Advanced Studies in Theoretical Physics
Vol. 8, 2014, no. 26, 1177 - 1183
HIKARI Ltd, www.m-hikari.com
http://dx.doi.org/10.12988/astp.2014.410139
Symmetric Properties for the Generalized
(h, q)-Tangent Polynomials
C. S. Ryoo
Department of Mathematics
Hannam University, Daejeon 306-791, Korea
c 2014 C. S. Ryoo. This is an open access article distributed under the
Copyright Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
Abstract
In this paper, we study the symmetry for the generalized (h, q)(h)
(h)
tangent numbers Tn,χ,q and polynomials Tn,χ,q (x). We give some interesting identities of the power sums and the generalized polynomials
(h)
Tn,χ,q (x) using the symmetric properties for the p-adic invariant integral
on Zp .
Mathematics Subject Classification: 11B68, 11S40, 11S80
Keywords: The generalized tangent numbers and polynomials, the generalized (h, q)-tangent numbers and polynomials, symmetric properties, p-adic
invariant integral on Zp
1
Introduction
Euler numbers, Euler polynomials, Bernoulli numbers, Bernoulli polynomials,
tangent numbers, and tangent polynomials possess many interesting properties
and arise in many areas of mathematics and physics(see [1-6]). Throughout
this paper we use the following notations. By Zp we denote the ring of padic rational integers, Q denotes the field of rational numbers, Qp denotes
the field of p-adic rational numbers, C denotes the complex number field,
and Cp denotes the completion of algebraic closure of Qp . Let νp be the
normalized exponential valuation of Cp with |p|p = p−νp (p) = p−1 . When one
1178
C. S. Ryoo
talks of q-extension, q is considered in many ways such as an indeterminate,
a complex number q ∈ C, or p-adic number q ∈ Cp . If q ∈ C one normally
1
assume that |q| < 1. If q ∈ Cp , we normally assume that |q − 1|p < p− p−1
so that q x = exp(x log q) for |x|p ≤ 1. Let U D(Zp ) be the space of uniformly
differentiable function on Zp . For g ∈ U D(Zp ) the fermionic p-adic invariant
integral on Zp is defined by Kim as follows:
Z
X
g(x)(−1)x , (see [1]).
(1.1)
I−1 (g) =
g(x)dµ−1 (x) = lim
N →∞
Zp
0≤x<pN
If we take gn (x) = g(x + n) in (1.1), then we see that
n−1
X
I−1 (gn ) = (−1) I−1 (g) + 2
(−1)n−1−l g(l).
n
(1.2)
l=0
Let a fixed positive integer d with (p, d) = 1, set
X = Xd = lim(Z/dpN Z), X1 = Zp ,
←−
[
X∗ =
N
a + dpZp ,
0<a<dp
(a,p)=1
a + dpN Zp = {x ∈ X | x ≡ a (mod dpN )},
where a ∈ Z satisfies the condition 0 ≤ a < dpN . It is easy to see that
Z
Z
g(x)dµ−1 (x).
g(x)dµ−1 (x) =
I−1 (g) =
X
(1.3)
Zp
Let χ be Dirichlet’s character with conductor d ∈ N with d ≡ 1(mod 2). In [5],
(h)
we introduced the generalized (h, q)-tangent numbers Tn,χ,q and polynomials
(h)
(h)
Tn,χ,q (x) attached to χ. The generalized (h, q)-tangent numbers Tn,χ,q attached
to χ are defined by the generating function:
P
∞
a ha 2at
n
X
2 d−1
(h) t
a=0 χ(a)(−1) q e
T
=
, cf. [5].
(1.4)
n,χ,q
q hd e2dt + 1
n!
n=0
(h)
We consider the generalized (h, q)-tangent polynomials Tn,χ,q,w (x) attached to
χ as follows:
!
Pd−1
∞
a ha 2at
X
2 a=0 χ(a)(−1) q e
tn
xt
(h)
e =
Tn,χ,q (x) .
(1.5)
q hd e2dt + 1
n!
n=0
Theorem 1.1 ([5]) For positive integers n and h ∈ Z, we have
Z
(h)
Tn,χ,q (x) =
χ(y)q hy (2y + x)n dµ−1 (y).
X
Symmetric properties for the generalized (h, q)-tangent polynomials
1179
Corollary 1.2 ([5]) For positive integers n and h ∈ Z, we have
Z
(h)
χ(y)q hy (2y)n dµ−1 (y).
Tn,χ,q =
X
Theorem 1.3 ([5]) For positive integers n and h ∈ Z, we have
n X
n (h)
(h)
Tn,χ,q (x) =
Tl,χ,q .
l
l=0
2
Symmetry for for the generalized (h, q)-tangent
polynomials
In this section, we assume that q ∈ Cp . We obtain some interesting identities of
(h)
the power sums and the generalized polynomials Tn,χ,q (x) using the symmetric
properties for the p-adic invariant integral on Zp . If n is odd from (1.2), we
have
n−1
X
I−1 (gn ) + I−1 (g) = 2
(−1)k g(k) (see [1], [2], [3], [5]).
(2.1)
k=0
It will be more convenient to write (3.1) as the equivalent integral form
Z
Z
n−1
X
(−1)k g(k).
(2.2)
g(x + n)dµ−1 (x) +
g(x)dµ−1 (x) = 2
Zp
Zp
k=0
hx 2xt
Substituting g(x) = χ(x)q e into the above, we obtain
Z
Z
h(x+n) (2x+2n)t
χ(x + n)q
e
dµ−1 (x) +
χ(x)q hx e2xt dµ−1 (x)
X
=2
X
n−1
X
(2.3)
(−1)j χ(j)q hj e2jt .
j=0
(h)
For k ∈ N ∪ {0}, let us define the power sums Tk,χ,q (n) as follows:
(h)
Tk,χ,q (n) =
n
X
(−1)l χ(l)q hl (2l)k .
(2.4)
l=0
After some elementary calculations, we have
m
Z
∞ Z
X
t
h(x+nd)
m
hx
m
χ(x)q
(2x + 2nd) dµ−1 (x) +
χ(x)q (2x) dµ−1 (x)
m!
X
X
m=0
!
∞
nd−1
X
X
tm
=
2
(−1)j χ(j)q hj (2j)m
m!
m=0
j=0
(2.5)
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C. S. Ryoo
From the above, we get
Z
Z
x+nd (2x+2nd))t
χ(x)ζ
e
dµ−1 (x) +
χ(x)ζ x e2xt dµ−1 (x)
X
X
R
2 X χ(x)ζ x e2xt dµ−1 (x)
.
= R ndx 2ndtx
ζ
e
dµ
(x)
−1
X
(2.6)
By using (2.5) and (2.6), we arrive at the following theorem:
Theorem 2.1 Let n be odd positive integer. Then we obtain
R
∞
tm
2 X χ(x)q hx e2xt dµ−1 (x) X
(h)
R
=
2T
(nd
−
1)
.
m,χ,q
m!
q hndx e2ndtx dµ−1 (x)
X
m=0
Let w1 and w2 be odd positive integers. Then we set
S(w1 , w2 ) =
R R
χ(x1 )χ(x2 )q h(w1 x1 +w2 x2 ) e(2w1 x1 +2w2 x2 +w1 w2 x)t dµ−1 (x1 )dµ−1 (x2 )
X X
R
.
q hw1 w2 dx e2w1 w2 dxt dµ−1 (x)
X
(2.7)
By Theorem 2.1 and (2.7), after elementary calculations, we obtain
!
!
∞
∞
m
m
X
X
1
t
t
(h)
(h)
S(w1 , w2 ) =
Tm,χ,qw1 (w2 x)w1m
2
T m,χ,qw2 (w1 d − 1)w2m
.
2 m=0
m!
m!
m=0
By using Cauchy product in the above, we have
S(w1 , w2 ) =
∞
X
m=0
m X
m (h)
(h)
Tj,χ,qw1 (w2 x)w1j Tm−j,χ,qw2 (w1 d − 1)w2m−j
j
j=0
!
tm
.
m!
(2.8)
From the symmetry of S(w1 , w2 ) in w1 and w2 , we also see that
!
!
∞
∞
m
m
X
t
t
1 X (h)
(h)
m
S(w1 , w2 ) =
T
2
Tm,χ,qw1 (w2 d − 1)w1m
.
w2 (w1 x)w2
2 m=0 m,χ,q
m!
m!
m=0
Thus we obtain
S(w1 , w2 ) =
∞
X
m X
m
m=0
j=0
j
!
(h)
(h)
Tj,χ,qw2 (w1 x)w2j Tm−j,χ,qw1 (w2 d − 1)w1m−j
tm
m!
(2.9)
tm
By comparing coefficients
in the both sides of (2.8) and (2.9), we arrive at
m!
the following theorem:
Symmetric properties for the generalized (h, q)-tangent polynomials
1181
Theorem 2.2 Let w1 and w2 be odd positive integers. Then we obtain
m X
m
(h)
(h)
w1m−j w2j Tj,χ,qw2 (w1 x)Tm−j,χ,qw1 (w2 d − 1)
j
m X
m j m−j (h)
(h)
=
w1 w2 Tj,χ,qw1 (w2 x)Tm−j,χ,qw2 (w1 d − 1),
j
j=0
j=0
(h)
(h)
where Tk,χ,q (x) and Tm,χ,q (k) denote the generalized (h, q)-tangent polynomials
and the alternating sums of powers of consecutive (h, q)-integers, respectively.
By Theorem 2.2 and Theorem 1.3, we have the following corollary.
Corollary 2.3 Let w1 and w2 be odd positive integers. Then we obtain
j m X
X
m
j
(h)
(h)
w1m−k w2j xj−k Tk,χ,qw2 Tm−j,χ,qw1 (w2 d − 1)
j
k
j m X
X
m
j
(h)
(h)
=
w1j w2m−k xj−k Tk,χ,qw1 Tm−j,χ,qw2 (w1 d − 1).
j
k
j=0 k=0
j=0 k=0
Now we will derive another interesting identities for the generalized (h, q)tangent polynomials using the symmetric property of S(w1 , w2 ). By (2.7),
after elementary calculations, we obtain
S(w1 , w2 )
!
Z
wX
1 d−1
1 w1 w2 xt
χ(x1 )q hw1 x1 e2x1 w1 t dµ−1 (x1 )
2
(−1)j χ(j)q w2 hj e2jw2 t
e
=
2
X
j=0

=
wX
1 d−1
(−1)j χ(j)q w2 hj
χ(x1 )q hw1 x1 e
2jw2
(w1 t)
w1
dµ−1 (x1 )

X
j=0
=
2x1 +w2 x+
Z
∞
X
wX
1 d−1
n=0
j=0
j
(−1) χ(j)q
w2 hj
(h)
Tn,χ,qw1
! n
2jw2
t
w2 x +
w1n
.
w1
n!
(2.10)
1182
C. S. Ryoo
By using the symmetry property in (2.10), we also have
S(w1 , w2 )
!
Z
wX
2 d−1
1 w1 w2 xt
(−1)j χ(j)q w1 hj e2jw1 t
2
χ(x2 )q hw2 x2 e2x2 w2 t dµ−1 (x2 )
e
=
2
X
j=0
2jw1
2x2 +w1 x+
(w2 t)
w2
χ(x2 )q hw2 x2 e
dµ−1 (x1 )

=
wX
2 d−1
(−1)j χ(j)q w1 hj
Z
X
j=0
=

∞
X
w
2 −1
X
n=0
j=0
(h)
(−1)j χ(j)q w1 hj Tn,χ,qw2
! n
2jw1
t
w1 x +
w2n
.
w2
n!
(2.11)
tn
By comparing coefficients
in the both sides of (2.10) and (2.11), we have
n!
the following theorem.
Theorem 2.4 Let w1 and w2 be odd positive integers. Then we have
wX
1 d−1
2jw2
(−1) χ(j)q
w2 x +
w1n
w1
j=0
wX
2 d−1
2jw1
j
w1 hj (h)
=
(−1) χ(j)q
Tn,χ,qw2 w1 x +
w2n .
w
2
j=0
j
w2 hj
(h)
Tn,χ,qw1
If we take x = 0 in Theorem 2.4, we also derive the interesting identity for the
generalized (h, q)-tangent numbers as follows:
wX
1 d−1
j
w2 hj
(h)
Tn,χ,qw1
j
w1 hj
(h)
Tn,χ,qw2
(−1) χ(j)q
2jw2
w1
2jw1
w2
j=0
=
wX
2 d−1
(−1) χ(j)q
j=0
w1n
w2n .
Letting q → 1 in Theorem 2.4, we can immediately have the generalized multiplication theorem for the generalized tangent polynomials(see, [3]). Observe
that if χ = χ0 , then Theorem 2.4 reduces to Theorem 3.4 in [2].
References
[1] T. Kim, q-Volkenborn integration, Russ. J. Math. Phys. 9(2002), 288-299.
Symmetric properties for the generalized (h, q)-tangent polynomials
1183
[2] C. S. Ryoo, Symmetric Properties for the (h, q)-Tangent Polynomials,
Adv. Studies Theor. Phys., 8(6)(2014), 259 - 265.
http://dx.doi.org/10.12988/astp.2014.413
[3] C. S. Ryoo, Symmetric Identities for the Generalized Tangent Polynomials
Associated with p-Adic Integral on Zp , Applied Mathematical Sciences
8(17)(2014), 829 - 835. http://dx.doi.org/10.12988/ams.2014.312704
[4] C. S. Ryoo, A Note on the Generalized Twisted Tangent Polynomials, Int. Journal of Math. Analysis 7(55)(2013), 2717 - 2722.
http://dx.doi.org/10.12988/ijma.2013.39214
[5] C. S. Ryoo, Generalized (h, q)-Tangent Numbers and Polynomials,
Applied Mathematical Sciences, Vol. 8, 2014, no.
17, Applied Mathematical Sciences, 8(49)(2014), 2439 - 2445.
http://dx.doi.org/10.12988/ams.2014.4295
[6] H. Shin, J. Zeng, The q-tangent and q-secant numbers via continued fractions, European J. Combin. 31(2010), 1689-1705.
Received: October 17, 2014; Published: December 12, 2014