Hindawi Publishing Corporation
International Journal of Combinatorics
Volume 2011, Article ID 432738, 12 pages
doi:10.1155/2011/432738
Research Article
Identities of Symmetry for Generalized
Euler Polynomials
Dae San Kim
Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea
Correspondence should be addressed to Dae San Kim, dskim@sogang.ac.kr
Received 10 January 2011; Accepted 15 February 2011
Academic Editor: Chı́nh T. Hoang
Copyright q 2011 Dae San Kim. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
We derive eight basic identities of symmetry in three variables related to generalized Euler
polynomials and alternating generalized power sums. All of these are new, since there have been
results only about identities of symmetry in two variables. The derivations of identities are based
on the p-adic fermionic integral expression of the generating function for the generalized Euler
polynomials and the quotient of integrals that can be expressed as the exponential generating
function for the alternating generalized power sums.
1. Introduction and Preliminaries
Let p be a fixed odd prime. Throughout this paper, p, p , and p will, respectively, denote
the ring of p-adic integers, the field of p-adic rational numbers, and the completion of the
algebraic closure of p . Let d be a fixed odd positive integer. Then we let
X Xd lim
←
−
N
and let π : X →
p
dpN
,
1.1
be the map given by the inverse limit of the natural maps
dpN
−→
pN
.
1.2
If g is a function on p, then we will use the same notation to denote the function g ◦ π. Let
∗
χ : /d ∗ → be a primitive Dirichlet character of conductor d. Then it will be pulled
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back to X via the natural map X → /d . Here we fix, once and for all, an imbedding
→ p , so that χ is regarded as a map of X to p cf. 1.
For a continuous function f : X → p , the p-adic fermionic integral of f is defined by
fzdμ−1 z lim
N
dp
−1
N→∞
X
f j −1j .
1.3
j0
Then it is easy to see that
fz 1dμ−1 z X
fzdμ−1 z 2f0.
1.4
X
More generally, we deduce from 1.4 that, for any odd positive integer n,
fz ndμ−1 z X
fzdμ−1 z 2
X
n−1
−1a fa
1.5
a0
and that, for any even positive integer n,
fz ndμ−1 z −
X
fzdμ−1 z 2
X
n−1
−1a−1 fa.
1.6
a0
Let | · |p be the normalized absolute value of p , such that |p|p 1/p, and let
E t ∈ p | |t|p < p−1/p−1 .
1.7
Then, for each fixed t ∈ E, the function ezt is analytic on p and hence considered as a function
on X, and, by applying 1.5 to f with fz χzezt , we get the p-adic integral expression of
the generating function for the generalized Euler numbers En,χ attached to χ:
χzezt dμ−1 z X
d−1
∞
tn
2 a
at
χae
E
−1
n,χ
n!
edt 1 a0
n0
1.8
t ∈ E.
So we have the following p-adic integral expression of the generating function for the
generalized Euler polynomials En,χ x attached to χ:
χzexzt dμ−1 z X
d−1
∞
tn
2ext a
at
χae
E
−1
x
n,χ
n!
edt 1 a0
n0
t ∈ E, x ∈
p
.
1.9
Also, from 1.4, we have
ezt dμ−1 z X
2
et 1
t ∈ E.
1.10
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3
Let Tk n, χ denote the kth alternating generalized power sum of the first n 1
nonnegative integers attached to χ, namely,
n
Tk n, χ −1a χaak −10 χ00k −11 χ11k · · · −1n χnnk .
1.11
a0
From 1.8, 1.10, and 1.11, one easily derives the following identities: for any odd positive
integer w,
d−1
χxext dμ−1 x ewdt 1 dt
−1a χaeat
wdyt
e
1
y
e
dμ
−1
a0
X
X
wd−1
−1a χaeat
1.12
1.13
a0
∞
tk
Tk wd − 1, χ
k!
k0
t ∈ E.
1.14
In what follows, we will always assume that the p-adic integrals of the various twisted
exponential functions on X are defined for t ∈ E cf. 1.7, and therefore it will not be
mentioned.
References 2–6 are some of the previous works on identities of symmetry in two
variables involving Bernoulli polynomials and power sums. On the other hand, for the first
time we were able to produce in 7 some identities of symmetry in three variables related
to Bernoulli polynomials and power sums and to extend in 8 to the case of generalized
Bernoulli polynomials and generalized power sums. Also, 4 is about identities of symmetry
in two variables for Euler polynomials and alternating power sums, and 9 is about those in
three variables for them.
In this paper, we will be able to produce 8 identities of symmetry in three variables
regarding generalized Euler polynomials and alternating generalized power sums. The case
of two variables was treated in 10.
The following is stated as Theorem 4.2 and an example of the full six symmetries in
w1 , w2 , w3 :
Ek,χ w1 y1 El,χ w2 y2 Tm w3 d − 1, χ w1lm w2km w3kl
n
k, l, m
n
Ek,χ w1 y1 El,χ w3 y2 Tm w2 d − 1, χ w1lm w3km w2kl
klmn k, l, m
n Ek,χ w2 y1 El,χ w1 y2 Tm w3 d − 1, χ w2lm w1km w3kl
k, l, m
klmn
klmn
klmn
n
k, l, m
Ek,χ w2 y1 El,χ w3 y2 Tm w1 d − 1, χ w2lm w3km w1kl
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n
Ek,χ w3 y1 El,χ w2 y2 Tm w1 d − 1, χ w3lm w2km w1kl
k, l, m
n
Ek,χ w3 y1 El,χ w1 y2 Tm w2 d − 1, χ w3lm w1km w2kl .
klmn k, l, m
klmn
1.15
The derivations of identities are based on the p-adic integral expression of the generating function for the generalized Euler polynomials in 1.9 and the quotient of integrals
in 1.12 that can be expressed as the exponential generating function for the alternating
generalized power sums. This abundance of symmetries would not be unearthed if such padic integral representations had not been available. We indebted this idea to paper 10.
2. Several Types of Quotients of p-Adic Fermionic Integrals
Here we will introduce several types of quotients of p-adic fermionic integrals on X or X 3
from which some interesting identities follow owing to the built-in symmetries in w1 , w2 ,
w3 . In the following, w1 , w2 , w3 are all positive integers, and all of the explicit expressions of
integrals in 2.2, 2.4, 2.6, and 2.8 are obtained from the identities in 1.8 and 1.10. To
ease notations, from now on, we will suppress μ−1 and denote, for example, dμ−1 x simply
by dx.
a Type Λi23 for i 0, 1, 2, 3:
I
Λi23
3−i
X3
χx1 χx2 χx3 ew2 w3 x1 w1 w3 x2 w1 w2 x3 w1 w2 w3 dw w w x t
i
e 1 2 3 4 dx4
X
j1
yj t
dx1 dx2 dx3
3−i
i
23−i ew1 w2 w3 j1 yj t edw1 w2 w3 t 1
dw w t
e 2 3 1 edw1 w3 t 1 edw1 w2 t 1
d−1
d−1
d−1
a
a
a
aw2 w3 t
aw1 w3 t
aw1 w2 t
.
×
−1 χae
−1 χae
−1 χae
a0
a0
2.1
2.2
a0
b Type Λi13 for i 0, 1, 2, 3:
I Λi13 X3
3−i
χx1 χx2 χx3 ew1 x1 w2 x2 w3 x3 w1 w2 w3 dw w w x t
i
1 2 3 4 dx
4
Xe
j1
yj t
dx1 dx2 dx3
3−i
i
23−i ew1 w2 w3 j1 yj t edw1 w2 w3 t 1
dw t
e 1 1 edw2 t 1 edw3 t 1
d−1
d−1
d−1
a
a
a
aw1 t
aw2 t
aw3 t
.
×
−1 χae
−1 χae
−1 χae
a0
a0
a0
2.3
2.4
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5
c-0 Type Λ012 :
I Λ012 X3
χx1 χx2 χx3 ew1 x1 w2 x2 w3 w3 w2 w3 yw1 w3 yw1 w2 yt dx1 dx2 dx3
8ew2 w3 w1 w3 w1 w2 yt
dw t
e 1 1 edw2 t 1 edw3 t 1
d−1
d−1
d−1
a
a
a
aw1 t
aw2 t
aw3 t
.
×
−1 χae
−1 χae
−1 χae
a0
a0
2.5
2.6
a0
c-1 Type Λ112 :
I Λ112 χx1 χx2 χx3 ew1 x1 w2 x2 w3 x3 t dx1 dx2 dx3
dw2 w3 z1 w1 w3 z2 w1 w2 z3 t dz z z
1 2 3
X3 e
dw w t
dw w t
dw w t
e 2 3 1 e 1 3 1 e 1 2 1
dw t
e 1 1 edw2 t 1 edw3 t 1
d−1
d−1
d−1
a
a
a
aw1 t
aw2 t
aw3 t
.
×
−1 χae
−1 χae
−1 χae
X3
a0
a0
2.7
2.8
a0
All of the above p-adic integrals of various types are invariant under all permutations
of w1 , w2 , w3 , as one can see either from p-adic integral representations in 2.1, 2.3, 2.5,
and 2.7 or from their explicit evaluations in 2.2, 2.4, 2.6, and 2.8.
3. Identities for Generalized Euler Polynomials
In the following, w1 , w2 , w3 are all odd positive integers except for a-0 and c-0, where they
are any positive integers. First, let’s consider Type Λi23 , for each i 0, 1, 2, 3. The following
results can be easily obtained from 1.9 and 1.12:
a-0
I Λ023
χx1 ew2 w3 x1 w1 y1 t dx1
X
X
χx2 ew1 w3 x2 w2 y2 t dx2
χx3 ew1 w2 x3 w3 y3 t dx3
X
∞ E
∞ E
∞ E
k,χ w1 y1
l,χ w2 y2
m,χ w3 y3
k
l
m
w2 w3 t
w1 w3 t
w1 w2 t
k!
l!
m!
m0
k0
l0
∞
n0
klmn
n
k, l, m
Ek,χ w1 y1 El,χ w2 y2 Em,χ w3 y3 w1lm w2km w3kl
tn
,
n!
3.1
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where the inner sum is over all nonnegative integers k, l, m with k l m n and
n!
n
.
k, l, m
k!l!m!
3.2
a-1 Here we write IΛ123 in two different ways:
1
I
Λ123
χx1 e
w2 w3 x1 w1 y1 t
dx1
χx2 e
w1 w3 x2 w2 y2 t
dx2 X
χx3 ew1 w2 x3 t dx3
edw1 w2 w3 x4 t dx4
X
X
X
∞
∞
w2 w3 tk
w1 w3 tl
w1 w2 tm
Tm w3 d−1, χ
Ek,χ w1 y1
El,χ w2 y2
k!
l!
m!
k0
l0
3.3
∞
n
n
t
Ek,χ w1 y1 El,χ w2 y2 Tm w3 d − 1, χ w1lm w2km w3kl
.
n!
n0 klmn k, l, m
3.4
2 Invoking 1.13, 3.3 can also be written as
w
3 d−1
I Λ123 χx1 ew2 w3 x1 w1 y1 t dx1
χx2 ew1 w3 x2 w2 y2 w2 /w3 at dx2
−1a χa
a0
X
X
∞
k ∞
l w1 y3 t
w2 y3 t
w2
a
Ek,χ w1 y1
El,χ w2 y2 a
−1 χa
k!
w3
l!
a0
k0
l0
w
d−1
∞
n
3
n
w2
tn
a
n
n−k k
.
a w1 w2
w3
Ek,χ w1 y1
−1 χaEn−k,χ w2 y2 w3
n!
n0
a0
k0 k
w
3 d−1
3.5
a-2 Here we write IΛ223 in three different ways:
1
χx2 ew1 w3 x2 t dx2 X χx3 ew1 w2 x3 t dx3
X
2
w2 w3 x1 w1 y1 t
I Λ23 χx1 e
dx1 dw w w x t
e 1 2 3 4 dx4
edw1 w2 w3 x4 t dx4
X
X
X
∞
∞
w2 w3 tk
w1 w3 tl
Ek,χ w1 y1
Tl w2 d − 1, χ
3.6
k!
l!
k0
l0
∞
w1 w2 tm
×
Tm w3 d − 1, χ
m!
m0
∞
n
lm km kl tn
.
Ek,χ w1 y1 Tl w2 d−1, χ Tm w3 d − 1, χ w1 w2 w3
n!
n0 klmn k, l, m
3.7
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2 Invoking 1.13, 3.6 can also be written as
w
2 d−1
χx3 ew1 w2 x3 t dx3
a
2
w2 w3 x1 w1 y1 w1 /w2 at
I Λ23 χx1 e
dx1 × X dw w w x t
−1 χa
e 1 2 3 4 dx4
X
a0
X
w
∞
∞
2 d−1
w1 w2 tl
w1
w2 w3 tk
a
Ek,χ w1 y1 a
Tl w3 d − 1, χ
−1 χa
w2
k!
l!
a0
k0
l0
3.8
w d−1
∞
n
2
n w1
tn
.
a Tn−k w3 d − 1, χ w1n−k w3k
w2n
−1a χaEk,χ w1 y1 w2
n!
n0
a0
k0 k
3.9
3 Invoking 1.13 once again, 3.8 can be written as
w
w
3 d−1
2 d−1
a
b
2
I Λ23 χx1 ew2 w3 x1 w1 y1 w1 /w2 aw1 /w3 bt dx1
−1 χa
−1 χb
a0
w
2 d−1
w
3 d−1
a
−1 χa
a0
∞
n0
X
b0
b0
n
w2 w3 ∞
w1
w1
w2 w3 tn
a
b
−1 χb En,χ w1 y1 w2
w3
n!
n0
w
2 d−1 w
3 d−1
ab
−1
a0
3.10
b
b0
n
w1
w1
t
.
χabEn,χ w1 y1 a
b
w2
w3
n!
3.11
a-3
w2 w3 x1 t
w1 w3 x2 t
dx1
dx2
χx3 ew1 w2 x3 t dx3
X χx1 e
X χx2 e
3
I Λ23 dw w w x t
× dw w w x t
× X dw w w x t
1 2 3 4 dx
1 2 3 4 dx
e 1 2 3 4 dx4
4
4
Xe
Xe
X∞
k
l
∞
w2 w3 t
w1 w3 t
Tk w1 d − 1, χ
Tl w2 d − 1, χ
3.12
k!
l!
k0
l0
∞
w1 w2 tm
Tm w3 d − 1, χ
×
m!
m0
∞
n
lm km kl tn
Tk w1 d−1, χ Tl w2 d−1, χ Tm w3 d−1, χ w1 w2 w3
.
n!
n0 klmn k, l, m
3.13
b For Type Λi13 i 0, 1, 2, 3, we may consider the analogous things to the ones in a0, a-1, a-2, and a-3. However, these do not lead us to new identities. Indeed, if
we substitute w2 w3 , w1 w3 , w1 w2 , respectively, for w1 , w2 , w3 in 2.1, this amounts
to replacing t by w1 w2 w3 t in 2.3. So, upon replacing w1 , w2 , w3 , respectively, by
w2 w3 , w1 w3 , w1 w2 and dividing by w1 w2 w3 n , in each of the expressions of 3.1,
3.4, 3.5, 3.7, 3.9–3.13, we will get the corresponding symmetric identities
for Type Λi13 i 0, 1, 2, 3.
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c-0
I
Λ012
w1 x1 w2 yt
χx1 e
dx1
χx2 e
dx2
χx3 ew3 x3 w1 yt dx3
X
X
∞
∞
∞
Ek,χ w2 y
El,χ w3 y
Em,χ w1 y
k
l
m
w1 t
w2 t
w3 t
3.14
k!
l!
m!
m0
k0
l0
∞
n
tn
.
Ek,χ w2 y El,χ w3 y Em,χ w1 y w1k w2l w3m
n!
n0 klmn k, l, m
w2 x2 w3 yt
X
c-1
I
Λ112
X
χx1 ew1 x1 t dx1
× X
χx2 ew2 x2 t dx2
× X
χx3 ew3 x3 t dx3
dw2 w3 z1 t dz
dw3 w1 z2 t dz
edw1 w2 z3 t dz3
1
2
Xe
Xe
X
k
l
∞
∞
∞
w1 t
w2 t
w3 tm
Tk w2 d−1, χ
Tl w3 d−1, χ
Tm w1 d−1, χ
k!
l!
m!
m0
k0
l0
∞
n
tn
Tk w2 d − 1, χ Tl w3 d − 1, χ Tm w1 d − 1, χ w1k w2l w3m
.
n!
n0 klmn k, l, m
3.15
4. Main Theorems
As we noted earlier in the last paragraph of Section 2, the various types of quotients of p-adic
fermionic integrals are invariant under any permutation of w1 , w2 , w3 . So the corresponding
expressions in Section 3 are also invariant under any permutation of w1 , w2 , w3 . Thus, our
results about identities of symmetry will be immediate consequences of this observation.
However, not all permutations of an expression in Section 3 yield distinct ones. In
fact, as these expressions are obtained by permuting w1 , w2 , w3 in a single one labeled by
them, they can be viewed as a group in a natural manner, and hence it is isomorphic to a
quotient of S3 . In particular, the number of possible distinct expressions is 1, 2, 3, or 6 a-0,
a-11, a-12, and a-22 give the full six identities of symmetry, a-21 and a-23
yield three identities of symmetry, and c-0 and c-1 give two identities of symmetry, while
the expression in a-3 yields no identities of symmetry.
Here we will just consider the cases of Theorems 4.4 and 4.8, leaving the others as easy
exercises for the reader. As for the case of Theorem 4.4, in addition to 4.11–4.13, we get
the following three ones:
n
Ek,χ w1 y1 Tl w3 d − 1, χ Tm w2 d − 1, χ w1lm w3km w2kl
k, l, m
n
Ek,χ w2 y1 Tl w1 d − 1, χ Tm w3 d − 1, χ w2lm w1km w3kl
klmn k, l, m
n
Ek,χ w3 y1 Tl w2 d − 1, χ Tm w1 d − 1, χ w3lm w2km w1kl .
klmn k, l, m
4.1
klmn
4.2
4.3
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But, by interchanging l and m, we see that 4.1, 4.2, and 4.3 are, respectively, equal to
4.11, 4.12, and 4.13. As to Theorem 4.8, in addition to 4.17 and 4.18, we have:
klmn
Tk w2 d − 1, χ Tl w3 d − 1, χ Tm w1 d − 1, χ w1k w2l w3m
n
k, l, m
n
k, l, m
n
4.4
Tk w3 d − 1, χ Tl w1 d − 1, χ Tm w2 d − 1, χ w2k w3l w1m
4.5
Tk w3 d − 1, χ Tl w2 d − 1, χ Tm w1 d − 1, χ w1k w3l w2m
4.6
Tk w2 d − 1, χ Tl w1 d − 1, χ Tm w3 d − 1, χ w3k w2l w1m .
4.7
klmn
k, l, m
n
klmn
klmn
k, l, m
However, 4.4 and 4.5 are equal to 4.17, as we can see by applying the permutations
k → l, l → m, m → k for 4.4 and k → m, l → k, m → l for 4.5. Similarly, we see that
4.6 and 4.7 are equal to 4.18, by applying permutations k → l, l → m, m → k for 4.6
and k → m, l → k, m → l for 4.7.
Theorem 4.1. Let w1 , w2 , w3 be any positive integers. Then one has
klmn
n
k, l, m
klmn
klmn
klmn
klmn
klmn
Ek,χ w1 y1 El,χ w2 y2 Em,χ w3 y3 w1lm w2km w3kl
n
k, l, m
n
k, l, m
n
k, l, m
n
k, l, m
n
k, l, m
Ek,χ w1 y1 El,χ w3 y2 Em,χ w2 y3 w1lm w3km w2kl
Ek,χ w2 y1 El,χ w1 y2 Em,χ w3 y3 w2lm w1km w3kl
4.8
Ek,χ w2 y1 El,χ w3 y2 Em,χ w1 y3 w2lm w3km w1kl
Ek,χ w3 y1 El,χ w1 y2 Em,χ w2 y3 w3lm w1km w2kl
Ek,χ w3 y1 El,χ w2 y2 Em,χ w1 y3 w3lm w2km w1kl .
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Theorem 4.2. Let w1 , w2 , w3 be any odd positive integers. Then one has
klmn
n
k, l, m
klmn
n
n
n
Ek,χ w1 y1 El,χ w3 y2 Tm w2 d − 1, χ w1lm w3km w2kl
Ek,χ w2 y1 El,χ w1 y2 Tm w3 d − 1, χ w2lm w1km w3kl
4.9
k, l, m
klmn
n
k, l, m
klmn
k, l, m
klmn
n
k, l, m
klmn
Ek,χ w1 y1 El,χ w2 y2 Tm w3 d − 1, χ w1lm w2km w3kl
k, l, m
Ek,χ
w2 y1 El,χ w3 y2 Tm w1 d − 1, χ w2lm w3km w1kl
Ek,χ w3 y1 El,χ w2 y2 Tm w1 d − 1, χ w3lm w2km w1kl
Ek,χ w3 y1 El,χ w1 y2 Tm w2 d − 1, χ w3lm w1km w2kl .
Theorem 4.3. Let w1 , w2 , w3 be any odd positive integers. Then one has
w1n
n
1 d−1
n
w
w2
Ek,χ w3 y1
a w3n−k w2k
−1a χaEn−k,χ w2 y2 w1
a0
k0 k
w1n
n
1 d−1
n
w
w3
Ek,χ w2 y1
a w2n−k w3k
−1a χaEn−k,χ w3 y2 w1
a0
k0 k
w2n
n
2 d−1
n
w
w1
a
Ek,χ w3 y1
a w3n−k w1k
−1 χaEn−k,χ w1 y2 w2
a0
k0 k
n
2 d−1
n
w
w3
n
Ek,χ w1 y1
w2
a w1n−k w3k
−1a χaEn−k,χ w3 y2 w2
a0
k0 k
w3n
n
3 d−1
n
w
w1
a w2n−k w1k
Ek,χ w2 y1
−1a χaEn−k,χ w1 y2 w3
a0
k0 k
w3n
n
3 d−1
n
w
w2
Ek,χ w1 y1
a w1n−k w2k .
−1a χaEn−k,χ w2 y2 w3
a0
k0 k
4.10
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11
Theorem 4.4. Let w1 , w2 , w3 be any odd positive integers. Then one has the following three
symmetries in w1 , w2 , w3 :
n
Ek,χ w1 y1 Tl w2 d − 1, χ Tm w3 d − 1, χ w1lm w2km w3kl
k, l, m
n
klmn
k, l, m
n
4.11
Ek,χ w2 y1 Tl w3 d − 1, χ Tm w1 d − 1, χ w2lm w3km w1kl
4.12
Ek,χ w3 y1 Tl w1 d − 1, χ Tm w2 d − 1, χ w3lm w1km w2kl .
4.13
klmn
klmn
k, l, m
Theorem 4.5. Let w1 , w2 , w3 be any odd positive integers. Then one has
w1n
w d−1
n
1
n w2
a Tn−k w3 d − 1, χ w2n−k w3k
−1a χaEk,χ w2 y1 w1
a0
k0 k
w1n
w d−1
n
1
n w3
a Tn−k w2 d − 1, χ w3n−k w2k
−1a χaEk,χ w3 y1 w1
a0
k0 k
w2n
w d−1
n
2
n w1
a Tn−k w3 d − 1, χ w1n−k w3k
−1a χaEk,χ w1 y1 w2
a0
k0 k
w d−1
2
n w3
n
w2
a Tn−k w1 d − 1, χ w3n−k w1k
−1a χaEk,χ w3 y1 w2
a0
k0 k
n
w3n
w d−1
n
3
n w1
a
a Tn−k w2 d − 1, χ w1n−k w2k
−1 χaEk,χ w1 y1 w3
a0
k0 k
w3n
w d−1
n
3
n w2
a
a Tn−k w1 d − 1, χ w2n−k w1k .
−1 χaEk,χ w2 y1 w
3
a0
k0 k
4.14
Theorem 4.6. Let w1 , w2 , w3 be any odd positive integers. Then one has the following three
symmetries in w1 , w2 , w3 :
n
w1 w2 w
1 d−1 w
2 d−1
ab
−1
a0
χabEn,χ
b0
w2 w3 n
w
3 d−1
2 d−1 w
a0
w3 w1 n
b0
w
3 d−1 w
1 d−1
a0
b0
w3
w3
w3 y1 a
b
w1
w2
w1
w1
a
b
−1ab χabEn,χ w1 y1 w2
w3
w2
w2
a
b .
−1ab χabEn,χ w2 y1 w3
w1
4.15
12
International Journal of Combinatorics
Theorem 4.7. Let w1 , w2 , w3 be any positive integers. Then one has the following two symmetries
in w1 , w2 , w3 :
klmn
n
Ek,χ w1 y El,χ w2 y Em,χ w3 y w3k w1l w2m
k, l, m
klmn
n
k, l, m
4.16
Ek,χ w1 y El,χ w3 y Em,χ w2 y w2k w1l w3m .
Theorem 4.8. Let w1 , w2 , w3 be any odd positive integers. Then one has the following two
symmetries in w1 , w2 , w3 :
klmn
klmn
n
k, l, m
n
Tk w1 d − 1, χ Tl w2 d − 1, χ Tm w3 d − 1, χ w3k w1l w2m ,
4.17
k, l, m
Tk w1 d − 1, χ Tl w3 d − 1, χ Tm w2 d − 1, χ w2k w1l w3m .
4.18
References
1 N. Koblitz, “A new proof of certain formulas for p-adic L-functions,” Duke Mathematical Journal, vol.
46, no. 2, pp. 455–468, 1979.
2 E. Y. Deeba and D. M. Rodriguez, “Stirling’s series and Bernoulli numbers,” The American Mathematical
Monthly, vol. 98, no. 5, pp. 423–426, 1991.
3 F. T. Howard, “Applications of a recurrence for the Bernoulli numbers,” Journal of Number Theory, vol.
52, no. 1, pp. 157–172, 1995.
4 T. Kim, “Symmetry p-adic invariant integral on p for Bernoulli and Euler polynomials,” Journal of
Difference Equations and Applications, vol. 14, no. 12, pp. 1267–1277, 2008.
5 H. J. H. Tuenter, “A symmetry of power sum polynomials and Bernoulli numbers,” The American
Mathematical Monthly, vol. 108, no. 3, pp. 258–261, 2001.
6 S.-l. Yang, “An identity of symmetry for the Bernoulli polynomials,” Discrete Mathematics, vol. 308,
no. 4, pp. 550–554, 2008.
7 D. S. Kim and K. H. Park, “Identities of symmetry for Bernoulli polynomials arising from quotients
of Volkenborn integrals invariant under S3 ,” submitted.
8 D. S. Kim, “Identities of symmetry for generalized Bernoulli polynomials,” submitted.
9 D. S. Kim and K. H. Park, “Identities of symmetry for Euler polynomials arising from quotients of
fermionic integrals invariant under S3 ,” Journal of Inequalities and Applications, Article ID 851521, 16
pages, 2010.
10 T. Kim, “A note on the generalized Euler numbers and polynomials,” http://arxiv.org/abs/
0907.4889.
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