Research Article
Some identities of degenerate q -Euler polynomials under the symmetry group of degree n
Taekyun Kima,b, D. V. Dolgyc,d, Lee-Chae Jange,∗, Hyuck-In Kwonb
aDepartment of Mathematics, College of Science, Tianjin Polytechnic University, Tianjin City, 300387, China.
bDepartment of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea.
cHanrimwon, Kwangwoon University, Seoul 139-701, Republic of Korea.
dInstitute of Natural Sciences, Far eastern Federal University, Vladivostok 690950, Russia.
eGraduate School of Education, Konkuk University, Seoul 143-701, Republic of Korea.
Communicated by S. H. Rim
Abstract
In this paper, we derive some interesting identities of symmetry for the degenerateq-Euler polynomials under the symmetry group of degreenarising from the fermionicp-adicq-integral onZp. c2016 all rights reserved.
Keywords: Identities of symmetry, degenerateq-Euler polynomial, symmetry group of degreen, fermionic p-adicq-integral.
2010 MSC: 11B68, 11S80, 05A19, 05A30.
1. Introduction
Letpbe a fixed odd prime number. Throughout this paper,Zp,Qp, andCp will denote the ring ofp-adic integers, the field ofp-adic rational numbers and the completion of the algebraic closure ofQp, respectively.
Let q be an indeterminate in Cp with |1−q|p < p−p−11 , where | · |p is the p-adic norm. As is known, the q-analogue of the numberx is defined as [x]q = 1−q1−qx. Letf(x) be continuous function onZp. The fermionic
∗Corresponding author
Email addresses: [email protected](Taekyun Kim),[email protected](D. V. Dolgy),[email protected](Lee-Chae Jang),[email protected](Hyuck-In Kwon)
Received 2016-07-30
p-adicq-integral on Zp is defined by Kim as follows (see [3–20]) I−q(f) =
Z
Zp
f(x)dµ−q(x) = lim
N→∞
pN−1
X
x=0
f(x)µ−q(x+pNZp)
= lim
N→∞
[2]q 2
pN−1
X
x=0
f(x)(−q)x
= lim
N→∞
1 +q 1 +qpN
pN−1
X
x=0
f(x)(−q)x.
Forλ, t∈Cp with |λ|p ≤1,|t|p< p−p−11 , the degenerate Euler polynomials are defined by Carlitz to be 2
(1 +λt)λ1 + 1(1 +λt)xλ =
∞
X
n=0
En,λ(x)tn
n!, (see [2, 12]). (1.1)
Note that limλ→0En,λ=En(x), whereEn(x) are ordinary Euler polynomials which are given by the gener- ating function to be
2
et+ 1ext=
∞
X
n=0
En(x)tn
n!, (see [1–20]). (1.2)
Recently, Kim proved the following equation:
Z
Zp
(1 +λt)x+yλ dµ−1(y) = 2
(1 +λt)1λ + 1(1 +λt)xλ =
∞
X
n=0
En,λ(x)tn
n!. (1.3)
Thus, by (1.3), we get
λn Z
Zp
x+y λ
n
dµ−1(y) =En,λ(x), (see [12]), (1.4) where R
Zpf(x)dµ−1(x) = limq→1R
Zpf(x)dµ−q(x) and (x)n = x(x−1)· · ·(x−n+ 1), (n ≥ 1), (x)0 = 1.
In [13], the degenerateq-Euler polynomials are defined by Kim as follows Z
Zp
(1 +λt)
[x+y]q
λ dµ−q(y) =
∞
X
n=0
En,λ,q(x)tn
n!. (1.5)
When x = 0, En,λ,q = En,λ,q(0) are called the degenerate q-Euler numbers. Note that limλ→0En,λ,q(x) = En,q(x), where En,q(x) are called the Carlitz’s q-Euler polynomials (see [10, 13]).
In this paper, we study some identities of symmetry for the degenerateq-Euler polynomials arising from the fermionicp-adicq-integral on Zp under symmetry group of degree n.
2. Symmetric identities for the degenerate q-Euler polynomials under Sn
Let Sn be the symmetry group of degree n and let w1, w2· · · , wn be odd positive integers. Then, we study the following integral equation for the fermionicp-adicq-integral on Zp:
Z
Zp
(1 +λt)
1 λ
Qn−1
j=1wj
y+ Qn j=1wj
x+wnPn−1 j=1
Qn−1
i=1i6=j wi
kj
q
dµ−qw1w2···wn−1(y)
= [2]qw1···wn−1
2 lim
N→∞
wn−1
X
m=0 pN−1
X
y=0
(1 +λt)
1 λ
Qn−1
j=1wj
m+wny
+ Qn j=1wj
x+wnPn j=1
Pn−1
i=1i6=j wi
kj
q
×(−1)m+wnyqw1w2···wn−1(m+wny).
(2.1)
From (2.1), we have 2
[2]qw1···wn−1
n−1
Y
l=1 wl−1
X
kl=0
(−1)Pn−1i=1 kiq
wnPn−1 j=1
Qn−1
i=1i6=j wi
kj
× Z
Zp
(1 +λt)
1 λ
Qn−1
j=1wj
y+ Qn j=1wj
x+wnPn−1 j=1
Qn−1
i=1i6=j wi
kj
q
dµ−qw1w2···wn−1(y)
= lim
N→∞
n−1
Y
l=1 wl−1
X
kl=0 wn−1
X
m=0 pN−1
X
y=0
(1 +λt)
1 λ
Qn−1
=1 wj
m+wny
+ Qn j=1wj
x+wnPn j=1
Qn−1
i=1i6=j wi
kj
q
×(−1)Pn−1i=1 ki+m+yq
wnPn−1 j=1
Qn−1
i=1i6=j wi
kj+ Qn−1 j=1wj
m+ Qn j=1wj
y
.
(2.2)
Thus, by (2.2), we note that I(w1, w2,· · · , wn) = 2
[2]qw1···wn−1
n−1
Y
l=1 wl−1
X
kl=0
(−1)Pn−1i=1 kiq
wnPn−1 j=1
Qn−1
i=1i6=j wi
kj
× Z
Zp
(1 +λt)
1 λ
Qn−1
j=1wj
y+ Qn j=1wj
x+wnPn−1 j=1
Qn−1
i=1i6=j wi
kj
q
dµ−qw1w2···wn−1(y) (2.3)
is invariant for any permutation σ in the symmetry group of degree n. Therefore, by (2.2), we obtain the following theorem.
Theorem 2.1. Forw1,· · ·, wn∈Nwithwi ≡1 (mod 2),i= 1,2,· · · , n, we note thatI(wσ(1), wσ(2),· · ·, wσ(n)) are the same for any σ∈Sn,(n≥1).
From the definition of [x]q, we note that h n−1Y
j=1
wj
y+
n
Y
j=1
wj
x+wn n−1
X
j=1 n−1
Y
i=1i6=j
wj
kj
i
q= hn−1Y
j=1
wj
i
q
h
y+wnx+wn n−1
X
j=1
kj wj
i
qw1w2···wn−1. (2.4) By (2.4), we get
Z
Zp
(1 +λt)
1 λ
Qn−1
j=1wj
y+ Qn j=1wj
x+wnPn−1 j=1
Qn−1
i=1i6=j wj
kj
q
dµ−qw1···wn−1(y)
= Z
Zp
1 + λ
Qn−1 j=1wj]q
n−1
Y
j=1
wj
qt 1λ
Qn−1
j=1wj
y+ Qn j=1wj
x+wnPn−1 j=1
Qn−1
i=1i6=j wj
kj
q
dµ−qw1···wn−1(y)
=
∞
X
m=0
hn−1Y
j=1
wjim
q Em, λ [Qn−1
j=1wj]q,qw1···wn−1
wnx+wn
n−1
X
j=1
kj
wj tn
n!, (n∈N).
(2.5)
Therefore, by Theorem 2.1 and (2.5), we obtain the following theorem.
Theorem 2.2. Let w1, w2,· · ·, wn be odd integers and let mbe a non-negative integer. Then, the following expressions
2 [2]qwσ(1)···wσ(n−1)
hn−1Y
j=1
wσ(j)im q
n−1
Y
l=1 wσ(l)−1
X
kl=0
(−1)Pn−1i=1 kiq
wσ(n)Pn−1 j=1
Qn−1
i=1i6=j wσ(i)
kj
× Em, λ
Qn−1
j=1wσ(j)
q
,qwσ(1)wσ(2)···wσ(n−1)
wσ(n)x+wσ(n)
n−1
X
j=1
kj
wσ(j)
are the same for any permutation σ in the symmetry group of degree n.
Now, we observe that
y+wnx+wn n−1
X
j=1
kj
wj
qw1w2···wn−1
= [wn]q hQn−1
j=1 wji
q
hn−1X
j=1 n−1
Y
i=1i6=j
wi kji
qwn+q
wnPn−1 j=1
Qn−1
i=1i6=j wi
kj
y+wnx
qw1···wn−1.
(2.6)
From (1.5), we have
∞
X
n=0
En,λ,q(x)tn n! =
Z
Zp
(1 +λt)
[x+y]q
λ dµ−q(y) =
∞
X
n=0
λn Z
Zp
[x+y]q
λ
n
dµ−q(y)tn
n!. (2.7)
By comparing the coefficients on the both sides of (2.7), we get En,λ,q(x) =λn
Z
Zp
[x+y]q λ
n
dµ−q(y), (n≥0), (2.8)
where
[x+y]q λ
n
=
n
X
l=0
S1(n, l)
[x+y]q λ
l
and S1(n, l) is the Stirling number of the first kind. From (2.8), we have Em, λ
Qn−1
j=1wj
q
,qw1···wn−1
wnx+wn
n−1
X
j=1
kj wj
=
λ Qn−1
j=1wj
q
mZ
Zp
y+wnx+wnPn−1 j=1
kj
wj
qw1···wn−1
λ
m
dµ−qw1···wn−1(y).
(2.9)
Now, by (2.6), we observe that λ
Qn−1 j=1 wj
q
m
y+wnx+wnPn−1 j=1
kj wj
qw1···wn−1
λ
m
=
λ Qn−1
j=1 wj
q
m1 λ
[wn]q
Qn−1 j=1wj
q
hn−1X
j=1 n−1
Y
i=1i6=j
wi kj
i
qwn+q
wnPn−1 j=1
Qn−1
i=1i6=j wi
kj
[y+wnx]qw1···wn−1
m
=
λ Qn−1
j=1 wj
q
m m
X
l=0
λ−lS1(m, l)
×
[wn]q Qn−1
j=1wj
q
hn−1X
j=1 n−1
Y
i=1i6=j
wi
kj
i
qwn +q
wnPn−1 j=1
Qn−1
i=1i6=j wi
kj
[y+wnx]qw1···wn−1
l
=
λ Qn−1
j=1 wj
q
m m X
l=0
λ−lS1(m, l)
l
X
i=0
[wn]q
Qn−1 j=1wj
q
l−ihn−1X
j=1 n−1
Y
i=1i6=j
wi kj
il−i qwn
×q
wniPn−1 j=1
Qn−1
i=1i6=j wi
kj
[y+wnx]iqw1···wn−1
l i
.
(2.10)
By (2.9) and (2.10), we get Em, λ
Qn−1
j=1wj
q
,qw1···wn−1
wnx+wn n−1
X
j=1
kj
wj
= 1
hQn−1 j=1wj
im q
m
X
l=0 l
X
i=0
l i
λm−lS1(m, l)
[wn]q Qn−1
j=1wj
q
l−i
hn−1X
j=1 n−1
Y
i=1i6=j
wi
kj
il−i qwn
×q
wniPn−1 j=1
Qn−1
i=1i6=j wi
kjZ
Zp
[y+wnx]iqw1···wn−1dµ−qw1···wn−1(y)
= 1
hQn−1 j=1wjim
q m
X
l=0 l
X
i=0
l i
λm−lS1(m, l)
[wn]q
Qn−1 j=1wj
q
l−i
hn−1X
j=1 n−1
Y
i=1i6=j
wi
kj
il−i qwn
×q
wniPn−1 j=1
Qn−1
i=1i6=j wi
kj
Ei,qw1···wn−1(wnx).
(2.11)
By Theorem 2.2 and (2.11), we get 2
[2]qw1···wn−1
hn−1Y
j=1
wjim q
n−1
Y
l=1 wl−1
X
kl=0
(−1)Pn−1i=1 kiq
wnPn−1 j=1
Qn−1
i=1i6=j wi
kj
× Em, λ
Qn−1
j=1wj
q
,qw1···wn−1
wnx+wn n−1
X
j=1
kj
wj
= 2
[2]qw1···wn−1
n−1
Y
l=1 wl−1
X
kl=0
(−1)Pn−1i=1 kiq
wnPn−1 j=1
Qn−1
i=1i6=j wi
kj
×
m
X
p=0 p
X
i=0
p i
λm−pS1(m, p)
[wn]q
Qn−1 j=1 wj
q
p−in−1 X
j=1 n−1
Y
i=1i6=j
wi kj
p−i qwn
×q
wniPn−1 j=1
Qn−1
i=1i6=j wi
kj
Ei,qw1w2···wn−1(wnx)
=
m
X
p=0 p
X
i=0
p i
λm−pS1(m, p)
[wn]q
Qn−1 j=1wj
q
p−i
Ei,qw1w2···wn−1(wnx)
×
n−1
Y
l=1 wl−1
X
kl=0
(−1)Pn−1i=1 kiq
(i+1)wnPn−1 j=1
Qn−1
i=1i6=j wi
kjn−1 X
j=1 n−1
Y
i=1i6=j
wi
kj
p−i
qwn
=
m
X
p=0 p
X
i=0
p i
λm−pS1(m, p)
[wn]q Qn−1
j=1wj
q
p−i
Ei,qw1w2···wn−1(wnx)Tn,q(p)wn(w1,· · ·, wn−1|i+ 1),
(2.12)
where
Tn,q(p)(w1,· · · , wn−1|i) =
n−1
Y
l=1 wl−1
X
kl=0
(−1)Pn−1t=1ktq
iPn−1 j=1
Qn−1
t=1t6=j wt
kjn−1 X
j=1 n−1
Y
t=1t6=j
wt kj
p−i
q
, (2.13)
and En,q(x) is the Carlitz’s q-Euler polynomials which are given by R
Zp[x+y]nqdµ−q(y) =En,q(x), (n≥0), (see [1, 6, 8]). Therefore, by (2.12), we obtain the following theorem.
Theorem 2.3. For w1, w2,· · · , wn ∈ N with wi ≡ 1 (mod 2), (i = 1,2,· · · , n), and m ≥ 0, the following expressions
m
X
p=0 p
X
i=0
p i
λm−pS1(m, p)
[wσ(n)]q
Qn−1 j=1wσ(j)
q
p−i
Ei,qwσ(1)wσ(2)···wσ(n−1)(wσ(n)x)T(p)
n,qwσ(n)(wσ(1),· · ·, wσ(n−1)|i+ 1) are the same for any permutation σ in the symmetry group of degree n.
Acknowledgment
This paper is supported by grant NO 14-11-00022 of Russian Scientific Fund.
References
[1] L. Carlitz,q-Bernoulli and Eulerian numbers, Trans. Amer. Math. Soc.,76(1954), 332–350. 1.2, 2 [2] L. Carlitz,Degenerate Stirling, Bernoulli and Eulerian numbers, Utilitas Math.,15(1979), 51–88. 1.1
[3] D. V. Dolgy, T. Kim, H. I. Kwon, J. J. Seo,Symmetric identities of degenerate q-Bernoulli polynomials under symmetry groupS3, Proc. Jangjeon Math. Soc.,19(2016), 1–9. 1
[4] Y. He, Symmetric identities for Carlitz’s q-Bernoulli numbers and polynomials, Adv. Difference Equ., 2013 (2013), 10 pages.
[5] T. Kim,Symmetryp-adic invariant integral onZpfor Bernoulli and Euler polynomials, J. Difference Equ. Appl., 14(2008), 1267–1277.
[6] T. Kim,Some identities on theq-Euler polynomials of higher-order andq-Stirling numbers by the fermionicp-adic integral onZp, Russ. J. Math. Phys.,16(2009), 484–491. 2
[7] T. Kim,Symmetry of power sum polynomials and multivariate fermionicp-adic invariant integral on Zp, Russ.
J. Math. Phys.,16(2009), 93–96.
[8] T. Kim,New approach toq-Euler polynomials of higher-order, Russ. J. Math. phys.,17(2010), 218–225. 2 [9] T. Kim,Symmetric identities of degenerate Bernoulli polynomials, Proc. Jangjeon Math. Soc.,18(2015), 593–
599.
[10] T. Kim, D. V. Dolgy, J. J. Seo,Identities of symmetry for degenerateq-Euler polynomials, Adv. Stud. Contemp.
Math.,25(2015), 577–582. 1
[11] Y. H. Kim, K. W. Hwang, Symmetry of power sum and twisted Bernoulli polynomials, Adv. Stud. Contemp.
Math.,18(2009), 127–133.
[12] D. S. Kim, T. Kim,Some identities of degenerate Euler polynomials arising fromp-adic fermionic integral onZp, Integral Transforms Spec. Funct.,26(2015), 295–302. 1.1, 1.4
[13] D. S. Kim, T. Kim,Symmetric identities of higher-order degenerateq-Euler polynomials, J. Nonlinear Sci. Appl., 9(2016), 443–451. 1, 1
[14] D. S. Kim, T. Kim,Some identities of symmetry forq-Bernoulli polynomials under symmetric group of degreen, Ars Combin.,126(2016), 435–441.
[15] D. S. Kim, T. Kim,Symmetry identities of higher-orderq-Euler polynomials under the symmetric group of degree four, J. Comput. Anal. Appl.,21(2016), 521–527.
[16] T. Kim, H. I. Kwon, J. J. Seo, Identities of symmetry for degenerate q-Bernoulli polynomials, Proc. Jangjeon Math. Soc.,18(2015), 495–499.
[17] D. S. Kim, N. Lee, J. Na, K. H. Park,Identities of symmetry for higher-order Euler polynomials in three variables (I), Adv. Stud Contemp. Math.,22(2012), 51–74.
[18] D. S. Kim, N. Lee, J. Na, K. H. Park,Abundant symmetry for higher-order Bernoulli polynomials (II), Proc.
Jangjeon Math. Soc.,16(2013), 359–378.
[19] H. I. Kwon, T. Kim, J. J. Seo,Some identities of symmetry for modified degenerate Frobenius-Euler polynomials, Adv. Stud. Contemp. Math.,26(2016), 299–305.
[20] E. J. Moon, S. H. Rim, J. H. Jin, S. J. Lee,On the symmetric properties of higher-order twistedq-Euler numbers and polynomials, Adv. Difference Equ.,2010(2010), 8 pages. 1, 1.2
