arXiv:1203.6399v2 [math.NT] 4 Apr 2012
EXPLICIT FORMULAS INVOLVING q-EULER NUMBERS AND POLYNOMIALS SERKAN ARACI, MEHMET ACIKGOZ, AND JONG JIN SEO Abstract. In this paper, we deal with q-Euler numbers and q-Bernoulli numbers. We derive some interesting relations for q-Euler numbers and polynomials by using their generating function and derivative operator. Also, we show between the q-Euler numbers and q-Bernoulli numbers via the p-adic q-integral in the p-adic integer ring.
1. PRELIMINARIES Imagine that p be a fixed odd prime number. Throughout this paper we use the following notations, by Zp denotes the ring of p-adic rational integers, Q denotes the field of rational numbers, Qp denotes the field of p-adic rational numbers, and Cp denotes the completion of algebraic closure of Qp . Let N be the set of natural numbers and N∗ = N ∪ {0}. The p-adic absolute value is defined by 1 |p|p = . p In this paper we assume |q − 1|p < 1 as an indeterminate. [x]q is a q-extension of x which is defined by [x]q =
1 − qx , 1−q
we note that limq→1 [x]q = x (see[1-12]). We say that f is a uniformly differentiable function at a point a ∈ Zp , if the difference quotient f (x) − f (y) Ff (x, y) = x−y has a limit f´(a) as (x, y) → (a, a) and denote this by f ∈ U D (Zp ). Let U D (Zp ) be the set of uniformly differentiable function on Zp . For f ∈ U D (Zp ), let us start with the expressions X X 1 f (ξ) µq ξ + pN Zp , f (ξ) q ξ = N [p ]q N N 0≤ξ 0. Γ (m + n)
=
As a result, we obtain the following theorem Theorem 1. For n ∈ N, we have max{k,m}
X
q
j=1
=
q
e Ek+m−j+1,q k j m + (−1) k+m−j+1 j j
m+1
(−1) (k + m + 1)
k+m k
− [2]q
ek+m+1,q E . k+m+1
Substituting m = k + 1 into Theorem 1, we readily get e k+1 X k E2k+2−j,q j k+1 q + (−1) j j 2k + 2 − j j=1 =
q
k e2k+2,q (−1) E . − [2] q 2k + 2 (2k + 2) 2k+1 k
By (2.1), it follows that max{k,m}
X j=0
k j m ek+m−j−1,q (x) (k + m − j) q + (−1) E j j m−1
= [2]q xk−1 (x − 1)
((k + m) x − k) .
Let m = k in (2.1), we see that k X k j k e2k−j,q (x) = [2] xk (x − 1)k q + (−1) E q j j j=0
Last from equality, we discover the following (2.12) [ k2 ] [ k2 ] X X k e k e2k−2j−1,q (x) = [2] xk (x − 1)k . [2]q E2k−2j,q (x) + (q − 1) E q 2j 2j + 1 j=0 j=0
KIM’S q-EULER NUMBERS AND POLYNOMIALS
5
Here [.] is Gauss’ symbol. Then, taking integral from 0 to 1 both sides of last equality, we get
− [2]q−1 [2]q
[ k2 ] e X k E2k−2j+1,q
2j 2k − 2j + 1
j=0
k
+ [2]q−1 (1 − q)
[ k2 ] X j=0
e k E2k−2j,q 2j + 1 2k − 2j
= [2]q (−1) B (k + 1, k + 1) k
=
[2]q (−1) (2k + 1)
2k k
.
Consequently, we derive the following theorem Theorem 2. The following identity
[2]q
(2.13)
[ k2 ] e X k E2k−2j+1,q
2j 2k − 2j + 1
j=0
+ (q − 1)
j=0
k+1
q (−1) (2k + 1)
=
2k k
is true.
[ k2 ] X
e k E2k−2j,q 2j + 1 2k − 2j
.
In view of (2.1) and (2.12), we discover the following applications:
(2.14)=
k+1 X j=0
=
=
k j k+1 e2k+1−j,q (x) q + (−1) E j j
e2k+1,q (x) + [2]q E
k+1 [X 2 ] k k k e2k+1−2j,q (x) q + + E 2j 2j 2j − 1 j=1
k+1 [X 2 ] k k k e2k−2j,q (x) + q − − E 2j + 1 2j + 1 2j j=0 [ k2 ] [ k2 ] X X k e q−1 k e2k−2j+1 (x) − E2k−2j,q (x) + E 2j 2j + 1 1 + q j=0 j=0
+ [2]q
[ k2 ] X k j=0
+ (q − 1)
2j
[ k2 ] X j=0
e2k+1−2j,q (x) + E
[ k2 ] X j=1
k e2k+1−2j,q (x) E 2j − 1
[ k2 ] k q−1 X k e e2k−2j+1 (x) E2k−2j,q (x) + E 2j + 1 1 + q j=0 2j + 1
By expressions (2.12) and (2.14), we have the following Theorem
6
S. ARACI, M. ACIKGOZ, AND J J. SEO
Theorem 3. For k ∈ N, we have [ k2 ] [ k2 ] X X k e k e2k+1−2j,q (x) E2k+1−2j,q (x) + E [2]q 2j 2j − 1 j=0 j=1
(2.15)
+ (q − 1)
[ k2 ] X
k 2j + 1
j=0
=
k
k
x (x − 1)
e2k−2j,q (x) + E
[2]q x − q
1 e E2k−2j+1 (x) 1+q
3. p-adic integral on Zp associated with Kim’s q-Euler polynomials In this section, we consider Kim’s q-Euler polynomials by means of p-adic qintegral on Zp . Now we start with the following assertion. Let m, k ∈ N, Then by (2.8), Z m xk (x − 1) dµ−q (x) I1 = [2]q Zp
= [2]q
m X m
l
l=0
= [2]q
m X m
l
l=0
(−1)m−l
Z
xl+k dµ−q (x)
Zp
m−l
(−1)
el+k,q E
On the other hand, right hand side of (2.8), k+m−j Z max{k,m} X k + m − j X k j m e q + (−1) I2 = Ek+m−j−l,q xl dµ−q (x) j j l Z p j=0 l=0
=
k+m−j X k + m − j k j m el,q ek+m−j−l,q E q + (−1) E j j l
max{k,m}
X j=0
l=0
Equating I1 and I2 , we get the following theorem
Theorem 4. For m, k ∈ N, we have k+m−j max{k,m} X k + m − j X k j m el,q ek+m−j−l,q E q + (−1) E j j l j=0 l=0 m X m m−l e = [2]q (−1) El+k,q . l l=0
Let us take fermionic p-adic q-inetgral on Zp left hand side of (2.15), we get Z k xk (x − 1) [2]q x − q dµ−q (x) I3 = Zp
=
[2]q
k X k l=0
=
[2]q
l
k X k l=0
l
k−l
(−1)
Z
k+l+1
x
dµ−q (x) − q
Zp
k−l
(−1)
ek+l+1,q − q E
k X k l=0
k X k l=0
l
l
k−l
(−1)
k−l
(−1)
ek+l,q E
Z
Zp
xk+l dµ−q (x)
7
KIM’S q-EULER NUMBERS AND POLYNOMIALS
In other word, we consider right hand side of (2.15) as follows: I4
[ k2 ] 2k−2j+1 Z X 2k − 2j + 1 X k e2k+1−2j−l,q xl dµ−q (x) E = [2]q 2j l Z p j=0 l=0
[ ] X k 2
+
j=1
[ ] X k 2
+
j=0
k 2j − 1 k 2j + 1
2k−2j+1 X l=0
"
Z 2k − 2j + 1 e E2k+1−2j−l,q xl dµ−q (x) l Zp
# R P2k−2j 2k−2j l e2k−2j−l,q (q − 1) j=0 E l Zp x dµ−q (x) R P2k−2j+1 2k−2j+1 e2k−2j−l+1 xl dµ−q (x) E + q−1 1+q
l=0
l
Zp
[ ] 2k−2j+1 X 2k − 2j + 1 X k el,q e2k+1−2j−l,q E = [2]q E 2j l j=0 k 2
l=0
+
[ k2 ] X j=1
+
[ k2 ] X j=0
k 2j − 1 k 2j + 1
2k−2j+1 X l=0
"
2k − 2j + 1 e el,q E2k+1−2j−l,q E l
# P2k−2j 2k−2j e el,q (q − 1) j=0 E2k−2j−l,q E l q−1 P2k−2j+1 2k−2j+1 e el,q + 1+q E2k−2j−l+1 E l=0 l
Equating I3 and I4 , we get the following theorem
Theorem 5. For k ∈ N, we have k i h X k k−l ek+l+1,q − q E ek+l,q [2]q E (−1) l l=0
=
[ k2 ] 2k−2j+1 X 2k − 2j + 1 X k el,q e2k+1−2j−l,q E E [2]q 2j l j=0 l=0
+
[ k2 ] X j=1
l=0
[ ] X k 2
+
2k−2j+1 X 2k − 2j + 1 k el,q e2k+1−2j−l,q E E 2j − 1 l
j=0
k 2j + 1
(
) P2k−2j 2k−2j e el,q (q − 1) j=0 E2k−2j−l,q E l P2k−2j+1 2k−2j+1 el,q e2k−2j−l+1 E + q−1 E l=0 1+q l
Now, we consider (2.8) and (2.1) by means of q-Volkenborn integral. Then, by (2.8), we see Z m xk (x − 1) dµq (x) [2]q Zp
=
[2]q
m X m l=0
=
[2]q
l
m X m l=0
l
m−l
(−1)
Z
xl+k dµq (x)
Zp
m−l
(−1)
el+k,q B
8
S. ARACI, M. ACIKGOZ, AND J J. SEO
On the other hand, k+m−j Z max{k,m} X k + m − j X k j m e q + (−1) xl dµq (x) Ek+m−j−l,q j j l Z p j=0 l=0
max{k,m}
X
=
j=0
k+m−j X k + m − j k j m el,q ek+m−j−l,q B q + (−1) E j j l l=0
Therefore, we get the following theorem Theorem 6. For m, k ∈ N, we have m X m m−l e [2]q (−1) Bl+k,q l l=0
=
max{k,m}
X
q
j=0
k+m−j X k + m − j k j m el,q ek+m−j−l,q B + (−1) E j j l l=0
By using fermionic p-adic q-integral on Zp left hand side of (2.15), we get Z k xk (x − 1) ([2] x − q) dµq (x) I5 = [2]q Zp
=
=
k X
Z Z k X k k k−l k−l k+l+1 [2]q (−1) x dµq (x) − q (−1) xk+l dµq (x) l l Z Z p p l=0 l=0 k k X X k k ek+l+1,q − q ek+l,q [2]q (−1)k−l B (−1)k−l B l l l=0
l=0
Also, we consider right hand side of (2.15) as follows: I6
= [2]q
[ k2 ] 2k−2j+1 X 2k − 2j + 1 X k j=0
+
[ k2 ] X j=1
+
[ k2 ] X j=0
= [2]q
j=0
[ ] X j=1
[ ] X k 2
+
k 2j − 1 k 2j + 1
l
l=0
2k−2j+1 X l=0
"
j=0
2j
k 2j − 1 k 2j + 1
xl dµq (x)
Zp
# R P2k−2j 2k−2j e (q − 1) j=0 E2k−2j−l,q Zp xl dµq (x) l R q−1 P2k−2j+1 2k−2j+1 e E2k−2j−l+1 Zp xl dµq (x) + 1+q l=0 l l
l=0
2k−2j+1 X l=0
"
Z
Z 2k − 2j + 1 e xl dµq (x) E2k+1−2j−l,q l Zp
[ k2 ] 2k−2j+1 X 2k − 2j + 1 X k
k 2
+
2j
e2k+1−2j−l,q E
el,q e2k+1−2j−l,q B E
2k − 2j + 1 e el,q E2k+1−2j−l,q B l
# P2k−2j 2k−2j e el,q (q − 1) j=0 B E 2k−2j−l,q l P2k−2j+1 2k−2j+1 el,q e2k−2j−l+1 B + q−1 E 1+q
l=0
l
Equating I5 and I6 , we get the following Corollary
KIM’S q-EULER NUMBERS AND POLYNOMIALS
9
Corollary 1. For k ∈ N, we get k i h X k k−l ek+l+1,q − q B ek+l,q [2]q B (−1) l l=0
= [2]q
[ k2 ] 2k−2j+1 X 2k − 2j + 1 X k 2j
j=0
+
[ k2 ] X j=1
[ ] X k 2
+
k 2j − 1
j=0
k 2j + 1
l
l=0
el,q e2k+1−2j−l,q B E
2k−2j+1 X 2k − 2j + 1 el,q e2k+1−2j−l,q B E l l=0
(
) P2k−2j 2k−2j e el,q (q − 1) j=0 B E 2k−2j−l,q l P2k−2j+1 2k−2j+1 el,q e2k−2j−l+1 B + q−1 E l=0 1+q l References
[1] Araci, S., Erdal, D., and Seo, J-J., A study on the fermionic p-adic q-integral representation on Zp associated with weighted q-Bernstein and q-Genocchi polynomials, Abstract and Applied Analysis, Volume 2011, Article ID 649248, 10 pages. [2] T. Kim, B. Lee, S. H. Lee, S-H. Rim, Identities for the Bernoulli and Euler numbers and polynomials, Accepted in Ars Combinatoria. [3] Kim, D., Kim, T., Lee, S-H., Dolgy, D-V., and Rim, S-H., Some new identities on the Bernoulli numbers and polynomials, Discrete Dynamics in Nature and Society, Volume 2011, Article ID 856132, 11 pages. [4] Kim, T., Choi, J., and Kim, Y-H., Some identities on the q-Bernoulli numbers and polynomials with weight 0, Abstract And Applied Analysis, Volume 2011, Article ID 361484, 8 pages. [5] Kim, T., On a q-analogue of the p-adic log gamma functions related integrals, J. Number Theory, 76 (1999) no. 2, 320-329. [6] Kim, T., and Choi, J., On the q-Euler numbers and polynomials with weight 0, Abstract and Applied Analysis, Volume 2012, ID 795304, 7 pages, doi:10.1155/2012/795304. [7] Kim, T., On the q-extension of Euler and Genocchi numbers, J. Math. Anal. Appl. 326 (2007) 1458-1465. [8] Kim, T., On the weighted q-Bernoulli numbers and polynomials, Advanced Studies in Contemporary Mathematics 21(2011), no.2, p. 207-215, http://arxiv.org/abs/1011.5305. [9] Kim, T., q-Volkenborn integration, Russ. J. Math. phys. 9(2002) , 288-299. [10] Kim, T., q-Euler numbers and polynomials associated with p-adic q-integrals, J. Nonlinear Math. Phys., 14 (2007), no. 1, 15–27. [11] Kim, T., New approach to q-Euler polynomials of higher order, Russ. J. Math. Phys., 17 (2010), no. 2, 218–225. [12] Kim, T., Some identities on the q-Euler polynomials of higher order and q-Stirling numbers by the fermionic p-adic integral on Zp , Russ. J. Math. Phys., 16 (2009), no.4,484–491. University of Gaziantep, Faculty of Science and Arts, Department of Mathematics, 27310 Gaziantep, TURKEY E-mail address:
[email protected] University of Gaziantep, Faculty of Science and Arts, Department of Mathematics, 27310 Gaziantep, TURKEY E-mail address:
[email protected] Department of Applied Mathematics, Pukyong National University, Busan 608-737, Republic of Korea E-mail address:
[email protected] (Corresponding author)
