Download Symmetric properties for the degenerate q

Global Journal of Pure and Applied Mathematics.
ISSN 0973-1768 Volume 12, Number 4 (2016), pp. 2819–2827
© Research India Publications
http://www.ripublication.com/gjpam.htm
Symmetric properties for the degenerate q-tangent
polynomials associated with p-adic integral on Zp
C. S. Ryoo
Department of Mathematics,
Hannam University, Daejeon 306-791, Korea
Abstract
In [5], we studied the degenerate q-tangent numbers and polynomials associated
with p-adic integral on Zp . In this paper, by using the symmetry of p-adic integral
on Zp , we give recurrence identities the degenerate q-tangent polynomials and the
generalized factorial sums.
AMS subject classification:
Keywords: Degenerate tangent numbers and polynomials, degenerate q-tangent
numbers and polynomials, generalized factorial sums.
1.
Introduction
L. Carlitz introduced the degenerate Bernoulli polynomials(see [1]). Feng Qi et al. [2]
studied the partially degenerate Bernoull polynomials of the first kind in p-adic field. T.
Kim studied the Barnes’ type multiple degenerate Bernoulli and Euler polynomials (see
[3]), Recently, Ryoo introduced the degenerate q-tangent numbers Tn,q (λ) and polynomials Tn,q (x, λ) (see [5]). In this paper, by using these numbers and polynomials, we
give some interesting relations between the generalized factorial sums and the degenerate
q-tangent polynomials.
Let p be a fixed odd prime number. Throughout this paper we use the following
notations. By Zp we denote the ring of p-adic rational integers, Qp denotes the field of
rational numbers, N denotes the set of natural numbers, C denotes the complex number
field, Cp denotes the completion of algebraic closure of Qp , N denotes the set of natural
numbers and Z+ = N ∪ {0}, and C denotes the set of complex numbers.
Let νp be the normalized exponential valuation of Cp with |p|p = p−νp (p) = p−1 .
When one 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 assumes that
2820
C. S. Ryoo
|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. For
1
g ∈ U D(Zp ) = {g|g : Zp → Cp is uniformly differentiable function},
the fermionic p-adic invariant integral on Zp is defined by Kim as follows:
I−1 (g) =
Zp
N −1
p
g(x)dµ−1 (x) = lim
N →∞
g(x)(−1)x ,
(see [2, 3]).
(1.1)
x=0
If we take g1 (x) = g(x + 1) in (1.1), then we see that
I−1 (g1 ) + I−1 (g) = 2g(0), (see [2, 3]).
(1.2)
We recall that the classical Stirling numbers of the first kind S1 (n, k) and S2 (n, k) are
defined by the relations (see [7])
(x)n =
n
S1 (n, k)x and x =
k
n
k=0
n
S2 (n, k)(x)k ,
k=0
respectively. Here (x)n = x(x −1) · · · (x −n+1) denotes the falling factorial polynomial
of order n. We also have
∞
∞
tn
tn
(et − 1)m
(log(1 + t))m
S2 (n, m) =
S1 (n, m) =
and
.
n!
m!
n!
m!
n=m
n=m
(1.3)
The generalized falling factorial (x|λ)n with increment λ is defined by
(x|λ)n =
n−1
(x − λk)
(1.4)
k=0
for positive integer n, with the convention (x|λ)0 = 1. We also need the binomial
theorem: for a variable x,
(1 + λt)x/λ =
∞
tn
(x|λ)n .
n!
(1.5)
n=0
For t, λ ∈ Zp such that |λt|p < p − p−1 , if we take g(x) = q x (1 + λt)2x/λ in (1.2), then
we easily see that
2
q x (1 + λt)2x/λ dµ−1 (x) =
.
q(1 + λt)2/λ + 1
Zp
1
Symmetric properties for the degenerate q-tangent polynomials
2821
Let us define the degenerate q-tangent numbers Tn,q (λ) and polynomials Tn,q (x, λ) as
follows:
∞
tn
y
2y/λ
(1.6)
q (1 + λt)
dµ−1 (y) =
Tn,q (λ) ,
n!
Zp
n=0
∞
tn
y
(2y+x)/λ
q (1 + λt)
dµ−1 (y) =
Tn,q (x, λ) .
(1.7)
n!
Zp
n=0
By (1.6) and (1.7), we obtain the following Witt’s formula.
Theorem 1.1. For n ≥ 0, we have
Tn,q (x, λ) =
n n
l=0
l
Tl,q (λ)(x|λ)n−l .
Theorem 1.2. For n ∈ Z+ , we have
q x (2x|λ)n dµ−1 (x) = Tn,q (λ),
Z
p
q y (x + 2y|λ)n dµ−1 (y) = Tn,q (x, λ).
Zp
Recently, many mathematicians have studied in the area of the q-analogues of the
degenerate Bernoulli umbers and polynomials, Euler numbers and polynomials, tangent
numbers and polynomials (see [2, 3, 5, 7]). Our aim in this paper is to obtain symmetric
properties for the degenerate q-tangent numbers and polynomials. We investigate some
properties which are related to degenerate q-tangent polynomials Tn,q (x, λ) and the
generalized factorial sums.
2. The alternating generalized factorial sums and
q-tangent polynomials
In this section, we assume that q ∈ C with |q| < 1. By using (1.6), we give the alternating
generalized factorial sums as follows:
∞
n=0
∞
2
tn
Tn,q (λ) =
(−1)n q n (1 + λt)2n/λ .
=
2
2/λ
n!
q(1 + λt) + 1
n=0
From the above, we obtain
−
=
∞
∞
(−1)n q n (1 + λt)(2n+2k)/λ +
(−1)n−k q (n−k) (1 + λt)2n/λ
n=0
k−1
(−1)n−k q (n−k) (1 + λt)2n/λ .
n=0
n=0
2822
C. S. Ryoo
By using (1.6) and (1.7), we obtain
∞
∞
1
1
tj
tj
−k −k
−
Tj,q (2k) + (−1) q
Tj,q
2
j! 2
j!
j =0
j =0
k−1
∞
tj
.
(−1)−k q −k
(−1)n q n (2n|λ)j
=
j!
j =0
By comparing coefficients of
n=0
tj
in the above equation, we obtain
j!
k−1
(−1)k+1 q k Tj,q (2k) + Tj,q
.
(−1)n q n (2n|λ)j =
2
n=0
By using the above equation we arrive at the following theorem:
Theorem 2.1. Let k be a positive integer and q ∈ C with |q| < 1. Then we obtain
Sj,q (k − 1, λ) =
k−1
(−1)k+1 q k Tj,q (2k) + Tj,q
.
(−1)n q n (2n|λ)j =
2
(2.1)
n=0
Remark 2.2. For the alternating generalized factorial sums, we have
lim Sj,q (k − 1) =
q→1
k−1
(−1)k+1 Tj (2k) + Tj
(−1)n (2n|λ)j =
,
2
n=0
where Tj (x) and Tj denote the tangent polynomials and the tangent numbers, respectively
(see [6]).
3.
Symmetry properties of the q-deformed fermionic integral on Zp
In this section, we assume that q ∈ Cp . In this section, we obtain recurrence identities
the degenerate q-tangent polynomials and the alternating generalized factorial sums. By
using (1.1), we have
I−1 (gn ) + (−1)
n−1
n−1
I−1 (g) = 2
(−1)n−1−k g(k),
k=0
where n ∈ N, gn (x) = g(x + n). If n is odd from the above, we obtain
n−1
(−1)n−1−k g(k) (see [2], [3], [4], [5]).
I−1 (gn ) + I−1 (g) = 2
k=0
(3.1)
Symmetric properties for the degenerate q-tangent polynomials
2823
It will be more convenient to write (3.1) as the equivalent integral form
Zp
g(x + n)dµ−1 (x) +
Zp
n−1
g(x)dµ−1 (x) = 2
(−1)n−1−k g(k).
(3.2)
k=0
Substituting g(x) = q x (1 + λt)2x/λ into the above, we obtain
(x+n)
(2x+2n)/λ
q
(1 + λt)
dµ−1 (x) +
q x (1 + λt)2x/λ dµ−1 (x)
Zp
Zp
(3.3)
n−1
(−1)j q j (1 + λt)2j/λ .
=2
j =0
After some calculations, we have
2
,
q x (1 + λt)2x/λ dµ−1 (x) =
q(1 + λt)2/λ + 1
Zp
q (x+n) (1 + λt)(2x+2n)/λ dµ−1 (x) = q n (1 + λt)2n/λ
Zp
By using (3.3) and (3.4), we have
(x+n)
(2x+2n)/λ
q
(1 + λt)
dµ−1 (x) +
Zp
=
2(1 + q n (1 + λt)2n/λ )
q(1 + λt)2/λ + 1
=
Zp
q x (1 + λt)2x/λ dµ−1 (x)
By (3.3), we obtain
∞
(x+n)
q
(2x + 2n|λ)m dµ−1 (x) +
=
q x (1 + λt)2x/λ dµ−1 (x)
(3.5)
2 Z q x (1 + λt)2x/λ dµ−1 (x)
p
.
nx
2nx/λ dµ (x)
−1
Zp q (1 + λt)
m=0
∞
m=0
Zp

2

n−1
tm
j j

(−1) q (2j |λ)m
m!
j =0
(3.4)
.
From the above, we get
(x+n)
(2x+2n)/λ
q
(1 + λt)
dµ−1 (x) +
Zp
Zp
2
.
q(1 + λt)2/λ + 1
Zp
q x (2x|λ)m dµ−1 (x)
tm
m!
2824
C. S. Ryoo
tm
in the above equation, we obtain
m!
m m
x
n
q x (2x|λ)m dµ−1 (x)
q (2x|λ)k dµ−1 (x) +
(2n|λ)m−k
q
k
Zp
Zp
By comparing coefficients
k=0
n−1
(−1)j q j (2j |λ)m
=2
j =0
By using (2.1), we have
m m
n
m−k
x
k
q (2x) dµ−1 (x) +
q x (2x)m dµ−1 (x)
(2n)
q
k
Zp
Zp
k=0
(3.6)
= 2Sm,q (n − 1, λ).
By using (3.5) and (3.6), we arrive at the following theorem:
Theorem 3.1. Let n be odd positive integer. Then we obtain
∞
2 Zp q x (1 + λt)2x/λ dµ−1 (x)
tm
.
2S
(n
−
1,
λ)
=
m,q
nx
2nx/λ dµ (x)
m!
−1
Zp q (1 + λt)
(3.7)
m=0
Let w1 and w2 be odd positive integers. By using (3.7), we have
(w1 x1 +w2 x2 ) (1 + λt)(w1 2x1 +w2 2x2 +w1 w2 x)/λ dµ (x )dµ (x )
−1 1
−1 2
Zp Zp q
w1 w2 x (1 + λt)2w1 w2 x/λ dµ (x)
−1
Zp q
(3.8)
2(1 + λt)w1 w2 x/λ q w1 w2 (1 + λt)2w1 w2 /λ + 1
= w
(q 1 (1 + λt)2w1 /λ + 1)(q w2 (1 + λt)2w2 /λ + 1)
By using (3.7) and (3.8), after elementary calculations, we obtain
1
a=
q w1 x1 (1 + λt)(w1 2x1 +w1 w2 x)/λ dµ−1 (x1 )
2 Zp
2 Zp q w2 x2 (1 + λt)2x2 w2 /λ dµ−1 (x2 )
× w1 w2 x (1 + λt)2w1 w2 x/λ dµ (x)
−1
Zp q
∞
∞
m
m
1
t
t
λ
λ
=
Tm,q w1 w2 x,
Sm,q w2 w1 − 1,
w1m
w2m
2
.
2
w1
m!
w2
m!
m=0
m=0
(3.9)
By using Cauchy product in the above, we have


m ∞
λ
λ
m
tm
j
m−j

Tj,q w1 w2 x,
w1 Sm−j,q w2 w1 − 1,
w2  .
a=
w1
w2
m!
j
m=0
j =0
(3.10)
Symmetric properties for the degenerate q-tangent polynomials
2825
By using the symmetry in (3.9), we have
1
a=
q w2 x2 (1 + λt)(w2 2x2 +w1 w2 x)/λ dµ−1 (x2 )
2 Zp
2 Zp q w1 x1 (1 + λt)2x1 w1 /λ dµ−1 (x1 )
× w1 w2 x (1 + λt)2w1 w2 x/λ dµ (x)
−1
Zp q
∞
∞
m
m
1
t
t
λ
λ
=
2
.
Tm,q w2 w1 x,
Sm,q w1 w2 − 1,
w2m
w1m
2
w2
m!
w1
m!
m=0
m=0
Thus we have
a=
∞
m=0


m λ
λ
m
tm
j
m−j

Tj,q w2 w1 x,
w2 Sm−j,q w1 w2 − 1,
w1 
w2
w1
m!
j
j =0
(3.11)
tm
in the both sides of (3.10) and (3.11), we arrive at the
By comparing coefficients
m!
following theorem:
Theorem 3.2. Let w1 and w2 be odd positive integers. Then we obtain
m m
λ
λ
j m−j
Tj,q w2 w1 x,
Sm−j,q w1 w2 − 1,
w2 w1
w2
w1
j
j =0
m λ
λ
m
j m−j
=
Tj,q w1 w2 x,
Sm−j,q w2 w1 − 1,
w1 w2 ,
w1
w2
j
j =0
where Tk,q (x) and Tm,q (k) denote the degenerate q-tangent polynomials and the alternating generalized factorial sums, respectively (see [5]).
By using Theorem 2, we have the following corollary:
Corollary 3.3. Let w1 and w2 be odd positive integers. Then we obtain
j m m j
j =0 k=0
=
j
k
m−j j
w1 w2
j m m j
j =0 k=0
j
k
λ
w1 x w2
j m−j
w1 w2
j −k
λ
w2 x w1
Tk,q w2
λ
Sm−j,q w1 (w2 − 1)
w2
j −k
Tk,q w1
λ
Sm−j,q w2 (w1 − 1).
w1
2826
C. S. Ryoo
By using (3.8), we have
1
a=
(1 + λt)w1 w2 x/λ
q w1 x1 (1 + λt)2x1 w1 /λ dµ−1 (x1 )
2
Zp
2 Zp q w2 x2 (1 + λt)2x2 w2 /λ dµ−1 (x2 )
× w1 w2 x (1 + λt)2w1 w2 x/λ dµ (x)
−1
Zp q
1
=
q w1 x1 (1 + λt)2x1 w1 /λ dµ−1 (x1 )
(1 + λt)w1 w2 x/λ
2
Zp


w
1 −1
× 2
(−1)j q w2 j (1 + λt)2j w2 /λ 
j =0
=
w
1 −1
(−1)j q w2 j
Zp
j =0
=
∞

w
1 −1

2j w2
2x1 +w2 x+
w1
q w1 x1 (1 + λt)
j =0
n=0
(w1 )/λ
dµ−1 (x1 )

(−1)j q w2 j Tn,q w1 w2 x +
(3.12)
2j w2 λ
tn
,
w1n  .
w1 w1
n!
By using the symmetry property in (3.12), we also have
1
q w2 x2 (1 + λt)2x2 w2 /λ dµ−1 (x2 )
(1 + λt)w1 w2 x/λ
a=
2
Zp
w
x
2 Zp q 1 1 (1 + λt)2x1 w1 /λ dµ−1 (x1 )
× w1 w2 x (1 + λt)2w1 w2 x/λ dµ (x)
−1
Zp q
1
=
(1 + λt)w1 w2 x/λ
q w2 x2 (1 + λt)2x2 w2 /λ dµ−1 (x2 )
2
Zp


w
2 −1
× 2
(−1)j q w1 j (1 + λt)2j w1 /λ 
j =0
=
w
2 −1
(−1)j q w1 j
j =0
=
∞
n=0

w
2 −1

j =0
Zp
2j w1
2x2 +w1 x+
w2
q w2 x2 (1 + λt)
(−1)j q w1 j Tn,q w2 w1 x +
(3.13)
(w2 )/λ
dµ−1 (x1 )

2j w1 λ
tn
,
w2n  .
w2 w2
n!
tn
By comparing coefficients in the both sides of (3.12) and (3.13), we have the following
n!
theorem.
Symmetric properties for the degenerate q-tangent polynomials
2827
Theorem 3.4. Let w1 and w2 be odd positive integers. Then we obtain
w
1 −1
j w2 j
(−1) q
T
n,q w1
j =0
=
w
2 −1
2j w2 λ
,
w2 x +
w1n
w1 w1
2j w1 λ
,
w2n .
w1 x +
w2 w2
j w1 j
(−1) q
Tn,q w2
j =0
Observe that if λ → 0, then Theorem 9 reduces to Theorem 3.4 in [4]. Substituting
w1 = 1 into Theorem 9, we have the following corollary.
Corollary 3.5. Let w2 be odd positive integer. Then we obtain
Tn,q (x) =
w2n
w
2 −1
j =0
(−1) q Tn,q w2
j j
x + 2j λ
,
.
w2
w2
Acknowledgment
This work was supported by 2016 Hannam University Research Fund
References
[1] L. Carlitz, Degenerate Stirling, Bernoulli and Eulerian numbers, Utilitas Math.
15(1979), 51–88.
[2] F. Qi, D. V. Dolgy, T. Kim, C. S. Ryoo, On the partially degenerate Bernoulli
polynomials of the first kind, Global Journal of Pure and Applied Mathematics,
11(2015), 2407–2412.
[3] T. Kim, Barnes’ type multiple degenerate Bernoulli and Euler polynomials, Appl.
Math. Comput. 258(2015), 556–564
[4] C. S. Ryoo, On the symmetric properties for the q-tangent polynomials, Int. Journal
of Math. Analysis, 7(2013), 3027–3033.
[5] C. S. Ryoo, On the degenerate q-tangent numbers and polynomials associated with
the p-adic integral on Zp , International Journal of Mathematical Analysis, 9(2015),
2571–2579.
[6] C. S. Ryoo, A Note on the Tangent Numbers and Polynomials, Adv. Studies Theor.
Phys., 7(9)(2013), 447–454.
[7] P. T.Young, Degenerate Bernoulli polynomials, generalized factorial sums, and their
applications, Journal of Number Theory, 128(2008), 738–758.