Identities of Symmetry for Generalized Euler Polynomials

Volume 2011, Article ID 432738,12pages 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,[email protected] Received 10 January 2011; Accepted 15 February 2011

Academic Editor: Ch´ınh T. Hoang

Copyrightq2011 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 thep-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

Letpbe a fixed odd prime. Throughout this paper, _p,_p, and_p will, respectively, denote the ring ofp-adic integers, the field of p-adic rational numbers, and the completion of the algebraic closure ofp. Letdbe a fixed odd positive integer. Then we let

XX_dlim_←−

N dp^N , 1.1

and letπ :X → pbe the map given by the inverse limit of the natural maps

dp^N −→

p^N . 1.2

Ifgis a function on p, then we will use the same notation to denote the functiong◦π. Let χ: /d ^∗ → ^∗be aprimitiveDirichlet character of conductord. Then it will be pulled

back toX via the natural map X → /d . Here we fix, once and for all, an imbedding

→ p, so thatχis regarded as a map ofXto_pcf.1.

For a continuous functionf:X → p, thep-adic fermionic integral offis defined by

X

fzdμ₋₁z lim

N→ ∞ dp^N−1

j0

f j

−1^j. 1.3

Then it is easy to see that

X

fz1dμ−1z

X

fzdμ−1z 2f0. 1.4

More generally, we deduce from1.4that, for any odd positive integern,

X

fzndμ−1z

X

fzdμ−1z 2

n−1

a0

−1^afa 1.5

and that, for any even positive integern,

X

fzndμ−1z−

X

fzdμ−1z 2 n−1 a0

−1^a−1fa. 1.6

Let| · |_pbe the normalized absolute value of_p, such that|p|p1/p, and let E

t∈p | |t|_p< p^−1/p−1

. 1.7

Then, for each fixedt∈E, the functione^ztis analytic on _pand hence considered as a function onX, and, by applying1.5tofwithfz χze^zt, we get thep-adic integral expression of the generating function for the generalized Euler numbersEn,χattached toχ:

X

χze^ztdμ₋₁z 2 e^dt1

d−1

a0

−1^aχae^at∞

n0

E_n,χtⁿ

n! t∈E. 1.8 So we have the following p-adic integral expression of the generating function for the generalized Euler polynomialsE_n,χxattached toχ:

X

χze^xztdμ₋₁z 2e^xt e^dt1

d−1

a0

−1^aχae^{at ∞}

n0

En,χxtⁿ n!

t∈E, x∈ p

. 1.9

Also, from1.4, we have

X

e^ztdμ₋₁z 2

e^t1 t∈E. 1.10

Let Tkn, χ denote the kth alternating generalized power sum of the first n 1 nonnegative integers attached toχ, namely,

T_k n, χ

ⁿ

a0

−1^aχaa^k −1⁰χ00^k −1¹χ11^k· · · −1ⁿχnn^k. 1.11

From1.8,1.10, and1.11, one easily derives the following identities: for any odd positive integerw,

Xχxe^xtdμ₋₁x

Xe^wdytdμ₋₁

y e^wdt1 e^dt1

d−1

a0

−1^aχae^at 1.12

^wd−1

a0

−1^aχae^at 1.13

^∞

k0

Tk

wd−1, χt^k

k! t∈E. 1.14

In what follows, we will always assume that thep-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.

References2–6 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 in7some 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,4is about identities of symmetry in two variables for Euler polynomials and alternating power sums, and9is 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 in10.

The following is stated asTheorem 4.2and an example of the full six symmetries in w₁,w₂,w₃:

klmn

n k, l, m

Ek,χ

w1y1

El,χ

w2y2

Tm

w3d−1, χ

w₁^lmw₂^kmw^kl₃

klmn

n k, l, m

Ek,χ

w1y1

El,χ

w3y2

Tm

w2d−1, χ

w^lm₁ w^km₃ w₂^kl

klmn

n k, l, m

E_k,χ

w₂y₁ E_l,χ

w₁y₂ T_m

w₃d−1, χ

w^lm₂ w^km₁ w^kl₃

klmn

n k, l, m

Ek,χ

w2y1

El,χ

w3y2

Tm

w1d−1, χ

w^lm₂ w^km₃ w₁^kl

klmn

n k, l, m

E_k,χ

w₃y₁ E_l,χ

w₂y₂ T_m

w₁d−1, χ

w^lm₃ w^km₂ w₁^kl

klmn

n k, l, m

E_k,χ

w₃y₁ E_l,χ

w₁y₂ T_m

w₂d−1, χ

w^lm₃ w^km₁ w₂^kl.

1.15

The derivations of identities are based on thep-adic integral expression of the gener- ating function for the generalized Euler polynomials in1.9and 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 suchp- adic integral representations had not been available. We indebted this idea to paper10.

2. Several Types of Quotients of p-Adic Fermionic Integrals

Here we will introduce several types of quotients ofp-adic fermionic integrals on X orX³ from which some interesting identities follow owing to the built-in symmetries inw₁,w₂, w3. In the following,w1,w2,w3are all positive integers, and all of the explicit expressions of integrals in2.2,2.4,2.6, and2.8are obtained from the identities in1.8and1.10. To ease notations, from now on, we will suppressμ₋₁ and denote, for example,dμ₋₁xsimply bydx.

aTypeΛⁱ₂₃fori0,1,2,3:

I Λⁱ₂₃

X³χx1χx2χx3e^{w2w3x1w1w3x2w1w2x3w1w2w33−ij1yjt}dx₁dx₂dx₃

Xe^dw1w2w3x4tdx4

_i 2.1

2³⁻ⁱe^{w1w2w33−ij1yjt}

e^dw1w2w3t1i

e^dw2w3t1

e^dw1w3t1

e^dw1w2t1

× ^d−1

a0

−1^aχae^{aw2w3t d−1}

a0

−1^aχae^{aw1w3t d−1}

a0

−1^aχae^aw1w2t

. 2.2

bTypeΛⁱ₁₃fori0,1,2,3:

I Λⁱ₁₃

X³χx1χx2χx3e^{w1x1w2x2w3x3w1w2w33−ij1yjt}dx1dx2dx3

Xe^dw1w2w3x4tdx4

_i 2.3

2³⁻ⁱe^{w1w2w33−ij1yjt}

e^dw1w2w3t1i

e^dw1t1

e^dw2t1

e^dw3t1

× ^d−1

a0

−1^aχae^{aw1t d−1}

a0

−1^aχae^{aw2t d−1}

a0

−1^aχae^aw3t

.

2.4

c-0TypeΛ⁰₁₂:

I Λ⁰₁₂

X³

χx1χx2χx3e^{w1x1w2x2w3w3w2w3yw1w3yw1w2yt}dx1dx2dx3 2.5

8e^{w2w3w1w3w1w2yt} e^dw1t1

e^dw2t1

e^dw3t1

× ^d−1

a0

−1^aχae^{aw1t d−1}

a0

−1^aχae^{aw2t d−1}

a0

−1^aχae^aw3t

.

2.6

c-1TypeΛ¹₁₂:

I Λ¹₁₂

X³χx1χx2χx3e^{w1x1w2x2w3x3t}dx₁dx₂dx₃

X³e^{dw2w3z1w1w3z2w1w2z3t}dz1z2z3

2.7

e^dw2w3t1

e^dw1w3t1

e^dw1w2t1 e^dw1t1

e^dw2t1

e^dw3t1

× ^d−1

a0

−1^aχae^{aw1t d−1}

a0

−1^aχae^{aw2t d−1}

a0

−1^aχae^aw3t

.

2.8

All of the abovep-adic integrals of various types are invariant under all permutations ofw1,w2,w3, as one can see either fromp-adic integral representations in2.1,2.3,2.5, and2.7or from their explicit evaluations in2.2,2.4,2.6, and2.8.

3. Identities for Generalized Euler Polynomials

In the following,w1,w2,w3are all odd positive integers except fora-0andc-0, where they are any positive integers. First, let’s consider TypeΛⁱ₂₃, for eachi 0,1,2,3. The following results can be easily obtained from1.9and1.12:

a-0

I Λ⁰₂₃

X

χx1e^w2w3x1w1y1tdx1

X

χx2e^w1w3x2w2y2tdx2

X

χx3e^w1w2x3w3y3tdx3

^∞

k0

E_k,χ w₁y₁

k! w2w₃t^{k ∞}

l0

E_l,χ w₂y₂

l! w1w₃t^{l ∞}

m0

E_m,χ w₃y₃

m! w1w₂t^m

^∞

n0

klmn

n k, l, m

Ek,χ

w1y1

El,χ

w2y2

Em,χ

w3y3

w^lm₁ w^km₂ w^kl₃ tⁿ

n!,

3.1

where the inner sum is over all nonnegative integersk,l,mwithklmnand n

k, l, m

n!

k!l!m!. 3.2

a-1Here we writeIΛ¹₂₃in two different ways:

1

I Λ¹₂₃

X

χx1e^w2w3x1w1y1tdx₁

X

χx2e^w1w3x2w2y2tdx₂

Xχx3e^w1w2x3tdx₃

Xe^dw1w2w3x4tdx4

^∞

k0

E_k,χ

w₁y₁w2w3t^k k!

∞ l0

E_l,χ

w₂y₂w1w3t^l l!

T_m

w₃d−1, χw1w2t^m m!

3.3 ^∞

n0

klmn

n k, l, m

E_k,χ

w₁y₁ E_l,χ

w₂y₂ T_m

w₃d−1, χ

w^lm₁ w^km₂ w^kl₃ tⁿ

n!. 3.4 2Invoking1.13,3.3can also be written as

I Λ¹₂₃

^w3d−1

a0

−1^aχa

X

χx1e^w2w3x1w1y1tdx1

X

χx2e^{w1w3x2w2y2w2/w3at}dx2

^w3d−1

a0

−1^aχa ^∞

k0

E_k,χ w₁y₁

w2y3t_k k!

∞ l0

E_l,χ

w₂y₂w₂ w3

a

w1y3t_l l!

^∞

n0

wⁿ₃ n k0

n k

Ek,χ

w1y1

^w3d−1

a0

−1^aχaE_n−k,χ

w2y2 w₂ w3a

w^n−k₁ w^k₂ tⁿ

n!. 3.5

a-2Here we writeIΛ²₂₃in three different ways:

1

I Λ²₂₃

X

χx1e^w2w3x1w1y1tdx1

Xχx2e^w1w3x2tdx2

Xe^dw1w2w3x4tdx₄

Xχx3e^w1w2x3tdx3

Xe^dw1w2w3x4tdx₄ ^∞

k0

Ek,χ

w1y1

w2w₃t^k k!

∞ l0

Tl

w2d−1, χw1w₃t^l l!

× ^∞

m0

Tm

w3d−1, χw1w2t^m m!

3.6

^∞

n0

klmn

n k, l, m

Ek,χ

w1y1

Tl

w2d−1, χ Tm

w3d−1, χ

w₁^lmw₂^kmw₃^kl tⁿ

n!. 3.7

2Invoking1.13,3.6can also be written as

I Λ²₂₃

^w2d−1

a0

−1^aχa

X

χx1e^{w2w3x1w1y1w1/w2at}dx₁×

Xχx3e^w1w2x3tdx₃

Xe^dw1w2w3x4tdx4

^w2d−1

a0

−1^aχa ^∞

k0

E_k,χ

w₁y₁w₁ w₂a

w2w₃t^k k!

∞ l0

T_l

w₃d−1, χw1w₂t^l l!

3.8 ^∞

n0

w₂ⁿ n k0

n k

_w

2d−1

a0

−1^aχaEk,χ

w₁y₁w1

w₂a

T_n−k

w₃d−1, χ

w^n−k₁ w₃^k tⁿ

n!. 3.9 3Invoking1.13once again,3.8can be written as

I Λ²₂₃

^w2d−1

a0

−1^aχa^w3d−1

b0

−1^bχb

X

χx1e^{w2w3x1w1y1w1/w2aw1/w3bt}dx1

^w2d−1

a0

−1^aχa^w3d−1

b0

−1^bχb^∞

n0

E_n,χ

w₁y₁w₁ w2

aw₁ w3

b

w2w₃tⁿ n!

3.10

^∞

n0

w2w3^nw2d−1

a0 w3d−1

b0

−1^abχabEn,χ

w1y1w1

w₂aw1

w₃b tⁿ

n!. 3.11 a-3

I Λ³₂₃

Xχx1e^w2w3x1tdx1

Xe^dw1w2w3x4tdx₄ ×

Xχx2e^w1w3x2tdx2

Xe^dw1w2w3x4tdx₄ ×

Xχx3e^w1w2x3tdx3

Xe^dw1w2w3x4tdx₄ ^∞

k0

Tk

w1d−1, χw2w3t^k k!

∞ l0

Tl

w2d−1, χw1w3t^l l!

× ^∞

m0

Tm

w3d−1, χw1w2t^m m!

3.12

^∞

n0

klmn

n k, l, m

T_k

w₁d−1, χ T_l

w₂d−1, χ T_m

w₃d−1, χ

w^lm₁ w^km₂ w^kl₃ tⁿ

n!. 3.13

bFor TypeΛⁱ₁₃i0,1,2,3, we may consider the analogous things to the ones ina- 0,a-1,a-2, anda-3. However, these do not lead us to new identities. Indeed, if we substitutew2w3,w1w3,w1w2, respectively, forw1,w2,w3in2.1, this amounts to replacingtbyw1w2w3tin2.3. So, upon replacingw1,w2,w3, respectively, by w₂w₃,w₁w₃,w₁w₂and dividing byw1w₂w₃ⁿ, in each of the expressions of3.1, 3.4,3.5,3.7,3.9–3.13, we will get the corresponding symmetric identities for TypeΛⁱ₁₃i0,1,2,3.

c-0

I Λ⁰₁₂

X

χx1e^w1x1w2ytdx1

X

χx2e^w2x2w3ytdx2

X

χx3e^w3x3w1ytdx3

^∞

k0

Ek,χ

w2y

k! w1t^{k ∞}

l0

El,χ

w3y

l! w2t^{l ∞}

m0

Em,χ

w1y m! w3t^m

^∞

n0

klmn

n k, l, m

Ek,χ

w2y El,χ

w3y Em,χ

w1y

w₁^kw^l₂w^m₃ tⁿ

n!.

3.14

c-1

I Λ¹₁₂

Xχx1e^w1x1tdx1

Xe^dw1w2z3tdz3

×

Xχx2e^w2x2tdx2

Xe^dw2w3z1tdz1

×

Xχx3e^w3x3tdx3

Xe^dw3w1z2tdz2

^∞

k0

Tk

w2d−1, χw1t^k k!

∞ l0

Tl

w3d−1, χw2t^l l!

∞ m0

Tm

w1d−1, χw3t^m m!

^∞

n0

klmn

n k, l, m

Tk

w2d−1, χ Tl

w3d−1, χ Tm

w1d−1, χ

w^k₁w₂^lw^m₃ tⁿ

n!. 3.15

4. Main Theorems

As we noted earlier in the last paragraph ofSection 2, the various types of quotients ofp-adic fermionic integrals are invariant under any permutation ofw₁,w₂,w₃. So the corresponding expressions inSection 3are also invariant under any permutation ofw1,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 3yield distinct ones. In fact, as these expressions are obtained by permutingw1,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 ofS₃. In particular, the number of possible distinct expressions is 1, 2, 3, or 6a-0, a-11,a-12, anda-22give the full six identities of symmetry,a-21anda-23 yield three identities of symmetry, andc-0andc-1give two identities of symmetry, while the expression ina-3yields no identities of symmetry.

Here we will just consider the cases of Theorems4.4and4.8, leaving the others as easy exercises for the reader. As for the case ofTheorem 4.4, in addition to4.11–4.13, we get the following three ones:

klmn

n k, l, m

E_k,χ

w₁y₁ T_l

w₃d−1, χ T_m

w₂d−1, χ

w^lm₁ w^km₃ w^kl₂ 4.1

klmn

n k, l, m

E_k,χ

w₂y₁ T_l

w₁d−1, χ T_m

w₃d−1, χ

w₂^lmw₁^kmw^kl₃ 4.2

klmn

n k, l, m

Ek,χ

w3y1

Tl

w2d−1, χ Tm

w1d−1, χ

w₃^lmw₂^kmw^kl₁ . 4.3

But, by interchanging land m, we see that4.1,4.2, and 4.3are, respectively, equal to 4.11,4.12, and4.13. As toTheorem 4.8, in addition to4.17and4.18, we have:

klmn

n k, l, m

Tk

w2d−1, χ Tl

w3d−1, χ Tm

w1d−1, χ

w^k₁w₂^lw^m₃ 4.4

klmn

n k, l, m

Tk

w3d−1, χ Tl

w1d−1, χ Tm

w2d−1, χ

w^k₂w^l₃w^m₁ 4.5

klmn

n k, l, m

T_k

w₃d−1, χ T_l

w₂d−1, χ T_m

w₁d−1, χ

w^k₁w^l₃w^m₂ 4.6

klmn

n k, l, m

T_k

w₂d−1, χ T_l

w₁d−1, χ T_m

w₃d−1, χ

w^k₃w^l₂w^m₁. 4.7

However, 4.4and 4.5are equal to 4.17, as we can see by applying the permutations k → l,l → m,m → kfor4.4andk → m,l → k,m → lfor4.5. Similarly, we see that 4.6and4.7are equal to4.18, by applying permutationsk → l,l → m,m → kfor4.6 andk → m,l → k,m → lfor4.7.

Theorem 4.1. Letw1,w2,w3be any positive integers. Then one has

klmn

n k, l, m

Ek,χ

w1y1

El,χ

w2y2

Em,χ

w3y3

w^lm₁ w^km₂ w^kl₃

klmn

n k, l, m

E_k,χ

w₁y₁ E_l,χ

w₃y₂ E_m,χ

w₂y₃

w₁^lmw^km₃ w^kl₂

klmn

n k, l, m

E_k,χ

w₂y₁ E_l,χ

w₁y₂ E_m,χ

w₃y₃

w₂^lmw^km₁ w^kl₃

klmn

n k, l, m

Ek,χ

w2y1

El,χ

w3y2

Em,χ

w1y3

w₂^lmw^km₃ w^kl₁

klmn

n k, l, m

Ek,χ

w3y1

El,χ

w1y2

Em,χ

w2y3

w₃^lmw^km₁ w^kl₂

klmn

n k, l,m

E_k,χ

w₃y₁ E_l,χ

w₂y₂ E_m,χ

w₁y₃

w^lm₃ w^km₂ w^kl₁ .

4.8

Theorem 4.2. Letw1,w2,w3be any odd positive integers. Then one has

klmn

n k, l, m

E_k,χ

w₁y₁ E_l,χ

w₂y₂ T_m

w₃d−1, χ

w^lm₁ w^km₂ w^kl₃

klmn

n k, l, m

Ek,χ

w1y1

El,χ

w3y2

Tm

w2d−1, χ

w₁^lmw₃^kmw^kl₂

klmn

n k, l, m

Ek,χ

w2y1

El,χ

w1y2

Tm

w3d−1, χ

w₂^lmw₁^kmw^kl₃

klmn

n k, l, m

Ek,χ

w2y1

El,χ

w3y2

Tm

w1d−1, χ

w₂^lmw₃^kmw^kl₁

klmn

n k, l, m

E_k,χ

w₃y₁ E_l,χ

w₂y₂ T_m

w₁d−1, χ

w₃^lmw₂^kmw^kl₁

klmn

n k, l, m

Ek,χ

w3y1

El,χ

w1y2

Tm

w2d−1, χ

w₃^lmw₁^kmw^kl₂ .

4.9

Theorem 4.3. Letw1,w2,w3be any odd positive integers. Then one has

w₁ⁿ n k0

n k

Ek,χ

w3y1

^w1d−1

a0

−1^aχaEn−k,χ

w2y2w2

w1a

w^n−k₃ w^k₂

w₁ⁿ n k0

n k

Ek,χ

w2y1

^w1d−1

a0

−1^aχaEn−k,χ

w3y2 w3

w₁a

w^n−k₂ w^k₃

w₂ⁿ n k0

n k

E_k,χ

w₃y₁^w2d−1

a0

−1^aχaEn−k,χ

w₁y₂ w₁ w2

a

w^n−k₃ w^k₁

w₂ⁿ n k0

n k

Ek,χ

w1y1

^w2d−1

a0

−1^aχaE_n−k,χ

w3y2 w₃ w2a

w^n−k₁ w^k₃

w₃ⁿ n k0

n k

E_k,χ

w₂y₁^w3d−1

a0

−1^aχaEn−k,χ

w₁y₂ w₁ w₃a

w^n−k₂ w^k₁

w₃ⁿ n k0

n k

E_k,χ

w₁y₁^w3d−1

a0

−1^aχaEn−k,χ

w₂y₂ w2

w₃a

w^n−k₁ w^k₂.

4.10