Journal of Contemporary Applied Mathematics V. 7, No 1, 2017, June ISSN 2222-5498
Theorems on Carlitz’s twisted q-Euler polynomials under S5 Ugur Duran, Mehmet Acikgoz
Abstract. In this paper, we consider Carlitz’s twisted q-Euler polynomials and present some new symmetric identities for Carlitz’s twisted q-Euler polynomials using the fermionic p-adic invariant integral on Zp under symmetric group of degree five. Key Words and Phrases: Symmetric identities; Carlitz’s twisted q-Euler polynomials; p-adic invariant integral on Zp ; Invariant under S5 . 2010 Mathematics Subject Classifications: 11B68, 05A19, 11S80, 05A30.
1. Introduction The Euler polynomials En (x) are defined by means of the Taylor expansion about t = 0: ∞ ∑ n=0
En (x)
tn 2 = t ext with (|t| < π). n! e +1
(1)
When x = 0, we have En (0) := En that are known as Euler numbers (see, e.g., [3], [5], [6], [7], [8], [9]). Let Z be the set of the all integers, C be the set of all complex numbers and p be chosen as a fixed odd prime number. Throughout this paper, the symbols Zp , Q, Qp and Cp shall denote topological closure of Z, the field of rational numbers, topological closure of Q and the field of p-adic completion of an algebraic closure of Qp , respectively. For d an odd positive number with (d, p) = 1, let X := Xd = lim Z/dpN Z and X1 = Zp ← − n
and { ( )} t + dpN Zp = x ∈ X | x ≡ t mod dpN where t ∈ Z lies in 0 ≤ t < dpN . See, for more details, [1,2, 4-9].
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Ugur Duran, Mehmet Acikgoz
The normalized absolute value according to the theory of p-adic analysis is given by |p|p = p−1 . The notation q can be considered as an indeterminate, a complex number q ∈ C with |q| < 1, or a p-adic number q ∈ Cp with |q − 1|p < p for |x|p ≤ 1.
1 − p−1
and q x = exp (x log q)
For fixed x, let us introduce the following notation [x]q =
1 − qx 1−q
(2)
which is known as q-number of x (or q-analogue of x), see [1-11]. Notice that as q → 1, the notation [x]q reduce to the x. For f ∈ U D (Zp ) = {f |f : Zp → Cp is uniformly differentiable function } , Kim [5] defined the p-adic invariant integral on Zp as follows: ∫ lim Iq (f ) = I−1 (f ) =
q→−1
Zp
f (x) dµ−1 (x) = lim
N →∞
N −1 p∑
f (x) (−1)x .
(3)
x=0
By Eq. (3), we have n
I−1 (fn ) = (−1) I−1 (f ) + 2
n−1 ∑
(−1)n−k−1 f (k)
k=0
where fn (x) implies f (x + n). For more details, take a look at the references [1], [2], [4], [5], [6], [7], [8], [9]. Let Tp =
∪ N ≥1
CpN = lim CpN , N →∞
{ } N where CpN = w : wp = 1 is the cyclic group of order pN . For w ∈ Tp , we show by ϕw : Zp → Cp the locally constant function x → wx . For q ∈ Cp with |1 − q|p < 1 and w ∈ Tp , the Carlitz’s twisted q-Euler polynomials are defined by the following p-adic invariant integral on Zp in [9], with respect to µ−1 : ∫ (4) wy [x + y]nq dµ−1 (y) (n ≥ 0) . En,q,w (x) = Zp
If we let x = 0 into the Eq. (4), we then get En,q,w (0) := En,q,w called as n-th Carlitz’s twisted q-Euler numbers.
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Theorems on Carlitz’s twisted q-Euler polynomials under S5
Notice that when w = 1 and q → 1 in the Eq. (4), we observe that ∫ En,q,w (x) → En (x) := (x + y)n dµ−1 (y) . Zp
Recently, symmetric identities of some special polynomials, such as q-Genocchi polynomials of higher order under third Dihedral group D3 in [1], q-Frobenious-Euler polynomials under symmetric group of degree five in [2], weighted q-Genocchi polynomials under the symmetric group of degree four in [4], Carlitz’s q-Euler polynomials invariant under the symmetric group of degree five in [6], have been studied extensively. In the next section, we give some new symmetric identities for Carlitz’s twisted q-Euler polynomials associated with the fermionic p-adic invariant integral on Zp symmetric group of degree five denoted by S5 .
2. Results on En,q,w (x) under S5 Let w ∈ Tp and wi are odd natural numbers where i ∈ {1, 2, 3, 4, 5}. By the Eqs. (3) and (4), we acquire ∫ ww1 w2 w3 w4 y (5) Zp
×e
[w1 w2 w3 w4 y+w1 w2 w3 w4 w5 x+w5 w4 w2 w3 i+w5 w4 w1 w3 j+w5 w4 w1 w2 k+w5 w3 w1 w2 s]q t
dµ−1 (y)
(6)
pN −1
∑
= lim
N →∞
(−1)y ww1 w2 w3 w4 y
(7)
y=0
×e[w1 w2 w3 w4 y+w1 w2 w3 w4 w5 x+w5 w4 w2 w3 i+w5 w4 w1 w3 j+w5 w4 w1 w2 k+w5 w3 w1 w2 s]q t = lim
N →∞
N −1 w∑ 5 −1 p∑
(−1)l+y ww1 w2 w3 w4 (l+w5 y)
l=0
y=0
×e
[w1 w2 w3 w4 (l+w5 y)+w1 w2 w3 w4 w5 x+w5 w4 w2 w3 i+w5 w4 w1 w3 j+w5 w4 w1 w2 k+w5 w3 w1 w2 s]q t
.
Taking w∑ 1 −1 w 2 −1 w 3 −1 w 4 −1 ∑ ∑ ∑
(−1)i+j+k+s ww5 w4 w2 w3 i+w5 w4 w1 w3 j+w5 w4 w1 w2 k+w5 w3 w1 w2 s
i=0
j=0 k=0 s=0
on the both sides of Eq. (6) gives w∑ 1 −1 w 2 −1 w 3 −1 w 4 −1 ∑ ∑ ∑
(−1)i+j+k+s ww5 w4 w2 w3 i+w5 w4 w1 w3 j+w5 w4 w1 w2 k+w5 w3 w1 w2 s
i=0
j=0 k=0 s=0
∫ ×
ww1 w2 w3 w4 y Zp
(8)
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Ugur Duran, Mehmet Acikgoz
×e[w1 w2 w3 w4 y+w1 w2 w3 w4 w5 x+w5 w4 w2 w3 i+w5 w4 w1 w3 j+w5 w4 w1 w2 k+w5 w3 w1 w2 s]q t dµ−1 (y) = lim
N →∞
×w
N −1 w∑ 1 −1 w 2 −1 w 3 −1 w 4 −1 w 5 −1 p∑ ∑ ∑ ∑ ∑
(−1)i+j+k+s+y+l
i=0
j=0 k=0 s=0
l=0
y=0
w1 w2 w3 w4 (l+w5 y)+w5 w4 w2 w3 i+w5 w4 w1 w3 j+w5 w4 w1 w2 k+w5 w3 w1 w2 s
×e[w1 w2 w3 w4 (l+w5 y)+w1 w2 w3 w4 w5 x+w5 w4 w2 w3 i+w5 w4 w1 w3 j+w5 w4 w1 w2 k+w5 w3 w1 w2 s]q t . Observe that the equation (8) is invariant for any permutation σ ∈ S5 . Therefore, we obtain the following theorem. Theorem 2.1. Let w ∈ Tp and wi are odd natural numbers where i ∈ {1, 2, 3, 4, 5}. Then the following wσ(1) −1 wσ(2) −1 wσ(3) −1 wσ(4) −1
∑
∑
∑
∑
(−1)i+j+k+s
s=0 j=0 k=0 wσ(5) wσ(4) wσ(2) wσ(3) i+wσ(5) wσ(4) wσ(1) wσ(3) j+wσ(5) wσ(4) wσ(1) wσ(2) k+wσ(5) wσ(3) wσ(1) wσ(2) s
i=0
×w ∫ ×
wwσ(1) wσ(2) wσ(3) wσ(4) (l+wσ(5) y)
Zp
( × exp [wσ(1) wσ(2) wσ(3) wσ(4) y + wσ(1) wσ(2) wσ(3) wσ(4) wσ(5) x + wσ(5) wσ(4) wσ(2) wσ(3) i ) +wσ(5) wσ(4) wσ(1) wσ(3) j + wσ(5) wσ(4) wσ(1) wσ(2) k + wσ(5) wσ(3) wσ(1) wσ(2) s]q t dµ−1 (y) holds true for any σ ∈ S5 . By using Eq. (2), we obtain the following equality: [w1 w2 w3 w4 y + w1 w2 w3 w4 w5 x + w5 w4 w2 w3 i + w5 w4 w1 w3 j + w5 w4 w1 w2 k + w5 w3 w1 w2 s]q (9)
] [ w5 w5 w5 w5 . i+ j+ k+ s = [w1 w2 w3 w4 ]q y + w5 x + w1 w2 w3 w4 qw1 w2 w3 w4 By virtue of Eqs. (6) and (9), we derive ∫ ww1 w2 w3 w4 y Zp
e
[w1 w2 w3 w4 y+w1 w2 w3 w4 w5 x+w5 w4 w2 w3 i+w5 w4 w1 w3 j+w5 w4 w1 w2 k+w5 w3 w1 w2 s]q t
=
∞ ∑ n=0
(∫ ×
dµ−1 (y)
(10)
[w1 w2 w3 w4 ]nq
) [ ]n w tn w w w 5 5 5 5 ww1 w2 w3 w4 y y + w5 x + dµ−1 (y) i+ j+ k+ s w1 w2 w3 w4 qw1 w2 w3 ww4 n! Zp
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Theorems on Carlitz’s twisted q-Euler polynomials under S5
=
∞ ∑
[w1 w2 w3 w4 ]nq En,qw1 w2 w3 w4 ,ww1 w2 w3 w4
n=0
( ) n w5 w5 w5 w5 t w5 x + i+ j+ k+ s . w1 w2 w3 w4 n!
On account of Eq. (10), we have ∫ ww1 w2 w3 w4 y
(11)
Zp
× [w1 w2 w3 w4 y + w1 w2 w3 w4 w5 x + w5 w4 w2 w3 i + w5 w4 w1 w3 j+ +w5 w4 w1 w2 k + w5 w3 w1 w2 s]nq dµ−1 (y) ) ( w5 w5 w5 w5 n w w w w w w w w i+ j+ k+ s , (n ≥ 0) . = [w1 w2 w3 w4 ]q En,q 1 2 3 4 ,w 1 2 3 4 w5 x + w1 w2 w3 w4 Herewith, from Theorem 2.1 and Eq. (11), we have the following theorem. Theorem 2.2. Let w ∈ Tp and wi are odd natural numbers where i ∈ {1, 2, 3, 4, 5}. The following expression [
w
wσ(1) wσ(2) wσ(3) wσ(4)
−1 wσ(2) −1 wσ(3) −1 wσ(4) −1
∑ ]n σ(1)
∑
∑
j=0
k=0
s=0
q i=0
×w
∑
(−1)i+j+k+s
wσ(5) wσ(4) wσ(2) wσ(3) i+wσ(5) wσ(4) wσ(1) wσ(3) j+wσ(5) wσ(4) wσ(1) wσ(2) k+wσ(5) wσ(3) wσ(1) wσ(2) s
(
×En,qwσ(1) wσ(2) wσ(3) wσ(4) ,wwσ(1) wσ(2) wσ(3) wσ(4)
wσ(5) x +
) wσ(5) wσ(5) wσ(5) wσ(5) i+ j+ k+ s wσ(1) wσ(2) wσ(3) wσ(4)
holds true for any σ ∈ S5 . We observe that ] [ ∫ w5 w5 w5 w5 n dµ−1 (y) i+ j+ k+ s (12) ww1 w2 w3 w4 y y + w5 x + w1 w2 w3 w4 qw1 w2 w3 w4 Zp ( )n−m n ( ) ∑ [w5 ]q n = [w2 w3 w4 i + w1 w3 w4 j + w1 w2 w4 k + w1 w2 w3 s]n−m q w5 m [w1 w2 w3 w4 ]q m=0 m(w5 w4 w2 w3 i+w5 w4 w1 w3 j+w5 w4 w1 w2 k+w5 w3 w1 w2 s)
×q ∫ Zp
=
ww1 w2 w3 w4 y [y + w5 x]m q w1 w2 w3 w4 dµ−1 (y)
n ( ) ∑ n
(
[w5 ]q
)n−m [w2 w3 w4 i + w1 w3 w4 j + w1 w2 w4 k + w1 w2 w3 s]n−m q w5
m [w1 w2 w3 w4 ]q m=0 m(w5 w4 w2 w3 i+w5 w4 w1 w3 j+w5 w4 w1 w2 k+w5 w3 w1 w2 s)
×q
Em,qw1 w2 w3 w4 ,ww1 w2 w3 w4 (w5 x) .
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Ugur Duran, Mehmet Acikgoz
In view of the Eq. (12), we acquire [w1 w2 w3 w4 ]nq
w∑ 4 −1 1 −1 w 2 −1 w 3 −1 w ∑ ∑ ∑
(−1)i+j+k+s ww5 w4 w2 w3 i+w5 w4 w1 w3 j+w5 w4 w1 w2 k+w5 w3 w1 w2 s
i=0
[
∫ × =
j=0 k=0 s=0
ww1 w2 w3 w4 y Zp
n ( ) ∑ n m=0
m
] w5 w5 w5 w5 n y + w5 x + i+ j+ k+ s dµ−1 (y) w1 w2 w3 w4 qw1 w2 w3 w4
n−m [w1 w2 w3 w4 ]m Em,qw1 w2 w3 w4 ,ww1 w2 w3 w4 (w5 x) q [w5 ]q
(13)
w∑ 1 −1 w 2 −1 ∑ i=0
j=0
w∑ 3 −1 w 4 −1 ∑
(−1)i+j+k+s ww5 w4 w2 w3 i+w5 w4 w1 w3 j+w5 w4 w1 w2 k+w5 w3 w1 w2 s
k=0 s=0 m(w5 w4 w2 w3 i+w5 w4 w1 w3 j+w5 w4 w1 w2 k+w5 w3 w1 w2 s)
×q
× [w2 w3 w4 i + w1 w3 w4 j + w1 w2 w4 k + w1 w2 w3 s]n−m q w5 ( ) n ∑ n n−m = [w1 w2 w3 w4 ]m Em,qw1 w2 w3 w4 ,ww1 w2 w3 w4 (w5 x) Un,qw5 ,ww5 (w1 , w2 , w3 , w4 | m), q [w5 ]q m m=0
where Un,q,w (w1 , w2 , w3 , w4 | m) =
(14)
w∑ 1 −1 w 2 −1 w 3 −1 w 4 −1 ∑ ∑ ∑
(−1)i+j+k+s ww2 w3 w4 i+w1 w3 w4 j+w1 w2 w4 k+w1 w2 w3 s
i=0
×q
j=0 k=0 s=0
m(w2 w3 w4 i+w1 w3 w4 j+w1 w2 w4 k+w1 w2 w3 s)
[w2 w3 w4 i + w1 w3 w4 j + w1 w2 w4 k + w1 w2 w3 s]n−m . q
As a result of the Eq. (13), we conclude the following theorem.
Theorem 2.3. Let w ∈ Tp and wi are odd natural numbers where i ∈ {1, 2, 3, 4, 5}. For n ≥ 0, the following n ( ) ∑ n [ m=0
m
wσ(1) wσ(2) wσ(3) wσ(4)
]m [ q
wσ(5)
]n−m q
( ) ×Em,qwσ(1) wσ(2) wσ(3) wσ(4) ,wwσ(1) wσ(2) wσ(3) wσ(4) wσ(5) x Un,qwσ(5) ,wwσ(5) (wσ(1) , wσ(2) , wσ(3) , wσ(4) | m) holds true for some σ ∈ S5 .
Theorems on Carlitz’s twisted q-Euler polynomials under S5
9
References [1] E. A˘gy¨ uz, M. Acikgoz, S. Araci, A symmetric identity on the q-Genocchi polynomials of higher order under third Dihedral group D3 , Proceedings of the Jangjeon Mathematical Society, 18 (2015), No. 2, pp. 177-187. [2] S. Araci, U. Duran, M. Acikgoz, Symmetric identities involving q-Frobenius-Euler polynomials under Sym (5), Turkish Journal of Analysis and Number Theory, Vol. 3, No. 3, pp. 90-93 (2015). [3] J. Choi, P. J. Anderson, H. M. Srivastava, Carlitz’s q-Bernoulli and q-Euler numbers and polynomials and a class of generalized q-Hurwitz zeta functions, Applied Mathematics and Computation, Volume 215, Issue 3 (2009), Pages 1185–1208. [4] U. Duran, M. Acikgoz, S. Araci, Symmetric identities involving weighted q-Genocchi polynomials under S4 , Proceedings of the Jangjeon Mathematical Society,18 (2015), No. 4, pp 455-465. [5] T. Kim, Some identities on the q-Euler polynomials of higher order and q-Stirling numbers by the fermionic p-adic integral on Zp , Russian Journal of Mathematical Physics, 484-491 (2009). [6] T. Kim and J. J. Seo, Some identities of symmetry for Carlitz-type q-Euler polynomials invariant under symmetric group of degree five, International Journal of Mathematical Analysis, Vol. 9 (2015), no. 37, 1815-1822. [7] H. Ozden, Y. Simsek, I. N. Cangul, Euler polynomials associated with p-adic q-Euler measure, General Mathematics Vol. 15, Nr. 2-3 (2007), 24-37. [8] C. S. Ryoo, Symmetric properties for Carlitz’s twisted (h,q)-Euler polynomials associated with p-adic q-integral on Zp , International Journal of Mathematical Analysis, Vol. 9 (2015), no. 40, 1947 - 1953. [9] C. S. Ryoo, Symmetric properties for Carlitz’s twisted q-Euler numbers and polynomials associated with p-adic integral on Zp , International Journal of Mathematical Analysis, Vol. 9 (2015), no. 83, 4129 - 4134. Ugur Duran Department of Mathematics, Faculty of Arts and Science, University of Gaziantep, TR-27310 Gaziantep, Turkey E-mail:
[email protected] Mehmet Acikgoz Department of Mathematics, Faculty of Arts and Science, University of Gaziantep, TR-27310 Gaziantep, Turkey E-mail:
[email protected]
Received 25 December 2016 Accepted 15 February 2017
