Identities of Symmetry for Euler Polynomials Arising from Quotients of Fermionic Integrals Invariant under S

Volume 2010, Article ID 851521,16pages doi:10.1155/2010/851521

Research Article

Identities of Symmetry for Euler Polynomials Arising from Quotients of Fermionic Integrals Invariant under S

Dae San Kim and Kyoung Ho Park

Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea

Correspondence should be addressed to Dae San Kim,[email protected] Received 16 February 2010; Accepted 13 April 2010

Academic Editor: Yeol J. E. Cho

Copyrightq2010 D. S. Kim and K. H. Park. 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 Euler polynomials and alternating power sums. These and most of their corollaries are new, since there have been results only about identities of symmetry in two variables. These abundances of symmetries shed new light even on the existing identities so as to yield some further interesting ones. The derivations of identities are based on thep-adic integral expression of the generating function for the Euler polynomials and the quotient of integrals that can be expressed as the exponential generating function for the alternating power sums.

1. Introduction and Preliminaries

Let p be a fixed odd prime. Throughout this paper, Zp,Qp,Cp will, respectively, denote the ring ofp-adic integers, the field ofp-adic rational numbers, and the completion of the algebraic closure ofQp. For a continuous functionf:Zp → Cp, thep-adic fermionic integral offis defined by

Zp

fzdμ₋₁z lim

N→ ∞ p^N−1

j0

f j

−1^j. 1.1

Then it is easy to see that

Zp

fz1dμ−1z

Zp

fzdμ−1z 2f0. 1.2

Let| |pbe the normalized absolute value ofCp, such that|p|p1/p, and let

E

t∈Cp| |t|_p< p^−1/p−1

. 1.3

Then, for each fixedt∈E, the functionfz e^ztis analytic onZp, and by applying1.2to thisf, we get thep-adic integral expression of the generating function for Euler numbersEn:

Zp

e^ztdμ₋₁z 2

e^t1 ^∞

n0

Entⁿ

n! t∈E. 1.4

So we have the followingp-adic integral expression of the generating function for the Euler polynomialsEnx:

Zp

e^xztdμ₋₁z 2

e^t1e^xt∞

n0

Enxtⁿ n!

t∈E, x∈Zp

. 1.5

Let Tkn, denote the alternating kth power sum of the firstn1 nonnegative integers, namely,

Tkn ⁿ

i0

−1ⁱi^k −1⁰0^k −1¹1^k −1²2^k· · · −1ⁿn^k. 1.6

In particular,

T0n

⎧⎨

⎩

1, ifn≡0mod 2,

0, ifn≡1mod 2, Tk0

⎧⎨

⎩

1, fork0,

0, fork >0. 1.7

From1.4and1.6, one easily derives the following identities: for any odd positive integer w,

Zpe^xtdμ−1x

Zpe^wytdμ₋₁

y ^w−1

i0

−1ⁱe^it∞

k0

Tkw−1t^k

k! t∈E. 1.8

In what follows, we will always assume that the p-adic fermionic integrals of the various exponential functions onZp are defined fort ∈ E cf.,1.3, and therefore it will not be mentioned.

Many authors have done much work on identities of symmetry involving Bernoulli polynomials or Euler polynomials orq-Bernoulli polynomials orq-Euler polynomials. We let the reader refer to the papers in1–20. In connection with Bernoulli polynomials and power sums, these results were generalized in21to obtain identities of symmetry involving three variables in contrast to the previous works involving just two variables.

In this paper, we will produce 8 basic identities of symmetry in three variablesw1,w2, w3 related to Euler polynomials and alternating power sumscf.,4.8,4.9,4.12,4.16,

4.20,4.23,4.25, and4.26. These and most of their corollaries seem to be new, since there have been results only about identities of symmetry in two variables in the literature.

These abundances of symmetries shed new light even on the existing identities. For instance, it has been known that1.9and1.10are equal and1.11and1.12are socf.,3, Theorems 5,7. In fact,1.9–1.12are all equal, as they can be derived from one and the same p- adic integral. Perhaps, this was neglected to mention in3. Also, we have a bunch of new identities in1.13–1.16. All of these were obtained as corollariescf.,Corollary 4.9, 4.12, 4.15to some of the basic identities by specializing the variablew3as 1. Those would not be unearthed if more symmetries had not been available.

Letw1,w2be any odd positive integers. Then we have n

k0

n k

Ek

w1y1

Tn−kw2−1w^n−k₁ w^k₂ 1.9

ⁿ

k0

n k

Ek

w2y1

T_n−kw1−1w^n−k₂ w^k₁ 1.10

wⁿ₁

w1−1 i0

−1ⁱEn

w2y1w2

w1i

1.11 wⁿ₂

w2−1 i0

−1ⁱEn

w1y1w1

w2i

1.12

klmn

n k, l, m

Ek

y1

Tlw1−1Tmw2−1w^km₁ w^kl₂ 1.13

wⁿ₁ n k0

n

k _w

1−1

i0

−1ⁱEk

y1 i

w1

Tn−kw2−1w^k₂ 1.14

wⁿ₂ n k0

n

k _w

2−1

i0

−1ⁱEk

y1 i

w2

Tn−kw1−1w^k₁ 1.15

w1w2^nw1−1

i0 w2−1

j0

−1^ijEn

y1 i

w1 j w2

. 1.16

The derivations of identities will be based on the p-adic integral expression of the generating function for the Euler polynomials in1.5and the quotient of integrals in1.8 that can be expressed as the exponential generating function for the alternating power sums.

We indebted this idea to the paper in3.

2. Several Types of Quotients of Fermionic Integrals

Here we will introduce several types of quotients ofp-adic fermionic integrals onZporZ³_p 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 identity in1.4.

aTypeΛⁱ₂₃(fori0,1,2,3). One has

I Λⁱ₂₃

Z³pe^{w2w3x1w1w3x2w1w2x3w1w2w33−ij1yjt}dμ−1x1dμ−1x2dμ−1x3

Zpe^w1w2w3x4tdμ−1x4i 2.1

2³⁻ⁱe^{w1w2w33−ij1yjt}

e^w1w2w3t1_i

e^w2w3t1e^w1w3t1e^w1w2t1 2.2

bTypeΛⁱ₁₃(fori0,1,2,3). One has

I Λⁱ₁₃

Z³pe^{w1x1w2x2w3x3w1w2w33−ij1yjt}dμ−1x1dμ−1x2dμ−1x3

Zpe^w1w2w3x4tdμ−1x4_i 2.3

2³⁻ⁱe^{w1w2w33−ij1yjt}

e^w1w2w3t1i

e^w1t1e^w2t1e^w3t1 2.4

c-0TypeΛ⁰₁₂. One has I

Λ⁰₁₂

Z³p

e^{w1x1w2x2w3x3w2w3yw1w3yw1w2yt}dμ−1x1dμ−1x2dμ−1x3 2.5

8e^{w2w3w1w3w1w2yt}

e^w1t1e^w2t1e^w3t1 2.6

c-1TypeΛ¹₁₂. One has I

Λ¹₁₂

Z³pe^{w1x1w2x2w3x3t}dμ₋₁x1dμ₋₁x2dμ₋₁x3

Z³pe^{w2w3z1w1w3z2w1w2z3t}dμ−1z1dμ−1z2dμ−1z3 2.7

e^w2w3t1

e^w1w3t1

e^w1w2t1

e^w1t1e^w2t1e^w3t1 . 2.8

All of the abovep-adic integrals of various types are invariant under all permutations ofw1, w2, w3as 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 Euler Polynomials

In the followingw1, w2, w3are all odd positive integers except fora-0andc-0, where they are any positive integers.

a-0First, let us consider Type Λⁱ₂₃, for eachi 0,1,2,3.The following results can be easily obtained from1.5and1.8:

I Λ⁰₂₃

Zp

e^w2w3x1w1y1tdμ−1x1

Zp

e^w1w3x2w2y2tdμ−1x2

Zp

e^w1w2x3w3y3tdμ−1x3

_∞

k0

Ek

w1y1

k! w2w3t^k _∞

l0

El

w2y2

l! w1w3t^l _∞

m0

Em

w3y3

m! w1w2t^m

^∞

n0

klmn

n k, l, m

Ek

w1y1

El

w2y2

Em

w3y3

w₁^lmw₂^kmw₃^kl tⁿ

n!,

3.1

where the inner sum is over all nonnegative integersk, l, m, withklmn, and n

k, l, m

n!

k!l!m!. 3.2

a-1Here we writeIΛ¹₂₃in two different ways:

1One has

I Λ¹₂₃

Zp

e^w2w3x1w1y1tdμ₋₁x1

×

Zp

e^w1w3x2w2y2tdμ−1x2×

Zpe^w1w2x3tdμ₋₁x3

Zpe^w1w2w3x4tdμ−1x4 3.3

_∞

k0

Ek

w1y1

w2w3t^k k!

_∞

l0

El

w2y2

w1w3t^l l!

_∞

m0

Tmw3−1w1w2t^m m!

^∞

n0

klmn

n

k, l, m

Ek

w1y1

El

w2y2

Tmw3−1w₁^lmw₂^kmw₃^kl tⁿ

n!.

3.4 2Invoking1.8,3.3can also be written as

I Λ¹₂₃

^w3−1

i0

−1ⁱ

Zp

e^w2w3x1w1y1t dμ₋₁x1

Zp

e^{w1w3x2w2y2w2/w3it}dμ₋₁x2

^w3−1

i0

−1ⁱ _∞

k0

Ek

w1y1

w2w3t^k k!

_∞

l0

El

w2y2w2

w3i

w1w3t^l l!

^∞

n0

wⁿ₃

n k0

n

k

Ek

w1y1

^w3−1

i0

−1ⁱEn−k

w2y2w2

w3i

w₁^n−kw₂^k tⁿ

n!.

3.5

a-2Here we writeIΛ²₂₃in three different ways:

1One has

I Λ²₂₃

Zp

e^w2w3x1w1y1tdμ−1x1

×

Zpe^w1w3x2tdμ₋₁x2

Zpe^w1w2w3x4tdμ₋₁x4×

Zpe^w1w2x3tdμ₋₁x3

Zpe^w1w2w3x4tdμ₋₁x4 3.6

_∞

k0

Ek

w1y1

w2w3t^k k!

_∞

l0

Tlw2−1w1w3t^l l!

_∞

m0

Tmw3−1w1w2t^m m!

^∞

n0

⎛

⎝

klmn

⎛

⎝ n k, l, m

⎞

⎠Ek

w1y1

Tlw2−1Tmw3−1w₁^lmw₂^kmw₃^kl

⎞

⎠tⁿ n!.

3.7 2Invoking1.8,3.6can also be written as

I Λ²₂₃

^w2−1

i0

−1ⁱ

Zp

e^{w2w3x1w1y1w1/w2it}dμ−1x1×

Zpe^w1w2x3tdμ−1x3

Zpe^w1w2w3x4tdμ−1x4 3.8

^w2−1

i0

−1ⁱ _∞

k0

Ek

w1y1w1

w2i

w2w3t^k k!

_∞

l0

Tlw3−1w1w2t^l l!

^∞

n0

⎛

⎝wⁿ₂ n k0

⎛

⎝n k

⎞

⎠^w2−1

i0

−1ⁱEk

w1y1w1

w2i

Tn−kw3−1w₁^n−kw^k₃

⎞

⎠tⁿ n!.

3.9

3Invoking1.8once again,3.8can be written as

I Λ²₂₃

^w2−1

i0 w3−1

j0

−1^ij

Zp

e^{w2w3x1w1y1w1/w2iw1/w3jt}dμ−1x1

^w2−1

i0 w3−1

j0

−1^ij∞

n0

En

w1y1w1

w2i w1

w3j

w2w3tⁿ n!

^∞

n0

⎛

⎝w2w3^nw2−1

i0 w3−1

j0

−1^ijEn

w1y1w1

w2iw1

w3j

⎞⎠tⁿ n!.

3.10

a-3One has

I Λ³₂₃

Zpe^w2w3x1tdμ−1x1

Zpe^w1w2w3x4tdμ−1x4×

Zpe^w1w3x2tdμ−1x2

Zpe^w1w2w3x4tdμ−1x4×

Zpe^w1w2x3tdμ−1x3

Zpe^w1w2w3x4tdμ−1x4

_∞

k0

Tkw1−1w2w3t^k k!

_∞

l0

Tlw2−1w1w3t^l l!

_∞

m0

Tmw3−1w1w2t^m m!

^∞

n0

klmn

⎛

⎝

⎛

⎝ n k, l, m

⎞

⎠Tkw1−1Tlw2−1Tmw3−1w^lm₁ w₂^kmw₃^kl

⎞

⎠tⁿ n!.

3.11

bFor TypeΛⁱ₁₃ i0,1,2,3, 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,w1w2, and then dividing byw1w2w3ⁿ, in each of the expressions of Theorem 4.1through Corollary 4.15, we will get the corresponding symmetric identities for TypeΛⁱ₁₃ i0,1,2,3.

c-0One has

I Λ⁰₁₂

Zp

e^w1x1w2ytdμ−1x1

Zp

e^w2x2w3ytdμ−1x2

Zp

e^w3x3w1ytdμ−1x3

_∞

n0

Ek

w2y k! w1t^k

_∞

l0

El

w3y l! w2t^l

_∞

m0

Em

w1y m! w3t^m

^∞

n0

klmn

n

k, l, m

Ek

w2y El

w3y Em

w1y

w₁^kw^l₂w₃^m tⁿ

n!.

3.12

c-1One has

Zpe^w1x1tdμ−1x1

Zpe^w1w2z3tdμ−1z3×

Zpe^w2x2tdμ−1x2

Zpe^w2w3z1tdμ−1z1×

Zpe^w3x3tdμ−1x3

Zpe^w3w1z2tdμ−1z2

_∞

k0

Tkw2−1w1t^k k!

_∞

l0

Tlw3−1w2t^l l!

_∞

m0

Tmw1−1w3t^m m!

^∞

n0

klmn

n

k, l, m

Tkw2−1Tlw3−1Tmw1−1w₁^kw^l₂w₃^m tⁿ

n!.

3.13

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 inSection 3yield distinct ones. In fact, as these expressions are obtained by permutingw1,w2,w3 in a single one labelled 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 numbers of possible distinct expressions are 1, 2, 3,or 6.a-0,a- 11,a-12, anda-22give the full six identities of symmetry,a-21anda-23yield 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 Theorems4.8and4.17leaving the others as easy exercises for the reader. As for the case ofTheorem 4.8, in addition to4.15–4.17, we get the following three ones:

klmn

n k, l, m

Ek

w1y1

Tlw3−1Tmw2−1w^lm₁ w^km₃ w^kl₂ , 4.1

klmn

n

k, l, m

Ek

w2y1

Tlw1−1Tmw3−1w₂^lmw₁^kmw₃^kl, 4.2

klmn

n k, l, m

Ek

w3y1

Tlw2−1Tmw1−1w^lm₃ w^km₂ w^kl₁ . 4.3

But, by interchanging land m, we see that4.1,4.2, and 4.3are, respectively, equal to 4.15,4.16, and4.17.

As to Theorem 17, in addition to4.26and4.27, we have

klmn

n k, l, m

Tkw2−1Tlw3−1Tmw1−1w^k₁w^l₂w₃^m, 4.4

klmn

n

k, l, m

Tkw3−1Tlw1−1Tmw2−1w₂^kw^l₃w₁^m, 4.5

klmn

n

k, l, m

Tkw3−1Tlw2−1Tmw1−1w₁^kw^l₃w₂^m, 4.6

klmn

n k, l, m

Tkw2−1Tlw1−1Tmw3−1w^k₃w^l₂w₁^m. 4.7

However, 4.4and 4.5are equal to 4.26, as we can see by applying the permutations k → l,l → m, andm → k for4.4andk → m,l → k, andm → l for4.5. Similarly,

we see that4.6and4.7are equal to4.27, by applying permutationsk → l,l → m, and m → kfor4.6andk → m,l → k, andm → lfor4.7.

Theorem 4.1. Letw1,w2,w3 be any positive integers. Then the following expression is invariant under any permutation ofw1,w2,w3, so that it gives us six symmetries:

klmn

n k, l, m

Ek

w1y1

El

w2y2

Em

w3y3

w^lm₁ w^km₂ w^kl₃

klmn

n k, l, m

Ek

w1y1

El

w3y2

Em

w2y3

w^lm₁ w^km₃ w^kl₂

klmn

n

k, l, m

Ek

w2y1

El

w1y2

Em

w3y3

w^lm₂ w^km₁ w^kl₃

klmn

n k, l, m

Ek

w2y1

El

w3y2

Em

w1y3

w^lm₂ w^km₃ w^kl₁

klmn

n k, l, m

Ek

w3y1

El

w1y2

Em

w2y3

w^lm₃ w^km₁ w^kl₂

klmn

n

k, l, m

Ek

w3y1

El

w2y2

Em

w1y3

w^lm₃ w^km₂ w^kl₁ .

4.8

Theorem 4.2. Letw1,w2,w3be any odd positive integers. Then the following expression is invariant under any permutation ofw1,w2,w3, so that it gives us six symmetries:

klmn

n k, l, m

Ek

w1y1

El

w2y2

Tmw3−1w^lm₁ w^km₂ w^kl₃

klmn

n

k, l, m

Ek

w1y1

El

w3y2

Tmw2−1w^lm₁ w^km₃ w^kl₂

klmn

n k, l, m

Ek

w2y1

El

w1y2

Tmw3−1w^lm₂ w^km₁ w^kl₃

klmn

n k, l, m

Ek

w2y1

El

w3y2

Tmw1−1w^lm₂ w^km₃ w^kl₁

klmn

n

k, l, m

Ek

w3y1

El

w2y2

Tmw1−1w^lm₃ w^km₂ w^kl₁

klmn

n

k, l, m

Ek

w3y1

El

w1y2

Tmw2−1w^lm₃ w^km₁ w^kl₂ .

4.9

Puttingw31 in4.9, we get the following corollary.

Corollary 4.3. Letw1,w2be any odd positive integers. Then one has

n k0

⎛

⎝n k

⎞

⎠Ek

w1y1

E_n−k w2y2

w^n−k₁ w^k₂

ⁿ

k0

⎛

⎝n k

⎞

⎠Ek

w2y1

En−k w1y2

w^n−k₂ w^k₁

klmn

⎛

⎝ n k, l, m

⎞

⎠Ek

y1

El

w2y2

Tmw1−1w^km₂ w^kl₁

klmn

⎛

⎝ n k, l, m

⎞

⎠Ek

w2y1

El

y2

Tmw1−1w^lm₂ w^kl₁

klmn

⎛

⎝ n k, l, m

⎞

⎠Ek

y1

El

w1y2

Tmw2−1w^km₁ w^kl₂

klmn

⎛

⎝ n k, l, m

⎞

⎠Ek

w1y1

El

y2

Tmw2−1w^lm₁ w^kl₂ .

4.10

Letting furtherw21 in4.10, we have the following corollary.

Corollary 4.4. Letw1be any odd positive integer. Then one has

n k0

⎛

⎝n k

⎞

⎠Ek

w1y1

En−k y2

w₁^n−k

ⁿ

k0

⎛

⎝n k

⎞

⎠Ek

y1

En−k w1y2

w^k₁

klmn

⎛

⎝ n k, l, m

⎞

⎠Ek

y1

El

y2

Tmw1−1w^kl₁ .

4.11

Theorem 4.5. Letw1, w2, w3be any odd positive integers. Then the following expression is invariant under any permutation ofw1,w2,w3, so that it gives us six symmetries:

wⁿ₁ n k0

n

k

Ek

w3y1

^w1−1

i0

−1ⁱEn−k

w2y2w2

w1i

w₃^n−kw₂^k

wⁿ₁ n k0

n

k

Ek

w2y1

^w1−1

i0

−1ⁱEn−k

w3y2w3

w1i

w^n−k₂ w^k₃

wⁿ₂ n k0

n k

Ek

w3y1

^w2−1

i0

−1ⁱEn−k

w1y2w1

w2i

w^n−k₃ w^k₁

wⁿ₂ n k0

n k

Ek

w1y1

^w2−1

i0

−1ⁱE_n−k

w3y2w3

w2i

w^n−k₁ w^k₃

wⁿ₃ n k0

n

k

Ek

w2y1

^w3−1

i0

−1ⁱEn−k

w1y2w1

w3i

w^n−k₂ w^k₁

wⁿ₃ n k0

n k

Ek

w1y1

^w3−1

i0

−1ⁱEn−k

w2y2w2

w3i

w^n−k₁ w^k₂.

4.12

Lettingw31 in4.12, we obtain alternative expressions for the identities in4.10.

Corollary 4.6. Letw1,w2be any odd positive integers. Then one has

n k0

n

k

Ek

w1y1

En−k w2y2

w^n−k₁ w^k₂

ⁿ

k0

n

k

Ek

w2y1

En−k w1y2

w^n−k₂ w^k₁

wⁿ₁ n k0

n k

Ek

y1

^w1−1

i0

−1ⁱEn−k

w2y2w2

w1i

w₂^k

wⁿ₁ n k0

n k

Ek

w2y1

^w1−1

i0

−1ⁱE_n−k

y2 i w1

w^n−k₂

wⁿ₂ n k0

n

k

Ek

y1

^w2−1

i0

−1ⁱEn−k

w1y2w1

w2i

w₁^k

wⁿ₂ n k0

n k

Ek

w1y1

^w2−1

i0

−1ⁱEn−k

y2 i

w2

w^n−k₁ .

4.13

Putting furtherw2 1 in4.13, we have the alternative expressions for the identities for4.11.