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$q$-Volkenborn Integration and Its Applications


q -VOLKENBORN INTEGRATION AND ITS APPLICATIONS

arXiv:math/0510524v1 [math.NT] 25 Oct 2005

TAEKYUN KIM Department of Mathematics Educations, Kongju National University, Kongju 314-701, S. Korea e-mail: tkim@kongju.ac.kr

Abstract. The main purpose of this paper is to present a systemic study of some families of multiple q -Euler numbers and polynomials. In particular, by using the q -Volkenborn integration on Zp , we construct p-adic q -Euler numbers and polynomials of higher order. We also de?ne new generating functions of multiple q -Euler numbers and polynomials. Furthermore, we construct Euler q -Zeta function.

1. Introduction For any complex number z , it is well known that the familiar Euler polynomials En (z ) are de?ned by means of the following generating function, cf.[3, 5, 6, 9, 13]: tn 2 zt e = E ( z ) , ( | t| < π ) . n et + 1 n ! n=0


F (z, t) =

(1)

We note that, by substituting z = 0 into (1), En (0) = En is the familiar n-th Euler number de?ned by G(t) = F (0, t) = 2 tn = E , (|t| < π ), cf.[4, 5]. n et + 1 n=0 n!


By the meaning of the generalization of En , Frobenius-Euler numbers and polynomials are also de?ned by tn 1 ? u xt tn 1?u = H ( u ) , and e = H ( u, x ) (u ∈ C with |u| > 1), cf.[16]. n n et ? u n=0 n! et ? u n ! n=0
Typeset by AMS-TEX
∞ ∞

1

Over ?ve decades ago, Calitz [2, 3] de?ned q -extension of Frobenius-Euler numbers and polynomials and proved properties analogous to those satis?ed Hn (u) and Hn (u, x). Recently, Satoh [14, 15] used these properties, especially the so-called distribution relation for the q -Frobenius-Euler polynomials, in order to construct the corresponding q -extension of the p-adic measure and to de?ne a q -extension of p-adic l-function lp,q (s, u). Let p be a ?xed odd prime in this paper. Throughout this paper, the symbols Z, Zp , Qp , C and Cp , denote the ring of rational integers, the ring of p-adic integers, the ?eld of p-adic numbers, the complex number ?eld, and the completion of the algebraic closure of Qp , respectively. Let νp (p) be the normalized exponential valuation of Cp with |p|p = p?νp (p) = p?1 . When one speaks of q -extension, q can be regarded as an indeterminate, a complex number q ∈ C, or a p-adic number q ∈ Cp ; it is always clear from the context. If q ∈ C, then one usually assumes that |q | < 1. If q ∈ Cp , then one usually assumes that 1 |q ? 1|p < p? p?1 , and hence q x = exp(x log q ) for x ∈ Zp . In this paper, we use the below notation [ x] q = 1 ? qx , 1?q (a : q )n = (1 ? a)(1 ? aq ) · · · (1 ? aq n?1 ), cf.[6, 7, 8, 9, 10, 11, 14 ].

Note that limq →1 [x]q = x for any x with |x|p ≤ 1 in the p-adic case. For a ?xed positive integer d with (p, d) = 1, set
N X = Xd = ← lim ? Z/dp , N

X1 = Zp , X ? =
0<a<dp (a,p)=1

a + dpZp ,

a + dpN Zp = {x ∈ X |x ≡ a (mod pN )}, where a ∈ Z satis?es the condition 0 ≤ a < dpN , (cf.[10,11]). We say that f is a uniformly di?erentiable function at a point a ∈ Zp , and write f ∈ U D(Zp ), if the di?erence quotients )?f (y ) have a limit f ′ (a) as (x, y ) → (a, a), cf.[11]. For f ∈ U D(Zp ), let us Ff (x, y ) = f (xx ?y begin with the expression 1 [pN ]
q 0≤j<pN

q j f (j ) =
0≤j<pN

f (j )?q (j + pN Zp ), cf.[7, 8, 9, 11],

which represents a q -analogue of Riemann sums for f . The integral of f on Zp is de?ned as the limit of those sums(as n → ∞) if this limit exists. The q -Volkenborn integral of a function f ∈ U D(Zp ) is de?ned by 1 Iq ( f ) = f (x)d?q (x) = f (x)d?q (x) = lim N →∞ [dpN ]q X Xd 2
dpN ?1

f ( x) q x .
x=0

Recently, we considered another construction of a q -Eulerian numbers, which are di?erent than Carlitz’s q -Eulerian numbers as follows, cf.[6, 12, 13]:
∞ ∞

Fq (x, t) = [2]q
n=0

(?1) q e

n n [n+x]q t

=
n=0

En,q (x)

tn . n!

Thus we have En,q [2]q = En,q (0) = (1 ? q )n [2]q (1 ? q )n
n n

l=0

n (?1)l , l 1 + q l+1

En,q (x) =

l=0

n (?1)l lx q , l 1 + q l+1

where n is a binomial coe?cient, cf.[13]. l Note that limq →1 En,q = En and limq →1 En,q (x) = En (x). In [12], we also proved that q -Eulerian polynomial En,q (x) can be represented by q -Volkenborn integral as follows: [ x + x1 ] k q d??q (x1 ) =
Xd q x [2] Zp

[ x + x1 ] k q d??q (x) = Ek,q (x),

for k, d ∈ N,

q x where ??q (x + pN Zp ) = N (?1) . 1+q p The purpose of this paper is to present a systemic study of some families of multiple q -Euler numbers and polynomials. In particular, by using the q -Volkenborn integration on Zp , we construct p-adic q -Euler numbers and polynomials of higher order. We also de?ne new generating function of these q -Euler numbers and polynomials of higher order. Furthermore, we construct Euler q -ζ -function. From section 2 to section 5, we assume that 1 q ∈ Cp with |1 ? q |p < p? p?1 .

2. q -Euler numbers and polynomials associated with an invariant p-adic q -integrals on Zp Let h ∈ Z, k ∈ N = {1, 2, 3, · · · }, and let us consider the extended higher-order q -Euler numbers as follows:
(h,k) = Em,q Zp

···
Zp k times

x1 (h?1)+···+xk (h?k) [ x1 + x2 + · · · + xk ] m d??q (x1 ) · · · d??q (xk ). q q

Then we have
(h,k) Em,q

[2]k q = (1 ? q )m

m

l=0

(?1)l m . l (?q h+l : q ?1 )k

3

From the de?nition of Em,q , we can easily derive the below:
(h,k) (h?1,k) Em,q = Em,q + (q ? 1)Em+1,q , (m ≥ 0). (h?1,k)

(h,k)

It is easy to show that ···
Zp Zp k+1 times m

q

k+1 j =1 (m?j )xj

d??q (x1 ) · · · d??q (xk+1 )

=
j =1

m (q ? 1)j j

k+1

···
Zp k+1 times

[
Zp l=1

? xl ] j q q

k j =1

jxj

d??q (x1 ) · · · d??q (xk+1 ),

(2)

and we also get ···
Zp Zp k+1 times

q

k+1 j =1 (m?j )xj

d??q (x1 ) · · · d??q (xk+1 ) =

+1 [2]k q . (?q m : q ?1 )k+1

(3)

From (2) and (3), we can derive the below proposition. Proposition 1. For m, k ∈ N, we have
m +1 [2]k m q j (0,k+1) (q ? 1) Ej,q = , j (?q m : q ?1 )k+1

j =0

(h,k) Em,q

[2]k q = (1 ? q )m

m

l=0

(?1)l m . l (?q h+l : q ?1 )k

Remark. Note that En,q = En,q , where En,q are the q -Euler numbers(see [13]). (h,k) From the de?nition of En,q , we can derive
i

(1,1)

j =0

i (h?1,k) (q ? 1)j Em?i+j,q = j

i?1 j =0

i?1 (h,k) (q ? 1)j Em+j ?i,q j

for m ≥ i. By simple calculation, we easily see that
m

j =0

m (h,1) (q ? 1)j Ej,q = j

q mx q (h?1)x d??q (x) =
Zp

[2]q . [2]q m+h
(0,h)

Furthermore, we can give the following relation for the q -Euler numbers, Em,q ,:
m

j =0

[2]k m q j (0,k) (q ? 1) Ej,q = . m j (?q : q ?1 )k 4

(4)

3. Polynomials En,q (x) We now de?ne the polynomials En,q (x) (in q x ) by
(0,k) En,q ( x) = Zp (0,k)

(0,k)

···
Zp k times

[ x1 + x2 + · · · + xk ] m q q

k j =1

jxj

d??q (x1 ) · · · d??q (xk ).

Thus, we have
m

(q ? 1)

m

(0,k) Em,q ( x)

=

[2]k q
j =0

1 m jx . q (?1)m?j j (?q : q ?1 )k j [2]k q , (?q m : q ?1 )k

(5)

It is not di?cult to show that ···
Zp Zp k times

q

m j =1 (m?j )xj +mx

d??q (x1 ) · · · d??q (xk ) = q mx

and
m

···
Zp Zp k times

q

m j =1 (m?j )xj +mx

d??q (x1 ) · · · d??q (xk ) =
j =0

m (0,k) (q ? 1)j Ej,q (x). j

Therefore we obtain the following. Lemma 2. For m, k ∈ N, we have
m

j =0

q mx [2]k m q j (0,k) (q ? 1) Ej,q (x) = , m ? j (?q : q 1 )k [2]k q = (1 ? q )m
m

(0,k) Em,q ( x)

j =0

1 m jx q (?1)j . j j (?q : q ?1 )k

(6)

Let l ∈ N with l ≡ 1 (mod 2). Then we get easily ?m ?
k

···

Zp

Zp

k times

?x +
l?1

j =1

xj ? q ?
q

k j =1

jxj

d??q (x1 ) · · · d??q (xk )

[l ]m q = k [l]?q

q?
i1 ,··· ,ik =0
k j =1 ij

k j =2 (j ?1)ij

· (?1)

···
Zp Zp k times

? ?

x+ l

k j =1 ij

k

+
j =1

xj ? q ?l
ql

?m

k j =1

jxj

d??q l (x1 ) · · · d??q l (xk ).

From this, we can derive the following “multiplication formula”: 5

Theorem 3. Let l be an odd positive integer. Then
(0,k) Em,q ( x) =

[l ]m q k [l]?q

l?1

q?
i1 ,··· ,ik =0

k j =2 (j ?1)ij

(?1)

k l=1 il

Em,q l (

(0,k)

x + i1 + · · · + ik ). l

(7)

Moreover,
(0,k) Em,q (lx)

[l ]m q = k [l]?q

l?1

q?
i1 ,··· ,ik =0

k j =2 (j ?1)ij

(?1)

k l=1 il

Em,q l (x +

(0,k)

i1 + · · · + ik ). l

(8)

From (4) and (5), we can also derive the below expression for En,q (x):
m (0,k) Em,q ( x)

(0,k)

=
i=0

m (0,k) m?i ix Ei,q [x]q q , i

(9)

whence also
m (0,k) Em,q (x

+ y) =
j =0

m m?i jy (0,k) [y ]q q Ej,q (x). j
(h,1)

(10)

4. Polynomials Em,q (x) Let us de?ne
(h,1) Em,q ( x) = Zp x1 (h?1) [ x + x1 ] m d??q (x1 ). q q

(11)

Then we have
(h,1) Em,q ( x)

[2]q = (1 ? q )m

m

l=0

1 m (?1)l q lx . l (1 + q l+h )

By simple calculation of q -Volkenvorn integral, we note that qx
Zp x1 (h?1) [ x + x1 ] m d??q (x1 ) = (q ? 1) q q Zp +1 x1 (h?2) [ x + x1 ] m q d??q (x1 ) q

+
Zp

x1 (h?2) [ x + x1 ] m d??q (x1 ). q q

Thus, we have
(h,1) (h?1,1) q x Em,q (x) = (q ? 1)Em+1,q (x) + Em,q ( x) . 6 (h?1,1)

(12)

It is easy to show that
m

[x +
Zp

x1 ] m q

q

(h?1)x1

d??q (x1 ) =
j =0

m m?j jx [ x] q q j

(h?1)x1 [ x1 ] j d??q (x1 ). qq Zp

This is equivalent to
m (h,1) Em,q ( x)

=
j =0

m m?j jx (h,1) [ x] q q Ej,q j
m

(h,1) = q x Eq + [ x] q

,

for m ≥ 1,
(h,1) n

where we use the technique method notation by replacing (Eq From (11), we can derive

) by En,q , symbolically. (13)

(h,1)

(h,1) (h,1) q h Em,q (x + 1) + Em,q (x) = [2]q [x]m q .

For x = 0 in (13), this gives
(h,1) q h qEm,q +1 m (h,1) + Em,q = δ0,k ,

(14)

where δ0,k is Kronecker symbol. By the simple calculation of q -Volkenborn integration, we easily see that [2]q q x1 (h?1) d??q (x1 ) = . [2]q h Zp Thus, we have E0,q
(h,1)

=

[2]q [2]q h .

From the de?nition of q -Euler polynomials, we can derive

?x1 (h?1) (h,1) [1 ? x + x1 ]m d??q (x1 ) = q m+h?1 (?1)m Em,q ( x) . q ?1 q Zp

Therefore we obtain the below “complementary formula”: Theorem 4. For m ∈ N, n ∈ Z, we have
(h,1) ( x) . Em,q ?1 (1 ? x) = (?1)m q m+h?1 Em,q (h,1)

(15)

In particular, for x = 1, we see that
(h,1) Em,q ?1(0) = (?1)m q m+h?1 Em,q (1) (h,1) = (?1)m?1 q m?1 Em,q , (h,1)

for m ≥ 1.

(16)

For l ∈ N with l ≡ 1 (mod 2), we have
x1 (h?1) q (h?1)x1 [x + x1 ]m d??q (x1 ) q q Zp

[l ]m q = [l]?q

l?1

q hi (?1)i
i=0 Zp

x+i + x1 l 7

m

q x1 (h?1)l d??q l (x1 ).
ql

Thus, we can also obtain the following:

Theorem 5. (Multiplication formula) For l ∈ N with l ≡ 1 (mod 2), we have [2]q m (h,1) x + i (h,1) q hi (?1)i Em,q l ( [l ]q ) = Em,q ( x) . [2]q l l i=0 Furthermore, i [2]q m (h,1) (h,1) q hi (?1)i Em,q l (x + ) = Em,q [l ]q (lx). [2]q l l i=0 5. Polynomials Em,q (x) and h = k It is now easy to combine the above results and de?ne the new polynomials as follows:
(h,k) Em,q ( x) = Zp (h,k) l?1 l?1

···
Zp k times

(h?1)x1 +···+(h?k)xk [ x + x1 + · · · + x k ] m d??q (x1 ) · · · d??q (xk ). q q

Thus, we note that
m

(q ? 1)

m

(h,k) Em,q ( x)

=
j =0

[2]k m q m?j xj . (?1) q j + h (?q : q ?1 )k j

(17)

We may now mention the following formulas which are easy to prove.
(h,k) (h,k) (h?1,k?1) q h Em,q (x + 1) + Em,q (x) = [2]q Em,q ( x) ,

(18)

and
(h+1,k) (h,k) q x Em,q (x) = (q ? 1)Em+1,q (x) + Em,q ( x) . (h,k)

(19)

Let l ∈ N with l ≡ 1 (mod 2). Then we note that
k

···
Zp Zp k times

[x +
j =1 l?1

xj ] m q q

k j =1 (h?j )xj

d??q (x1 ) · · · d??q (xk )

[l ]m q = k [l]?q ·
Zp

qh
i1 ,··· ,ik =0

k j =1 ij ?

k j =2 (j ?1)ij

(?1)

k j =1 ij

···
Zp k times

? ?

x+ l

k j =1 ij

k

+
j =1

xj ? ( q l )
ql

?m

k j =1 (h?j )xj

d??q l (x1 ) · · · d??q l (xk ).

Therefore we obtain the following: 8

Theorem 6. ( Distribution for q-Euler polynomials) For l ∈ N with l ≡ 1 (mod 2). Then we have (20) l?1 [l ]m k k k i1 + · · · + ik q (h,k) (h,k) q h j=1 ij ? j=2 (j ?1)ij (?1) j=1 ij Em,q l x + . Em,q (lx) = k l [l]?q i ,·,i =0
1 k

It is interesting to consider the case h = k , which leads to the desired extension of the q -Euler numbers of higher order, cf.[1]. We shall denote the polynomials in this special (k) (k,k) case by Em,q (x) := Em,q (x). Then we have
m

(q ? 1) and

m

(k) Em,q ( x)

=
j =0

[2]k m q m?j jx (?1) q , j + k j (?q : q ?1 )k

(21)

(k) ( x) . Em,q ?1 (k ? x) = (?1)m q m+(2) Em,q (k)
k

(22)

For x = k in (22), we see that
(k) (k ). Em,q ?1 (0) = (?1)m q m+(2) Em,q (k)
k

(23)

From (18), we can derive the below formula:
(k) (k) (k?1) q k Em,q (x + 1) + Em,q (x) = [2]q Em,q ( x) .

(24)

Putting x = 0 in (17), we obtain
m

(q ? 1) Note that
m

m

(k) Em,q

=
i=0

[2]k m q m?i . (?1) i + k (?q : q ?1 )k i

(25)

i=0

m (q ? 1)i i

···
Zp Zp k times

[ x1 + · · · + xk ] i qq

k ?1 j =1 (k ?j )xj

d??q (x1 ) · · · d??q (xk )

=

[2]k q m + k (?q : q ?1 )

.
k

From this, we can easily derive
m

i=0

[2]k m (k) q (q ? 1)i Ei,q = i (?q m+k : q ?1 )k 9

(26)

and so it follows
(k) (k) Em,q ( x) = ( q x E q + [ x] q ) m , m ≥ 1 , (k) (k)

(27)

where we use the technique method notation by replacing (Eq )n by En,q , symbolically. In particular, from (24), we have
(k) (k) (k?1) q k (qEq + 1)m + Em,q = [2]q Em,q .

(28)

It is easy to see that ···
Zp Zp k times

q (k?1)x1 +···+xk?1 d??q (x1 ) · · · d??q (xk ) =

[2]k q . (?q k : q ?1 )k

Thus, we note that E0,q =

(k)

[2]k q k (?q :q ?1 )k

.

6. Generating function for q -Euler polynomials An obvious generating function for q -Euler polynomials is obtained, from (17), by

t 1 ?q [2]k qe

j =0 ∞

(?1)j q jx (?q j +h : q ?1 )k
n

1 1?q

j

tj j! (29)

=
n=0

(h,k) t En,q

n!

.

From (17), we can also derive the below formula:
(h,k+1) (h,k) (h,k+1) q h?k Em,q (x + 1) = [2]q Em,q (x) ? Em,q ( x) .

(30)

Again from (21) and (25), we get easily
k

···
Zp Zp k times m

[x +
j =1

xj ] m q q

k ?1 j =1 (k ?j )xj

d??q (x1 ) · · · d??q (xk )

=
j =0

m xj q j

k?1

···
Zp Zp k times

[ xk ] j q [x

+
j =1

?j xj ] n q q

k ?1 l=1 (k +j ?l)xl

d??q (x1 ) · · · d??q (xk ).

10

Thus, we note that
m (k) Em,q ( x)

=
j =0

m xj (1) (k+j,k?1) q Ej,q Em?j,q ( x) . j

(31)

Take x = 0 in (31), we have
m (k) Em,q

=
i=0

m (1) (k+j,k?1) Ej,q Em?j,q . i

(32)

So, for k = 2,
(2) Em,q =

m

i=0

m (j +2,1) Ej,q Em?j,q . i

It is not di?cult to show that
h hx [ x] m q q d??q (x) Zp

=
j =0

h (q ? 1)j j

+j [ x] m d??q (x), for h ∈ N . q Zp

From this, we can derive the below:
h (h+1,1) Em,q

=
j =0

h (q ? 1)j Em+j,q , h ∈ N. j

(33)

By (32) and (33), we easily see that
m (2) Em,q

=
j =0

m Ej,q j

j +1

i=0

j+1 (q ? 1)i Em?j +i,q . i

(34)

By (34), for q = 1, we note that
m (2) Em

=
j =0

m Ej Em?j , where j

2 t e +1

k



=
n=0

(k) En

tn . n!

It is easy to show that
m

[ x + x1 + · · · + x k ] m q =
j =0

m m?j j (x1 +x) [ x1 + x] q q [ x2 + · · · + xk ] j q. j

11

By using this, we get easily
k

···
Zp Zp k times m

[x +
j =1

xj ] m q q

k ?1 j =1 (k ?j )xj

d??q (x1 ) · · · d??q (xk )

=
j =0

m jx q j

m?j (k+j ?1)x1 [ x + x1 ] q q d??q (x1 ) Zp
k ?1 j =2 (k ?j )xj

·
Zp

···
Zp k?1 times

[ x2 + · · · + xk ] j qq

d??q (x2 ) · · · d??q (xk ).

Therefore we obtain the following: Theorem 7. For m, k ∈ N, we have
m (k) Em,q ( x) = j =0

m jx (k+j,1) (k?1) q Em?j,q (x)Ej,q . j

(35)

Indeed for x = 0,
m (k) Em,q

=
j =0

m (k+j,1) (k?1) Em?j,q Ej,q j
k+j

m

=
j =0

m (k?1) Ej,q j

(q ? 1)i
j =0

k+j?1 (1) Em?j +i,q . i

(36)

As for q = 1, we get the below formula
m (k) Em

=
j =0

m (k?1) (1) Ej Em?j . j

7. q -Euler zeta function in C In this section, we assume that q ∈ C with |q | < 1. From section 4, we note that [2]q = (1 ? q )m
∞ m

(h,1) Em,q ( x)

l=0

1 m lx q (?1)l l 1 + q l+h (37)

= [2]q
n=0

(?1)n q nh [n + x]n q.

Thus, we can de?ne q -Euler zeta function: 12

De?ntion 8. For s, q ∈ C with |q | < 1, de?ne
h ζE,q (s, x)

= [2]q

(?1)n q nh , s [ n + x ] q n=0



where x ∈ R with 0 < x ≤ 1.
h Note that ζE,q (?m, x) = Em,q (x), for m ∈ N. Let ∞ (h,1)

Fq (t, x) =
n=0

(h,1) En,q ( x)

tn . n!

Then we have


Fq (t, x) = [2]q e = [2]q

t 1 ?q

(?1)n q hn e?
n=0

q n+x 1 ?q

t



(?1)n q hn e[n+x]q t , for h ∈ Z.
n=0

Therefore we obtain the following Lemma 9. For h ∈ Z, we have


Fq (t, x) = [2]q
n=0 ∞

(?1)n q hn e[n+x]q t
(h,1) En,q ( x)

=
n=0

tn . n!

(38)

Let Γ(s) be the gamma function. Then we easily see that 1 Γ(s)
∞ 0 h ts?1 Fq (?t, x)dt = ζE,q (s, x),

for s ∈ C.

(39)

From (38) and (39), we can also derive the below Eq.(40):
h (h,1) ζE,q (?n, x) = En,q ( x) ,

(40)

for n ∈ N.

13

References
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] E. Boros, K. Elbassioni, V. Gurvich, L. Khachiyan, K. Makino, An intersection inequality for discrete distributions and related generation problems, Lecture Notes in Comput. Sci. 2719 (2003), 543-555. L. Carlitz, q -Bernoulli numbers and polynomials, Duke Math. J. 15 (1948), 987-1000. L. Carlitz, q -Bernoulli and Eulerian numbers, Trans. Amer. Math. Soc. 76 (1954), 332-350. E. Deeba and D. Rodriguez,, Stirling’s series and Bernoulli numbers, Amer. Math. Monthly 98 (1991), 423-426. T. Howard, Applications of a recurrences formula for the Bernoulli numbers, J. Number Theory 52 (1995), 157-172. T. Kim, L. C. Jang, H. K. Park, A note on q -Euler and Genocchi numbers, Proc. Japan Academy 77 (2001), 139-141. T. Kim, Power series and asymptotic series associated with the q -analog of the two-variable p-adic L-function, Russ. J. Math. Phys. 12 (2005), 189-196. T. Kim, Non-Archimedean q -integrals associated with multiple Changhee q -Bernoulli polynomials, Russ. J. Math. Phys. 10 (2003), 91-98. T. Kim, An invariant p-adic integral associated with Daehee numbers, Integral Trnasforms and special functions 13 (2002), 65-69. T. Kim, On a q -analogue of the p-adic log gamma functions and related integrals, J. Number Theory 76 (1999), 320-329. T. Kim, q -Volkenborn integration, Russ. J. Math. Phys. 9 (2002), 288-299. T. Kim, A note on q -Volkenborn integration, Proc. Jangjeon Math. Soc. 8 (2005), 13-17. T. Kim, q -Euler and Genocchi numbers, arXiv: math. NT/0506278 vol 1 14 June (2005). J. Satoh, q -analogue of Riemann’s ζ -function and q -Euler numbers, J. Number Theory 31 (1989), 346-362. J. Satoh, Sums products of two q -Bernoulli numbers, J. Number Theory 74 (1999), 173-180. J. Shiratani and S. Yamamoto, On a p-adic interpolating function for the Euler numbers and its derivatives, Mem. Fac. Sci. Kyushu Univ. Math. 39 (1985), 113-125.

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