SYMMETRIC IDENTITIES OF THE q-EULER POLYNOMIALS
arXiv:1310.1658v1 [math.NT] 7 Oct 2013
DAE SAN KIM AND TAEKYUN KIM
Abstract. In this paper, we study some symmetric identities of q-Euler numbers and polynomials. From these properties, we derive several identities of q-Euler numbers and polynomials.
1. Introduction The Euler polynomials are defined by the generating function to be ∞ X 2 tn xt E(x)t e = e = E (x) , (see [2 − 5]). n et + 1 n! n=0
(1.1)
with the usual convention about replacing E n (x) by En (x).
When x = 0, En = En (0) are called the Euler numbers. Let q ∈ C with |q| < 1. x For any complex number x, the q-analogue of x is defined by [x]q = 1−q 1−q . Note that limq→1 [x]q = x. Recently, T. Kim introduced a q-extension of Euler polynomials as follows: Fq (t, x) = [2]q
∞ X
n n [n+x]q t
(−1) q e
n=0
=
∞ X
En,q (x)
n=0
tn , (see[7, 8]). n!
(1.2)
When x = 0, En,q = En,q (0) are called the q-Euler numbers. From (1.2), we note that En,q (x) = (q x Eq + [x]q )n n X n xl = q El,q [x]qn−l , (see [7, 8]), l
(1.3)
l=0
with the usual convention about replacing Eql by El,q . In [8], Kim introduced q-Euler zeta function as follows: Z ∞ 1 ts−1 Fq (−t, x)dt Γ(s) 0 ∞ X (−1)n q n = [2]q , [n + x]sq n=0
ζE,q (s, x) =
where x 6= 0, −1, −2, . . . , and s ∈ C.
1
(1.4)
2
DAE SAN KIM AND TAEKYUN KIM
From (1.4), we have ζE,q (−m, x) = Em,q (x),
(1.5)
where m ∈ Z≥0 . Recently, Y. He gave some interesting symmetric identities of Carlitz’s q-Bernoulli numbers and polynomials. In this paper, we study some new symmetries of the qEuler numbers and polynomials, which is the answer to an open question for the symmetric identities of Carlitz’s type q-Euler numbers and polynomials in [5]. By using our symmetries for the q-Euler polynomials we can obtain some identities between q-Euler numbers and polynomials. 2. Symmetric identities of q-Euler polynomials In this section, we assume that a, b ∈ N with a ≡ 1 (mod 2) and b ≡ 1 (mod 2). First, we observe that ∞ X (−1)n q na bj 1 ζE,qa (s, bx + ) = bj s [2]qa a n=0 [n + bx + a ]qa
∞ X b−1 X q an (−1)n [a]sq (−1)i+bn q a(i+bn) s = = [a] . q [bj + abx + an]sq [ab(x + n) + bj + ai]sq n=0 n=0 i=0 ∞ X
(2.1)
Thus, by (2.1), we get a−1 X b−1 X ∞ a−1 X [b]sq X q ai+bj+abn (−1)i+n+j bj (−1)j q bj ζE,qa (s, bx + ) = [b]sq [a]sq . [2]qa j=0 a [ab(x + n) + bj + ai]sq j=0 i=0 n=0
(2.2)
By the same method as (2.2), we get b−1 [a]sq X aj (−1)j q aj ζE,qb (s, ax + ) [2]qb j=0 b
=
[a]sq [b]sq
b−1 a−1 ∞ X XX q bi+aj+abn (−1)i+n+j . [ab(x + n) + aj + bi]sq j=0 i=0 n=0
(2.3)
Therefore, by (2.2) and (2.3), we obtain the following theorem. Theorem 2.1. For a, b ∈ N with a ≡ 1 (mod 2), b ≡ 1 (mod 2), [2]qb [b]sq
b−1 a−1 X X bj aj (−1)j q aj ζE,qb (s, ax + ). (−1)j q bj ζE,qa (s, bx + ) = [2]qa [a]sq a b j=0 j=0
By (1.5) and Theorem 2.1, we obtain the following theorem. Theorem 2.2. For n ∈ Z≥0 and a, b ∈ N with a ≡ 1 (mod 2), b ≡ 1 (mod 2), we have [2]qb [a]nq
b−1 a−1 X X aj bj (−1)j q aj En,qb (ax + ). (−1)j q bj En,qa (bx + ) = [2]qa [b]nq a b j=0 j=0
SYMMETRIC IDENTITIES OF THE q-EULER POLYNOMIALS
3
From (1.3), we note that En,q (x + y) = (q x+y Eq + [x + y]q )n = (q x+y Eq + q x [y]q + [x]q )n = (q x (q y Eq + [y]q ) + [x]q )n n X n xi y q (q Eq + [y]q )i [x]qn−i = i i=0 n X n xi q Ei,q (y)[x]qn−i . = i i=0
(2.4)
Therefore, by (2.4), we obtain the following proposition. Proposition 2.3. For n ≥ 0, we have n X n xi q Ei,q (y)[x]qn−i i i=0 n X n (n−i)x q En−i,q (y)[x]iq . = i i=0
En,q (x + y) =
Now, we observe that a−1 X bj (−1)j q bj En,qa (bx + ) a j=0
n−i n a−1 X X n ia( bj ) bj j bj a q (−1) q Ei,qa (bx) = i a qa i=0 j=0
i n a−1 X X n (n−i)bj bj j bj q En−i,qa (bx) (−1) q = i a qa i=0 j=0 =
i n X n [b]q i=0
=
i
[a]q
i n X n [b]q i=0
∗ where Sn,i,q (a) =
i
[a]q
En−i,qa (bx)
a−1 X
(2.5)
(−1)j q bj(n+1−i) [j]iqb
j=0
∗ En−i,qa (bx)Sn,i,q b (a),
Pa−1
j (n+1−i)j [j]iq . j=0 (−1) q
From (2.5), we can derive [2]qb [a]nq
a−1 X
(−1)j q bj En,qa (bx +
j=0
n X bj n ∗ [a]qn−i [b]iq En−i,qa (bx)Sn,i,q ) = [2]qb b (a). a i i=0
(2.6)
4
DAE SAN KIM AND TAEKYUN KIM
By the same method as (2.6), we get [2]
qa
[b]nq
n X n aj ∗ a [b]qn−i [a]iq En−i,qb (ax)Sn,i,q (−1) q En,qb (ax + ) = [2]q a (b). b i i=0 j=0
b−1 X
j aj
(2.7)
Therefore, by Theorem 2.2, (2.6) and (2.7), we obtain the following theorem. Theorem 2.4. For n ∈ Z≥0 and a, b ∈ N with a ≡ 1 (mod 2), b ≡ 1 (mod 2), we have n n X X n n ∗ ∗ [b]qn−i [a]iq En−i,qb (ax)Sn,i,q [a]qn−i [b]iq En−i,qa (bx)Sn,i,q [2]qb a (b), b (a) = [2]q a i i i=0 i=0 Pa−1 ∗ where Sn,i,q (a) = j=0 (−1)j q (n+1−i)j [j]iq . It is easy to show that
[x]q u + q x [y + m]q (u + v) = [x + y + m]q (u + v) − [x]q v. Thus, by (2.8), we get ∞ ∞ X X x e[x]q u q m (−1)m e[y+m]q q (u+v) = e−[x]q v q m (−1)m q [x+y+m]q (u+v) . m=0
(2.8)
(2.9)
m=0
The left hand side of (2.9) multiplied by [2]q is given by ∞ X
[2]q e[x]q u
q m (−1)m e[y+m]q q
x
(u+v)
m=0 ∞ X
= e[x]q u
q nx En,q (y)
n=0
! uk v n = q Ek+n,q (y) l! k! n! l=0 k=0 n=0 ! m ∞ X ∞ X X m (k+n)x um v n m−k . q Ek+n,q (y)[x]q = m! n! k m=0 n=0 ∞ X
u [x]lq
l
!
(u + v)n n!
∞ X ∞ X
(2.10)
(k+n)x
k=0
The right hand side of (2.9) multiplied by [2]q is given by [2]q e−[x]q v
∞ X
(−1)m q m e[x+y+m]q (u+v)
m=0
= e−[x]q v
∞ X
En,q (x + y)
n=0
!
(u + v)n n!
! um v k Em+k,q (x + y) v = l! m! k! m=0 k=0 l=0 ! n ∞ X ∞ X X um v n n n−k Em+k,q (x + y)(−[x]q ) = k m! n! n=0 m=0 k=0 ! n ∞ X ∞ X X n um v n . Em+k,q (x + y)q (n−k)x [−x]qn−k = m! n! k n=0 m=0 ∞ X (−[x]q )l
k=0
l
∞ ∞ X X
(2.11)
SYMMETRIC IDENTITIES OF THE q-EULER POLYNOMIALS
5
Therefore, by (2.10) and (2.11), we get m n X X m (n+k)x n (n−k)x q En+k,q (y)[x]m−k = q Em+k,q (x + y)[−x]qn−k (2.12) q k k k=0
k=0
References [1] Y. He, Symmetric identities for Calitz’s q-Bernoulli numbers and polynomials, Advances in Difference Equations 2013(2013), 246, doi:10.1186/1687-1847-2013-246. [2] D. S. Kim, Symmetry identities for generalized twisted Euler polynomials twisted by unramified roots of unity, Proc. Jangjeon Math. Soc. 15(2012), no. 3, 303-316. [3] D. S. Kim, N. Lee, J. Na and K. H. Park, Identities of symmetry for higher-order Euler polynomials in three variables (I), Adv. Stud. Contemp. Math. 22 (2012), no. 1, 51-74. [4] D. S. Kim, N. Lee, J. Na and K. H. Park, Identities of symmetry for higher-order Euler polynomials in three variables (II), J. Math. Anal. Appl. 379 (2011), no. 1, 388-400. [5] T. Kim, Symmetry p-adic invariant integral on Zp for Bernoulli and Euler polynomials, J. Difference Equ. Appl. 14(2008), no.12, 1267-1277. [6] T. Kim, An identity of the symmetry for the Frobenius-Euler polynomials associated with the fermionic p-adic invariant q-integrals on Zp , Rocky Mountain J. Math. 41 (2011), no. 1, 239-247. [7] T. Kim, Analytic continuation of q-Euler numbers and polynomials, Applied mathematics Letters 21(2008), 1320-1323. [8] T. Kim, q-Euler numbers and polynomials associated with p-adic q-integrals, Journal of Nonlinear Mathematical Physics, 14(2007), no.1, 15-27. Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea. E-mail address:
[email protected] Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea E-mail address:
[email protected]
