Symmetric identities of higher-order degenerate q−Euler polynomials

J. Nonlinear Sci. Appl. 9 (2016), 443–451 Research Article

Symmetric identities of higher-order degenerate q−Euler polynomials

Dae San Kim^a, Taekyun Kim^b,∗

aDepartment of Mathematics, Sogang University, Seoul 121-742, Republic of Korea.

bDepartment of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea.

Communicated by Seog-Hoon Rim

Abstract

In this paper, we study the higher-order degenerateq-Euler polynomials and give some identities of symmetry on these polynomials derived from symmetric properties for certain multivariate fermionicp-adicq-integrals on Zp. c2016 All rights reserved.

Keywords: Symmetry, identity, higher-order degenerateq-Euler polynomial.

2010 MSC: 11B75, 11B83, 11S80.

1. Introduction

Let p be an odd prime number. Throughout this paper, Zp, Qp and Cp will denote the ring of p-adic integers, the field of p-adic rational numbers and the completion of the algebraic closure of Qp. Thep-adic norm is normalized so that |p|_p = ¹_p. Let q be an indeterminate in Cp such that |1−q|_p < p^−p−11 . The q-analogue of the numberx is defined as [x]_q = ^1−q_1−q^x. Note that limq→1[x]_q=x. Let f(x) be a continuous functional Zp. Then, the fermionicp-adicq-integral on Zp is defined by Kim as

Z

Zp

f(x)dµ−q(x) = lim

N→∞

1 [p^N]_−q

p^N−1

X

x=0

f(x) (−q)^x (1.1)

= [2]_q 2 lim

N→∞

P^N−1

X

x=0

f(x)q^x(−1)^x, (see [12, 14]),

∗Corresponding author

Email addresses: [email protected](Dae San Kim),[email protected](Taekyun Kim) Received 2015-09-10

where [x]_−q = ^1−(−q)_1+q ^x. Note that

q→1lim Z

Zp

f(x)dµ−q(x) = lim

N→∞

p^N−1

X

x=0

f(x) (−1)^x (1.2)

= Z

Z^p

f(x)dµ−1(x) is the ordinary fermonicp-adic integral onZp.

From (1.1), we can easily derive the following equation:

qⁿ Z

Zp

f(x+n)dµ−q(x) + (−1)ⁿ⁻¹ Z

Zp

f(x)dµ−q(x) = [2]_q

n−1

X

x=0

f(x) (−1)^n−1−x, (1.3) and

q Z

Zp

f(x+ 1)dµ−q(x) + Z

Zp

f(x)dµ−q(x) = [2]_qf(0), (see [14]). (1.4) As is well known, the higher-order Euler polynomials are defined by the generating function

2 e^t+ 1

r

e^xt=

∞

X

n=0

E_n^(r)(x)tⁿ

n!. (1.5)

When x= 0,En^(r) =En^(r)(0) are called the higher-order Euler numbers (see [1]–[23]).

From (1.2), we note that Z

Zp

· · · Z

Zp

e^{(x1+···+xr+x)t}dµ−1(x1)· · ·dµ−1(xr) = 2

e^t+ 1 r

e^xt

=

∞

X

n=0

E_n^(r)(x)tⁿ n!.

Carlitz considered q-Bernoulli numbers defined by the recurrence relation β_0,q = 1, q(qβ_q+ 1)ⁿ−β_n,q =

(1, ifn= 1,

0, ifn >1, (1.6)

with the usual convention about replacingβ_qⁿ byβn,q (see [4]).

In [12, 14], Kim defined Carlitz’s type q-Euler numbers given by

E_0,q = 1, q(qE_q+ 1)ⁿ− E_n,q= [2]_qδ0,n, (1.7) whereδn,k is the Kronecker’s symbol.

Recently, the higher-orderq-Euler polynomials are defined by the multivariate fermionicp-adicq-integral on Zp

Z

Zp

· · · Z

Zp

e^{[x1+···+xr+x]qt}dµ−q(x1)· · ·dµ−q(xr) =

∞

X

n=0

E_n,q^(r)(x)tⁿ

n!, (see [14]). (1.8) When x = 0, E_n,q^(r) = E_n,q^(r)(0) are called the higher-order q-Euler numbers. In particular, r = 1, then E_n,q⁽¹⁾(x) =E_n,q(x).

From (1.8), we have Z

Zp

· · · Z

Zp

[x₁+· · ·+x_r+x]ⁿ_qdµ−q(x₁)· · ·dµ−q(x_r) =E_n,q^(r)(x) (1.9)

= 1 (1−q)ⁿ

n

X

l=0

n l

(−1)^l

[2]_q 1 +q^l+1

^r q^lx

= [2]^r_q

∞

X

m1,...,mr=0

(−q)^m1+···+mr[m1+· · ·+mr+x]ⁿ_q , wherer∈Nand n≥0.

By (1.9), we get the generating function of the higher-order q-Euler polynomials as follows:

[2]^r_q

∞

X

m1,...,mr=0

(−q)^m1+···+mre^{[m1+···+mr]qt} (1.10)

=

∞

X

n=0

E_n,q^(r)(x)tⁿ

n!, (see [14, 15]).

Carlitz introduced the higher-order degenerate Euler polynomials given by the generating function 2

(1 +λt)^1λ+ 1

!r

(1 +λt)^λx =

∞

X

n=0

E_n^(r)(x|λ)tⁿ

n!. (1.11)

When x = 0, E_n^(r)(x) = E_n^(r)(0|λ) are called the higher-order degenerate Euler numbers (see [5]). In particular,r= 1, E_n⁽¹⁾(x|λ) =E_n(x|λ) are called degenerate Euler polynomials.

Note that limλ→0E_n^(r)(x|λ) =E^(r)n (x), (n≥0).

In this paper, we study the higher-order degenerate q-Euler polynomials and give some identities of symmetry on these polynomials derived from symmetric properties for certain multivariate fermionicp-adic q-integrals onZp.

2. Symmetric identities of higher-order degenerate q-Euler polynomials

Let λ, t∈Cp be such that|λt|_p < p^−p−11 . From (1.2) and (1.3), we note that Z

Zp

· · · Z

Zp

(1 +λt)

x1+···+xr+x

λ dµ−1(x1)· · ·dµ−1(xr) = 2 (1 +λt)^λ1 + 1

!r

(1 +λt)^xλ (2.1)

=

∞

X

n=0

E_n^(r)(x|λ)tⁿ n!.

In view of (1.8), we define the higher-order degenerate q-Euler polynomials by the generating function as

Z

Zp

· · · Z

Zp

(1 +λt)^{λ1[x1+···+xr+x]q}dµ−q(x₁)· · ·dµ−q(x_r) =

∞

X

n=0

E_n,q,λ^(r) (x)tⁿ

n!. (2.2)

Thus, by (2.2), we get

λ→0limE_n,q,λ^(r) (x) =E_n,q^(r)(x), (n≥0). From (2.2), we can derive

Z

Zp

· · · Z

Zp

[x1+· · ·+xr+x]_q

n,λdµ−q(x1)· · ·dµ−q(xr) =E_n,q,λ^(r) (x), (n≥0), (2.3) where

[x]_q

n,λ= [x]_q

[x]_q−λ [x]_q−2λ

· · ·

[x]_q−(n−1)λ

, (n≥1)

and

[x]_q

0,λ= 1.

By (2.3), we get E_n,q,λ^(r) (x) =

Z

Zp

· · · Z

Zp

[x₁+· · ·+x_r]_q

n,λdµ−q(x₁)· · ·dµ−q(x_r) (2.4)

=

n

X

l=0

S1(n, l)λ^n−l Z

Zp

· · · Z

Zp

[x1+· · ·+xr+x]^l_qdµ−q(x1)· · ·dµ−q(xr)

=

n

X

l=0

S1(n, l)λ^n−lE_l,q^(r)(x), whereS₁(n, l) is the Stirling number of the first kind.

From (1.9) and (2.4), we have E_n,q,λ^(r) (x) = [2]^r_q

n

X

l=0

∞

X

m1,...,mr=0

(−q)^m1+···+mr[m1+· · ·+mr+x]^l_qS1(n, l)λ^n−l. (2.5)

Therefore, by (2.4), we obtain the following theorem.

Theorem 2.1. For n≥0, we have E_n,q,λ^(r) (x) =

n

X

l=0

S1(n, l)λ^n−lE_l,q^(r)(x)

= [2]^r_q

n

X

l=0

∞

X

m1,...,mr=0

(−q)^m1+···+mr[m1+· · ·+mr+x]^l_qS1(n, l)λ^n−l.

Now, we observe that [2]^r_q

n

X

l=0

∞

X

m1,...,mr=0

(−q)^m1+···+mr[m1+· · ·+mr+x]^l_qS1(n, l)λ^n−l (2.6)

= [2]^r_q

∞

X

m1,...,mr=0

(−q)^m1+···+mr

[m₁+· · ·+m_r+x]_q

n,λ.

Thus, by (2.6), we get

∞

X

n=0

E_n,q,λ^(r) (x)tⁿ n!= [2]^r_q

∞

X

m1,...,mr=0

(−q)^m1+···+mr

∞

X

n=0

[m1+· · ·+mr+x]_q

n,λ

n! tⁿ (2.7)

= [2]^r_q

∞

X

m1,...,mr=0

(−q)^m1+···+mr(1 +λt)

[m1+···+mr+x]q

λ .

Therefore, by (2.7), we obtain the following theorem.

Theorem 2.2. For r∈N, we have

∞

X

n=0

E_n,q,λ^(r) (x)tⁿ n! = [2]^r_q

∞

X

m1,...,mr=0

(−q)^m1+···+mr(1 +λt)

[m1+···+mr+x]q

λ .

By replacing tby ¹_λ e^λt−1

in (2.2), we get Z

Zp

e^{[x1+···+xr+x]qt}dµ−q(x1)· · ·dµ−q(xr) =

∞

X

m=0

E_m,q,λ^(r) (x)λ^−m e^λt−1m

m! (2.8)

=

∞

X

n=0 n

X

m=0

λ^n−mE_m,q,λ^(r) (x)S₂(n, m)

!tⁿ n!, whereS₂(n, m) is the Stirling number of the second kind.

Therefore, by (1.8) and (2.8), we obtain the following theorem.

Theorem 2.3. For n≥0, we have

E_n,q^(r)(x) =

n

X

m=0

λ^n−mE_m,q,λ^(r) (x)S₂(n, m). Let w₁, w₂ ∈N be such thatw₁ ≡1, w₂ ≡1 (mod 2). Then, by (2.2), we get

1 [w₁]^r_−q

Z

Z^p

· · · Z

Z^p

(1 +λt)

[^w1w2x+w2Pr

l=1jl+w1Pr l=1yl]_q

λ dµ−q^w1(y1)· · ·dµ−q^w1 (yr) (2.9)

= 1

[w1]^r_−q lim

N→∞

p^N−1

X

y1,...,yr=0

1 [p^N]^r_−qw1

(1 +λt)

[w1w2x+w2Pr

l=1jl+w1Pr l=1yl]_q

λ

×(−q^w1)^y1+···+yr

= 1

[w₁]^r_−q lim

N→∞

1 [w2p^N]^r_−qw1

w2p^N−1

X

y1,...,yr=0

(1 +λt)

[^w1w2x+w2Pr

l=1jl+w1Pr l=1yl]_q

λ

×(−q)^{w1y1+···+w1yr}

= [2]^r_q 2^r lim

N→∞

w2−1

X

i1,i2,...,ir=0 p^N−1

X

y1,...,yr=0

(1 +λt)

[w1w2x+w2Pr

l=1jl+w1Pr

l=1(_il+w2yl)]_q

λ

×(−1)^y1+···+yrq^{w1(i1+w2y1)+w1(i2+w2y2)+···+w1(ir+w2yr)}×(−1)^i1+···+ir

= [2]^r_q 2^r

w2−1

X

i1,...,ir=0

(−1)^Prl=1ilq^w1Prl=1il

× lim

N→∞

p^N−1

X

y1,...,yr=0

(1 +λt)

[^w1w2x+w2Pr

l=1jl+w1Pr

l=1(il+w2yl)]_q

λ

×(−1)^y1+···+yrq^{w1w2y1+w1w2y2+···+w1w2yr}. From (2.9), we note that

1 [w1]^r_−q

w1−1

X

j1,...,jr=0

q^w2Prl=1jl(−1)^Prl=1jl (2.10)

× Z

Zp

· · · Z

Zp

(1 +λt)

[w1w2x+w2Pr

l=1jl+w1Pr l=1yl]_q

λ dµ−q^w1 (y1)· · ·dµ−q^w1 (yr)

= [2]^r_q 2^r lim

N→∞

w2−1

X

i1,...,ir=0 w1−1

X

j1,...,jr=0 p^N−1

X

y1,...,yr=0

(−1)^{Prl=1(jl+il+yl)}

×q^{w1Prl=1il+w2Prl=1jl+w1w2Prl=1yl}

×(1 +λt)

1

λ[^w1w2x+w2Prl=1jl+w1Pr

l=1il+w1w2Pr l=1yl]_q. On the other hand,

1 [w₂]^r_−q

w2−1

X

j1,...,jr=0

q^w1Prl=1jl(−1)^Prl=1jl (2.11)

× Z

Z^p

· · · Z

Z^p

(1 +λt)

[^w1w2x+w1Pr

l=1jl+w2Pr l=1yl]_q

λ dµ−q^w2 (y1)· · ·dµ−q^w2 (yr)

= [2]^r_q 2^r lim

N→∞

w1−1

X

i1,...,ir=0 w2−1

X

j1,...,jr=0 p^N−1

X

y1,...,yr=0

(−1)^{Prl=1(il+jl+yl)}

×q^{w1Prl=1jl+w2Prl=1il+w1w2Prl=1yl}

×(1 +λt)

1

λ[^w1w2x+w1Pr

l=1jl+w2Pr

l=1il+w1w2Pr l=1yl]_q. Therefore, by (2.10) and (2.11), we obtain the following theorem.

Theorem 2.4. Let w1, w2 ∈Nsuch that w1≡1 (mod 2) andw2 ≡1 (mod 2). Then, we have 1

[w₁]^r_−q

w1−1

X

j1,...,jr=0

q^w2Prl=1jl(−1)^Prl=1jl

× Z

Z^p

· · · Z

Z^p

(1 +λt)

[^w1w2x+w2Pr

l=1jl+w1Pr l=1yl]_q

λ dµ−q^w1(y1)· · ·dµ−q^w1(yr)

= 1

[w₂]^r_−q

w2−1

X

j1,...,jr=0

q^w1Prl=1jl(−1)^Prl=1jl

× Z

Z^p

· · · Z

Z^p

(1 +λt)

[^w1w2x+w1Pr

l=1jl+w2Pr l=1yl]_q

λ dµ−q^w2(y1)· · ·dµ−q^w2(yr). We observe that

"

w₁w₂x+

r

X

l=1

j_lw₂+

r

X

l=1

y_lw₁

#

q

= [w₁]_q

"

w₂x+ w₂ w₁

r

X

l=1

j_l+

r

X

l=1

y_l

#

q^w1

. (2.12)

From (2.12), we have Z

Zp

· · · Z

Zp

(1 +λt)

1

λ[^w1w2x+Pr

l=1jlw2+Pr l=1ylw1]_q

dµ−q^w1 (y1)· · ·dµ−q^w1(yr) (2.13)

= Z

Zp

· · · Z

Zp

(1 +λt)

[w1]q λ

h w2x+^w_w²

1

Pr

l=1j_l+Pr l=1y_li

qw1 dµ−q^w1 (y₁)· · ·dµ−q^w1(y_r)

=

∞

X

n=0

E^(r)

n,q^w1,_[w^λ

1]q

w2x+w2

w1

(j1+· · ·+jr)

[w1]ⁿ_q tⁿ n!.

Therefore, by Theorem 2.4, (2.12) and (2.13), we obtain the following theorem.

Theorem 2.5. For n≥0, w1, w2 ∈Nwith w1≡1 (mod 2),w2 ≡1 (mod 2), we have [w₁]ⁿ_q

[w₁]^r_−q

w1−1

X

j1,...,jr=0

(−1)^j1+···+jrq^{w2(j1+···+jr)}E^(r)

n,q^w1,_[w^λ

1]q

w2x+w2

w₁ (j1+· · ·+jr)

= [w₂]ⁿ_q [w₂]^r_−q

w2−1

X

j1,...,jr=0

(−1)^j1+···+jrq^{w1(j1+···+jr)}E^(r)

n,q^w2,_[w^λ

2]q

w1x+w1

w₂ (j1+· · ·+jr)

.

From (2.3), we have Z

Zp

· · · Z

Zp

1 [w₁]_q

!n

[w₁]_q

"

w₂x+ w₂ w₁

r

X

l=1

j_l+

r

X

l=1

y_l

#

q^w1





n,λ

(2.14)

×dµ_−q^w1 (y1)· · ·dµ−q^w1 (yr)

=

n

X

l=0

S₁(n, l) λ [w₁]_q

!n−l

Z

Zp

· · · Z

Zp

"

w₂x+w₂ w₁

r

X

l=1

j_l+

r

X

l=1

y_l

#l

q^w1

×dµ_−q^w1 (y₁)· · ·dµ−q^w1 (y_r)

=

n

X

l=0

S₁(n, l) λ [w1]_q

!n−l l

X

i=0

l i

[w2]_q [w1]_q

!i

[j₁+· · ·+j_r]ⁱ_qw2q^{w2(l−i)Prk=1jk}

× Z

Zp

· · · Z

Zp

[w₂x+y₁+· · ·+y_r]^l−i_qw1dµ−q^w1(y₁)· · ·dµ−q^w1(y_r)

=

n

X

l=0 l

X

i=0

S1(n, l)λ^n−l[w1]^l−n−i_q [w2]ⁱ_q[j1+· · ·+jr]ⁱ_qw2

×q^{w2(l−i)Prk=1jk} l

i

E_l−i,q^(r) w1 (w₂x). Thus, by (2.14), we get

[w₁]ⁿ_q [w₁]^r_−q

w1−1

X

j1,...,jr=0

q^w2Prl=1jl(−1)^Prl=1jl 1 [w₁]_q

!n

(2.15)

× Z

Zp

· · · Z

Zp



[w₁]_q

"

w₂x+w₂ w₁

r

X

l=1

j_l+

r

X

l=1

y_l

#

q^w1





n,λ

dµ−q^w1 (y₁)· · ·dµ−q^w1 (y_r)

= 1

[w₁]^r_−q

n

X

l=0 l

X

i=0

l i

S1(n, l)λ^n−l[w1]^l−i_q [w2]ⁱ_qE_l−i,q^(r) w1 (w2x) ˜T_l+1,i^(r) (w1 |q^w2), where

T˜_n,i^(r)(w|q) =

w−1

X

j1,...,jr=0

(−1)^j1+···+jr[j₁+· · ·+j_r]ⁱ_qq^{(n−i)Prl=1jl}. (2.16) On the other hand,

[w₂]ⁿ_q [w₂]^r_−q

w2−1

X

j1,...,jr=0

q^w1Prl=1jl(−1)^Prl=1jl λ [w₂]_q

!n

(2.17)

× Z

Zp

· · · Z

Zp



 [w₂]_q

λ

"

w₁x+w1

w₂

r

X

l=1

j_l+

r

X

l=1

y_l

#

q^w2





n,λ

dµ−q^w2 (y₁)· · ·dµ−q^w2 (y_r)

= 1

[w₂]^r_−q

n

X

l=0 l

X

i=0

l i

S1(n, l)λ^n−l[w2]^l−i_q [w1]ⁱ_qE_l−i,q^(r) w2 (w1x) ˜T_l+1,i^(r) (w2|q^w1).

Therefore, by (2.15) and (2.17), we obtain the following theorem.

Theorem 2.6. For n≥0, w₁, w₂ ∈Nwith w₁≡1 (mod 2),w₂ ≡1 (mod 2), we have 1

[w1]^r_−q

n

X

l=0 l

X

i=0

l i

S₁(n, l)λ^n−l[w₁]^l−i_q [w₂]ⁱ_qE_l−i,q^(r) w1 (w₂x) ˜T_l+1,i^(r) (w₁ |q^w2)

= 1

[w2]^r_−q

n

X

l=0 l

X

i=0

l i

S₁(n, l)λ^n−l[w₂]^l−i_q [w₁]ⁱ_qE_l−i,q^(r) w2 (w₁x) ˜T_l+1,i^(r) (w₂ |q^w1). Remark 2.7. If we take λ→0, then we get

1 [w₁]^r_−q

n

X

i=0

n i

[w1]ⁿ⁻ⁱ_q [w2]ⁱ_qE_n−i,q^(r) w1 (w2x) ˜T_n+1,i^(r) (w1 |q^w2)

= 1

[w2]^r_−q

n

X

i=0

n i

[w₂]ⁿ⁻ⁱ_q [w₁]ⁱ_qE_n−i,q^(r) w2 (w₁x) ˜T_n+1,i^(r) (w₂ |q^w1).

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