n).
Let Tp =
Cpn = lim
Z/pnZ, where Cpn =
w ∈ X : wp
is the cyclic group of order
pn. For w ∈ Tp, let φw : Zp → Cp denote the locally constant function defined by t→ w
For f (t) = φw (t) e
zt, we obtain
φw (t) e
ztdµ−1 (z) =
wez + 1
using (2.1) and (2.2), and
χ (t)φw (t) e
ztdµ−1 (t) = 2
χ (i)φw (i) e
wdedz + 1
using (2.1) and (2.3) (cf. [8]). As a consequence, the twisted Euler numbers and generalized twisted
Euler numbers associated with χ can respectively be defined by
wez + 1
, and 2
χ (i)φw (i) e
wdedz + 1
En,w,χ
from which
tnφw (t) dµ−1 (t) = En,w, and
χ (t) tnφw (t) dµ−1 (t) = En,w,χ
follow.
Twisted extension of (h, q)-Euler zeta function is defined by
E,q,w (s, x) = [2]q
wkqhk
[k + x]
with h, s ∈ C, Re(s) > 1 and x 6=negative integer or zero. For n ∈ Z, n > 0 and h ∈ Z, this
function gives
E,q,w (−n, x) = E
n,q,w (x) ,
where E
n,q,w (x) are the twisted q-Euler polynomials defined as
E(h)n,q,w (x) =
q(h−1)tφw (t) [x+ t]
q dµ−q (t)
= [2]q
wkqhk [k + x]
by using p-adic invariant q-integral on X in the fermionic sense (cf. [13], [16]). The following
expressions for twisted (h, q)-Euler polynomials can be verified from the defining equalities:
E(h)n,q,w (x) =
(1− q)
1 + wqh+j
, (2.4)
E(h)n,q,w (x) =
[2]qd
qhawa (−1)
n,qd,wd
, (2.5)
where n, d ∈ N with d is odd. From (2.4), the twisted (h, q)-Euler polynomials can be determined
explicitly. A few of them are
0,q,w (x) =
1 + q
1 + wqh
1,q,w (x) =
1 + q
1 + wqh
1 + wqh+1
2,q,w (x) =
1 + q
(1− q)
1 + wqh
1 + wqh+1
1 + wqh+2
For x = 0, E
n,q,w (0) = E
n,q,w are called twisted (h, q)-Euler numbers. Thus we can write
E(h)n,q,w (x) =
qxj [x]
j,q,w.
Let χ be a Dirichlet character of conductor d with d ∈ N and d is odd. Then the generalized
twisted (h, q)-Euler numbers associated with χ are defined as
E(h)n,q,w,χ =
χ (t) q(h−1)tφw (t) [t]
q dµ−q (t) .
These numbers arise at the negative integer values of the twisted (h, q)-Euler-l-function which is
defined by
l(h)q,w (s, χ) = [2]q
χ (k) (−1)
wkqhk
with h, s ∈ C, Re(s) > 1. Indeed, for n ∈ Z, n > 0 and h ∈ Z, we have
l(h)q,w (−n, χ) = E
n,q,w,χ
(cf. [13], [16]).
We conclude this section by stating the distribution property for generalized twisted (h, q)-Euler
numbers associated with χ, which will take a major role in constructing a measure in the next
section.
For n, d ∈ N with d is odd, we have
E(h)n,q,w,χ =
[2]qd
qhawaχ (a) (−1)
n,qd,wd
3. p-adic Twisted (h, q)-l-Functions
In this section we first focus on defining a p-adic invariant measure, which is apparently an
important tool to construct p-adic twisted (h, q)-Euler-l-function in the sense of p-adic invariant
q-integral. We afterwards give the definition of p-adic twisted (h, q)-Euler-l-function, together with
Witt’s type formulas for twisted and generalized twisted (h, q)-Euler numbers.
Throughout, we assume that ξ is the rth root of unity with (r, pd) = 1, where p is an odd prime
and d is an odd natural number. If (r, pd) = 1, it has been known that |1− ξ|p > 1 (see [15], [19])
and ξ lies in the cyclic group Cpn =
w : wp
. The following theorem plays a crucial role in
constructing p-adic generalized twisted (h, q)-Euler-l-function on X.
Theorem 3.1 Let q ∈ Cp with |1− q|p < p
−1/(p−1) and ξ is the rth root of unity with |1− ξ|p > 1.
For N ∈ Z, n ∈ Z, n > 0, let µ
n,ξ,q be defined as
n,ξ,q
a+ dpNZp
ξaqhaE
n,qdp
Then µ
n,ξ,q extends uniquely to a measure on X.
Proof. In order to show that µ
n,ξ,q is a measure on X, we need to show that it is a distribution
and is bounded on X.
To show it is a distribution on X, we check the equality
n,ξ,q
a+ idpN + dpN+1Zp
n,ξ,q
a+ dpNZp
Beginning the calculation from right hand side yields
n,ξ,q
a+ idpN + dpN+1Zp
dpN+1
a+idpN
ξa+idp
qh(a+idp
n,qdp
a+ idpN
dpN+1
ξaqha [p]
n,(qdpN )
,(ξdpN )
ξaqhaE
n,qdp
n,ξ,q
a+ dpNZp
where we have used (2.5).
To present boundedness, we use equation (2.4) to expand the polynomial E
n,qdp
so that
n,ξ,q
a+ dpNZp
(1− q)
n (−1)
ξaqha
1 + ξdp
N+jdpN
Now, since d is an odd natural number and p is an odd prime, we have
N+jdpN
1, so by induction on j, we obtain
n,ξ,q
a+ dpNZp
for a constant M . This is what we require, so the proof is completed.
Let χ be a Dirichlet character with conductor d. Then we can express the generalized twisted
(h, q)-Euler numbers associated with χ as an integral over X, by using the measure µ
n,ξ,q.
Lemma 3.2 For n ∈ Z, n > 0, we have
χ (t) dµ
n,ξ,q (t) = E
n,q,ξ,χ.
Proof. From the definition of p-adic invariant integral, we have
χ (t) dµ
n,ξ,q (t) = lim
dpN−1
χ (c)
ξcqhcE
n,qdp
Writing c = a+ dm with a = 0, 1, . . . , d− 1 and m = 0, 1, 2, . . ., we get
χ (t) dµ
n,ξ,q (t) = [d]
[2]qd
χ (a) (−1)
ξaqha
× lim
[2]qd
(qd)p
n,(qd)p
,(ξd)p
= [d]
[2]qd
χ (a) (−1)
ξaqhaE
n,qd,ξd
Assuming χ (0) = 0 and by the fact that χ (d) = 0, last expression equals E
n,ξ,q,χ, and the proof
is completed.
Since it is impossible to have a non-zero translation-invariantmeasure on X, µ
n,ξ,q is not invariant
under translation, but satisfies the following:
Lemma 3.3 For a compact-open subset U of X, we have
n,ξ,q (pU) = [p]
[2]qp
n,ξp,qp (U) .
Proof. Let U = a+ dpNZp be the compact-open subset of X. Then
n,ξ,q (pU) = µ
n,ξ,q
pa+ dpN+1Zp
dpN+1
ξpaqhpaE
n,qdp
dpN+1
[2]qp
[2]qp
(qp)dp
n,(qp)dp
,(ξp)dp
[2]qp
n,ξp,qp
a+ dpNZp
= [p]
[2]qp
n,ξp,qp (U) ,
which is the desired result.
Next, we give a relation between µ
n,ξ,q and µ−q.
Lemma 3.4 For any n ∈ Z, n > 0, we have
n,ξ,q (t) = q
(h−1)tξt [t]
q dµ−q (t) .
Proof. From the definition of µ
n,ξ,q and expansion of twisted (h, q)-Euler polynomials, we have
n,ξ,q
a+ dpNZp
(1− q)
n (−1)
ξaqha
1 + ξdp
N+jdpN
By the same method presented in [7], we obtain
n,ξ,q
a+ dpNZp
(1− q)
n (−1)
ξaqha
1 + q
ξaq(h−1)a [a]
q (−1)
qa = q(h−1)aξa [a]
q lim
1−(−qdpN )
1−(−q)
= q(h−1)aξa [a]
q lim
a+ dpNZp
We thus have
n,ξ,q (t) = q
(h−1)tξt [t]
q dµ−q (t) ,
the desired result.
Let ω denote the Teichmüller character mod p. For an arbitrary character χ and n ∈ Z, let
χn = χω
−n in the sense of product of characters. For t ∈ X∗ = X − pX, we set 〈t〉q = [t]q /ω (t).
Since
〈t〉q − 1
< p−1/(p−1), 〈t〉
q is defined by exp
slogp 〈t〉q
for |s|p 6 1, where logp is the
Iwasawa p-adic logarithm function ([3]). For |1− q|p < p
−1/(p−1), we have 〈t〉
q ≡ 1
modpN
We now define p-adic generalized twisted (h, q)-Euler-l-function.
Definition 3.5 For s ∈ Zp,
p,q,ξ (s, χ) =
(h−1)tξtdµ−q (t) .
The values of this function at non-positive integers are given by the following theorem:
Theorem 3.6 For any n ∈ Z, n > 0,
p,q,ξ (−n, χ) = E
n,q,ξ,χn
− χn (p) [p]
[2]qp
n,qp,ξp,χn
Proof.
p,q,ξ (−n, χ) =
(h−1)tξtdµ−q (t) =
χn (t) [t]
(h−1)tξtdµ−q (t)
χn (t) dµ
n,ξ,q (t) =
χn (t) dµ
n,ξ,q (t)−
χn (t) dµ
n,ξ,q (t)
n,q,ξ,χn
− χn (p) [p]
[2]qp
n,qp,ξp,χn
where Lemma 3.2, Lemma 3.3 and Lemma 3.4 are used.
This theorem will be mainly used in the next section, where certain applications of p-adic
generalized twisted (h, q)-Euler-l-function are given.
4. Kummer Congruences for Generalized Twisted (h, q)-Euler Numbers
This section is devoted to an application of the p-adic generalized twisted (h, q)-Euler-l-function
to an important number theoretic concept, congruences systems. In particular, we derive Kummer-
type congruences for generalized twisted (h, q)-Euler numbers by using p-adic integral representa-
tion of p-adic generalized twisted (h, q)-Euler-l-function and Theorem 3.6.
In the sequel, we assume that q ∈ Cp with |1− q|p < 1. Then q ≡ 1 (modZp). For t ∈ X
we have [t]q ≡ t (modZp), thus 〈t〉q ≡ 1 (modpZp). For a positive integer c, the forward difference
operator ∆c acts on a sequence {am} by ∆cam = am+c − am. The powers ∆
c of ∆c are defined
by ∆0c =identity and for any positive integer k, ∆
c = ∆c ◦∆
c . Thus
∆kcam =
am+jc.
For simplicity in the notation, we write
n,q,ξ,χn
n,q,ξ,χn
− χn (p) [p]
[2]qp
n,qp,ξp,χn
Theorem 4.1 For n ∈ Z, n > 0 and c ≡ 0 (mod (p− 1)), we have
n,q,ξ,χn
modpkZp
Proof. Since ∆kc is a linear operator, by Theorem 3.6 we have
n,q,ξ,χn
= ∆kc l
p,q,ξ (−n, χ) = ∆
(h−1)tξtdµ−q (t)
(h−1)tξtdµ−q (t)
(h−1)tξt
q − 1
dµ−q (t) .
Now, 〈t〉q ≡ 1 (modpZp), which implies that 〈t〉
q ≡ 1 (modpZp) since c ≡ 0 (mod (p− 1)), and thus
q − 1
modpkZp
Therefore
∆kc l
p,q,ξ (−n, χ) ≡ 0
modpkZp
from which the result follows.
Theorem 4.2 Let n and n′ be positive integers such that n ≡ n′ (mod (p− 1)). Then, we have
n,q,ξ,χn
n′,q,ξ,χn′
(modpZp) .
Proof. Without loss of generality, let n > n′. Then
p,q,ξ (−n, χ)− l
p,q,ξ (−n
′, χ) =
(h−1)tξt
q − 1
dµ−q (t) .
Since n− n′ ≡ 0 (mod (p− 1)), we have 〈t〉
q − 1 ≡ 0 (modpZp), which entails the result.
Acknowledgment: This work was supported by Akdeniz University Scientific Research Projects
Unit.
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	Introduction
	Generalized Twisted q-Euler Numbers
	p-adic Twisted ( h,q) -l-Functions
	Kummer Congruences for Generalized Twisted ( h,q) -Euler Numbers

  The purpose of this paper is to construct the p-adic twisted
(h,q)-Euler-l-function, which interpolates the twisted generalized twisted
Euler numbers attached to chi at a negative integer.

Introduction
Let N, Z, Q, R and C denote, respectively, sets of positive integer, integer, rational, real and
complex numbers as usual. Let p be a fixed odd prime number and x ∈ Q. Then there exists
integers m, n and νp (x) such that x = p
νp(x)m/n and p does not divide either m or n. Let |·|p
be defined such that |x|p = p
−νp(x) and |0|p = 0. Then |·|p is a valuation on Q which satisfies the
non-Archimedean property
|x+ y|p 6 max
|x|p , |y|p
Completion ofQ with respect to |·|p is denoted by Qp and called the field of p-adic rational numbers.
But Qp itself is not complete with respect to |·|p. Cp is the completion of the algebraic closure of
Qp and Zp =
x ∈ Qp : |x|p 6 1
is called the ring of p-adic rational integers (see [14], [17]).
Let d be a fixed positive odd integer and let
X = Xd = lim←−
Z/dpNZ
, X1 = Zp,
0<a<dp
(a,p)=1
a+ dpNZp
a+ dpNZp =
x ∈ X : x ≡ a
moddpN
where N ∈ N and a ∈ Z with 0 6 a < dpN ([1], [5], [12], [18]).
When talking about q-extensions, q can variously be considered as an indeterminate, a complex
number q ∈ C or a p-adic number q ∈ Cp. If q ∈ C, we normally assume that |q| < 1. If q ∈ Cp,
we assume that |1− q|p < p
−1/(p−1) so that for |x|p 6 1, we have q
x =exp(xlogq) ([1], [2], [5], [6],
http://arxiv.org/abs/0704.1397v1
[7]). We use the notations
[x]q =
1− qx
and [x]
1− (−q)
1 + q
We say that f is uniformly differentiable function at a point a ∈ X, and denote this property by
f ∈ UD (X), if the quotient of the differences
f (x)− f (y)
has a limit l = f ′ (a) as (x, y)→ (a, a). For f ∈ UD (X), the p-adic invariant q-integral on X was
defined by
Iq (f) =
f (t) dµq (t) = lim
[dpN ]q
dpN−1
f (a) qa
(cf. [5], [7]), where for any positive integer N
a+ dpNZp
[dpN ]q
(cf. [5], [6], [7]).
The concept of twisted has been applied by many authors to certain functions which interpolate
certain number sequences. In [15], Koblitz defined twisted Dirichlet L-function which interpolates
twisted Bernoulli numbers in the field of complex numbers. In [20], Simsek constructed a q-
analogue of the twisted L-function interpolating q-twisted Bernoulli numbers. Kim et.al. [12]
derived a p-adic analogue of the twisted L-function by using p-adic invariant integrals. By using
the definition of h-extension of p-adic q-L-function which is constructed by Kim [11], Simsek
[22, 23] and Jang [4] defined twisted p-adic generalized (h, q)-L-function. In [18], Satoh derived
p-adic interpolation function for q-Frobenius-Euler numbers. Simsek [21] gave twisted extensions
of q-Frobenius-Euler numbers and their interpolating function q-twisted l-series. In [1], Cenkci
et.al. constructed generalized p-adic twisted l-function in p-adic number field. Recently, Kim and
Rim [13] defined twisted q-Euler numbers by using p-adic invariant integral on Zp in the fermionic
sense. In that paper, they raised the following question: Find a p-adic analogue of the q-twisted l-
function which interpolates E
(h,1)
n,ξ,q,χ, the generalized twisted q-Euler numbers attached to χ [8], [10].
In a forthcoming paper, Rim et.al. [16] answered this question by constructing partial (h, q)-zeta
function motivating from a method of Washington [24, 25].
In this paper, we construct p-adic generalized twisted (h, q)-Euler-l-function by employing p-adic
invariant measure on p-adic number field. This is the answer of the part of the question posed in
[13]. This way of derivation of p-adic generalized twisted (h, q)-Euler-l-function is different from
that of [16], and leads an explicit integral representation for this function. As an application of
the derived integral representation, we obtain general congruences systems for generalized twisted
q-Euler numbers associated with χ, containing Kummer-type congruences.
2. Generalized Twisted q-Euler Numbers
In this section, we give a brief summary of the concepts p-adic q-integrals and Euler numbers
and polynomials. Let UD (X) be the set of all uniformly differentiable functions on X. For any
f ∈ UD (X), Kim defined a q-analogue of an integral with respect to a p-adic invariant measure in
[5, 7] which was called p-adic q-integral. The p-adic q-integral was defined as follows:
Iq (f) =
f (t) dµq (t) = lim
[dpN ]q
dpN−1
f (a) qa.
Note that
I1 (f) = lim
Iq (f) =
f (t) dµ1 (t) = lim
dpN−1
f (a)
is the Volkenborn integral (see [17]).
The Euler zeta function ζE (s) is defined by means of
ζE (s) = 2
for s ∈ C with Re(s) > 1 (cf. [8]). For a Dirichlet character χ with conductor d, d ∈ N, d is odd,
the l-function associated with χ is defined as ([8])
l (s, χ) = 2
χ (k) (−1)
for s ∈ C with Re(s) > 1. This function can be analytically continued to whole complex plane,
except s = 1 when χ = 1; and when χ = 1, it reduces to Euler zeta function ζE (s). In [9],
(h, q)-extension of Euler zeta function is defined by
E,q (s, x) = [2]q
[k + x]
with s, h ∈ C, Re(s) > 1 and x 6=negative integer or zero. (h, q)-Euler polynomials are defined by
the p-adic q-integral as
E(h)n,q (x) =
q(h−1)t [t+ x]
q dµ−q (t) ,
for h ∈ Z. E
n,q (0) = E
n,q are called (h, q)-Euler numbers. In [9], it has been shown that for
n ∈ Z, n > 0
E,q (−n, x) = E
n,q (x) ,
thus we have
E(h)n,q (x) = [2]q
qhk [k + x]
from which the following entails:
E(h)n,q (x) =
(1− q)
1 + qh+j
In [8, 9], (h, q)-extension of the l-function associated with χ is defined by
l(h)q (s, χ) = [2]q
χ (k) (−1)
for h, s ∈ C with Re(s) > 1. The negative integer values of s are determined explicitly by
l(h)q (−n, χ) = E
n,q,χ,
for n ∈ Z, n > 0 where E
n,q,χ are the generalized (h, q)-Euler numbers associated with χ defined
E(h)n,q,χ =
χ (t) q(h−1)t [t]
q dµ−q (t)
= [2]q
χ (k) (−1)
qhk [k]
Now assume that q ∈ Cp with |1− q|p < 1. From the definition of p-adic invariant q integral on
X, Kim [8] defined the integral
I−1 (f) = lim
Iq (f) =
f (t) dµ−1 (t) (2.1)
for f ∈ UD (X). Note that
I−1 (f1) + I−1 (f) = 2f (0) , (2.2)
where f1 (t) = f (t+ 1). Repeated application of last formula yields
I−1 (fn) = (−1)
I−1 (f) + 2
n−1−j
f (j) , (2.3)
with fn (t) = f (t+ n).
Let Tp =
Cpn = lim
Z/pnZ, where Cpn =
w ∈ X : wp
is the cyclic group of order
pn. For w ∈ Tp, let φw : Zp → Cp denote the locally constant function defined by t→ w
For f (t) = φw (t) e
zt, we obtain
φw (t) e
ztdµ−1 (z) =
wez + 1
using (2.1) and (2.2), and
χ (t)φw (t) e
ztdµ−1 (t) = 2
χ (i)φw (i) e
wdedz + 1
using (2.1) and (2.3) (cf. [8]). As a consequence, the twisted Euler numbers and generalized twisted
Euler numbers associated with χ can respectively be defined by
wez + 1
, and 2
χ (i)φw (i) e
wdedz + 1
En,w,χ
from which
tnφw (t) dµ−1 (t) = En,w, and
χ (t) tnφw (t) dµ−1 (t) = En,w,χ
follow.
Twisted extension of (h, q)-Euler zeta function is defined by
E,q,w (s, x) = [2]q
wkqhk
[k + x]
with h, s ∈ C, Re(s) > 1 and x 6=negative integer or zero. For n ∈ Z, n > 0 and h ∈ Z, this
function gives
E,q,w (−n, x) = E
n,q,w (x) ,
where E
n,q,w (x) are the twisted q-Euler polynomials defined as
E(h)n,q,w (x) =
q(h−1)tφw (t) [x+ t]
q dµ−q (t)
= [2]q
wkqhk [k + x]
by using p-adic invariant q-integral on X in the fermionic sense (cf. [13], [16]). The following
expressions for twisted (h, q)-Euler polynomials can be verified from the defining equalities:
E(h)n,q,w (x) =
(1− q)
1 + wqh+j
, (2.4)
E(h)n,q,w (x) =
[2]qd
qhawa (−1)
n,qd,wd
, (2.5)
where n, d ∈ N with d is odd. From (2.4), the twisted (h, q)-Euler polynomials can be determined
explicitly. A few of them are
0,q,w (x) =
1 + q
1 + wqh
1,q,w (x) =
1 + q
1 + wqh
1 + wqh+1
2,q,w (x) =
1 + q
(1− q)
1 + wqh
1 + wqh+1
1 + wqh+2
For x = 0, E
n,q,w (0) = E
n,q,w are called twisted (h, q)-Euler numbers. Thus we can write
E(h)n,q,w (x) =
qxj [x]
j,q,w.
Let χ be a Dirichlet character of conductor d with d ∈ N and d is odd. Then the generalized
twisted (h, q)-Euler numbers associated with χ are defined as
E(h)n,q,w,χ =
χ (t) q(h−1)tφw (t) [t]
q dµ−q (t) .
These numbers arise at the negative integer values of the twisted (h, q)-Euler-l-function which is
defined by
l(h)q,w (s, χ) = [2]q
χ (k) (−1)
wkqhk
with h, s ∈ C, Re(s) > 1. Indeed, for n ∈ Z, n > 0 and h ∈ Z, we have
l(h)q,w (−n, χ) = E
n,q,w,χ
(cf. [13], [16]).
We conclude this section by stating the distribution property for generalized twisted (h, q)-Euler
numbers associated with χ, which will take a major role in constructing a measure in the next
section.
For n, d ∈ N with d is odd, we have
E(h)n,q,w,χ =
[2]qd
qhawaχ (a) (−1)
n,qd,wd
3. p-adic Twisted (h, q)-l-Functions
In this section we first focus on defining a p-adic invariant measure, which is apparently an
important tool to construct p-adic twisted (h, q)-Euler-l-function in the sense of p-adic invariant
q-integral. We afterwards give the definition of p-adic twisted (h, q)-Euler-l-function, together with
Witt’s type formulas for twisted and generalized twisted (h, q)-Euler numbers.
Throughout, we assume that ξ is the rth root of unity with (r, pd) = 1, where p is an odd prime
and d is an odd natural number. If (r, pd) = 1, it has been known that |1− ξ|p > 1 (see [15], [19])
and ξ lies in the cyclic group Cpn =
w : wp
. The following theorem plays a crucial role in
constructing p-adic generalized twisted (h, q)-Euler-l-function on X.
Theorem 3.1 Let q ∈ Cp with |1− q|p < p
−1/(p−1) and ξ is the rth root of unity with |1− ξ|p > 1.
For N ∈ Z, n ∈ Z, n > 0, let µ
n,ξ,q be defined as
n,ξ,q
a+ dpNZp
ξaqhaE
n,qdp
Then µ
n,ξ,q extends uniquely to a measure on X.
Proof. In order to show that µ
n,ξ,q is a measure on X, we need to show that it is a distribution
and is bounded on X.
To show it is a distribution on X, we check the equality
n,ξ,q
a+ idpN + dpN+1Zp
n,ξ,q
a+ dpNZp
Beginning the calculation from right hand side yields
n,ξ,q
a+ idpN + dpN+1Zp
dpN+1
a+idpN
ξa+idp
qh(a+idp
n,qdp
a+ idpN
dpN+1
ξaqha [p]
n,(qdpN )
,(ξdpN )
ξaqhaE
n,qdp
n,ξ,q
a+ dpNZp
where we have used (2.5).
To present boundedness, we use equation (2.4) to expand the polynomial E
n,qdp
so that
n,ξ,q
a+ dpNZp
(1− q)
n (−1)
ξaqha
1 + ξdp
N+jdpN
Now, since d is an odd natural number and p is an odd prime, we have
N+jdpN
1, so by induction on j, we obtain
n,ξ,q
a+ dpNZp
for a constant M . This is what we require, so the proof is completed.
Let χ be a Dirichlet character with conductor d. Then we can express the generalized twisted
(h, q)-Euler numbers associated with χ as an integral over X, by using the measure µ
n,ξ,q.
Lemma 3.2 For n ∈ Z, n > 0, we have
χ (t) dµ
n,ξ,q (t) = E
n,q,ξ,χ.
Proof. From the definition of p-adic invariant integral, we have
χ (t) dµ
n,ξ,q (t) = lim
dpN−1
χ (c)
ξcqhcE
n,qdp
Writing c = a+ dm with a = 0, 1, . . . , d− 1 and m = 0, 1, 2, . . ., we get
χ (t) dµ
n,ξ,q (t) = [d]
[2]qd
χ (a) (−1)
ξaqha
× lim
[2]qd
(qd)p
n,(qd)p
,(ξd)p
= [d]
[2]qd
χ (a) (−1)
ξaqhaE
n,qd,ξd
Assuming χ (0) = 0 and by the fact that χ (d) = 0, last expression equals E
n,ξ,q,χ, and the proof
is completed.
Since it is impossible to have a non-zero translation-invariantmeasure on X, µ
n,ξ,q is not invariant
under translation, but satisfies the following:
Lemma 3.3 For a compact-open subset U of X, we have
n,ξ,q (pU) = [p]
[2]qp
n,ξp,qp (U) .
Proof. Let U = a+ dpNZp be the compact-open subset of X. Then
n,ξ,q (pU) = µ
n,ξ,q
pa+ dpN+1Zp
dpN+1
ξpaqhpaE
n,qdp
dpN+1
[2]qp
[2]qp
(qp)dp
n,(qp)dp
,(ξp)dp
[2]qp
n,ξp,qp
a+ dpNZp
= [p]
[2]qp
n,ξp,qp (U) ,
which is the desired result.
Next, we give a relation between µ
n,ξ,q and µ−q.
Lemma 3.4 For any n ∈ Z, n > 0, we have
n,ξ,q (t) = q
(h−1)tξt [t]
q dµ−q (t) .
Proof. From the definition of µ
n,ξ,q and expansion of twisted (h, q)-Euler polynomials, we have
n,ξ,q
a+ dpNZp
(1− q)
n (−1)
ξaqha
1 + ξdp
N+jdpN
By the same method presented in [7], we obtain
n,ξ,q
a+ dpNZp
(1− q)
n (−1)
ξaqha
1 + q
ξaq(h−1)a [a]
q (−1)
qa = q(h−1)aξa [a]
q lim
1−(−qdpN )
1−(−q)
= q(h−1)aξa [a]
q lim
a+ dpNZp
We thus have
n,ξ,q (t) = q
(h−1)tξt [t]
q dµ−q (t) ,
the desired result.
Let ω denote the Teichmüller character mod p. For an arbitrary character χ and n ∈ Z, let
χn = χω
−n in the sense of product of characters. For t ∈ X∗ = X − pX, we set 〈t〉q = [t]q /ω (t).
Since
〈t〉q − 1
< p−1/(p−1), 〈t〉
q is defined by exp
slogp 〈t〉q
for |s|p 6 1, where logp is the
Iwasawa p-adic logarithm function ([3]). For |1− q|p < p
−1/(p−1), we have 〈t〉
q ≡ 1
modpN
We now define p-adic generalized twisted (h, q)-Euler-l-function.
Definition 3.5 For s ∈ Zp,
p,q,ξ (s, χ) =
(h−1)tξtdµ−q (t) .
The values of this function at non-positive integers are given by the following theorem:
Theorem 3.6 For any n ∈ Z, n > 0,
p,q,ξ (−n, χ) = E
n,q,ξ,χn
− χn (p) [p]
[2]qp
n,qp,ξp,χn
Proof.
p,q,ξ (−n, χ) =
(h−1)tξtdµ−q (t) =
χn (t) [t]
(h−1)tξtdµ−q (t)
χn (t) dµ
n,ξ,q (t) =
χn (t) dµ
n,ξ,q (t)−
χn (t) dµ
n,ξ,q (t)
n,q,ξ,χn
− χn (p) [p]
[2]qp
n,qp,ξp,χn
where Lemma 3.2, Lemma 3.3 and Lemma 3.4 are used.
This theorem will be mainly used in the next section, where certain applications of p-adic
generalized twisted (h, q)-Euler-l-function are given.
4. Kummer Congruences for Generalized Twisted (h, q)-Euler Numbers
This section is devoted to an application of the p-adic generalized twisted (h, q)-Euler-l-function
to an important number theoretic concept, congruences systems. In particular, we derive Kummer-
type congruences for generalized twisted (h, q)-Euler numbers by using p-adic integral representa-
tion of p-adic generalized twisted (h, q)-Euler-l-function and Theorem 3.6.
In the sequel, we assume that q ∈ Cp with |1− q|p < 1. Then q ≡ 1 (modZp). For t ∈ X
we have [t]q ≡ t (modZp), thus 〈t〉q ≡ 1 (modpZp). For a positive integer c, the forward difference
operator ∆c acts on a sequence {am} by ∆cam = am+c − am. The powers ∆
c of ∆c are defined
by ∆0c =identity and for any positive integer k, ∆
c = ∆c ◦∆
c . Thus
∆kcam =
am+jc.
For simplicity in the notation, we write
n,q,ξ,χn
n,q,ξ,χn
− χn (p) [p]
[2]qp
n,qp,ξp,χn
Theorem 4.1 For n ∈ Z, n > 0 and c ≡ 0 (mod (p− 1)), we have
n,q,ξ,χn
modpkZp
Proof. Since ∆kc is a linear operator, by Theorem 3.6 we have
n,q,ξ,χn
= ∆kc l
p,q,ξ (−n, χ) = ∆
(h−1)tξtdµ−q (t)
(h−1)tξtdµ−q (t)
(h−1)tξt
q − 1
dµ−q (t) .
Now, 〈t〉q ≡ 1 (modpZp), which implies that 〈t〉
q ≡ 1 (modpZp) since c ≡ 0 (mod (p− 1)), and thus
q − 1
modpkZp
Therefore
∆kc l
p,q,ξ (−n, χ) ≡ 0
modpkZp
from which the result follows.
Theorem 4.2 Let n and n′ be positive integers such that n ≡ n′ (mod (p− 1)). Then, we have
n,q,ξ,χn
n′,q,ξ,χn′
(modpZp) .
Proof. Without loss of generality, let n > n′. Then
p,q,ξ (−n, χ)− l
p,q,ξ (−n
′, χ) =
(h−1)tξt
q − 1
dµ−q (t) .
Since n− n′ ≡ 0 (mod (p− 1)), we have 〈t〉
q − 1 ≡ 0 (modpZp), which entails the result.
Acknowledgment: This work was supported by Akdeniz University Scientific Research Projects
Unit.
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	Introduction
	Generalized Twisted q-Euler Numbers
	p-adic Twisted ( h,q) -l-Functions
	Kummer Congruences for Generalized Twisted ( h,q) -Euler Numbers
