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On the Identities of Symmetry for the
-Euler Polynomials of Higher Order
Advances in Difference Equations volume 2009, Article number: 273545 (2009)
Abstract
The main purpose of this paper is to investigate several further interesting properties of symmetry for the multivariate -adic fermionic integral on
. From these symmetries, we can derive some recurrence identities for the
-Euler polynomials of higher order, which are closely related to the Frobenius-Euler polynomials of higher order. By using our identities of symmetry for the
-Euler polynomials of higher order, we can obtain many identities related to the Frobenius-Euler polynomials of higher order.
1. Introduction/Definition
Let be a fixed odd prime number. Throughout this paper,
and
will, respectively, denote the ring of
-adic rational integer, the field of
-adic rational numbers, the complex number field, and the completion of algebraic closure of
. Let
be the normalized exponential valuation of
with
. Let
be the space of uniformly differentiable functions on
. For
,
with
, the fermionic
-adic
-integral on
is defined as

(see [1]). Let us define the fermionic -adic invariant integral on
as follows:

(see [1–8]). From (1.2), we have

(see [9, 10]), where . For
with
, let
. Then, we define the
-Euler numbers as follows:

where are called the
-Euler numbers. We can show that

where are the Frobenius-Euler numbers. By comparing the coefficients on both sides of (1.4) and (1.5), we see that

Now, we also define the -Euler polynomials as follows:

In the viewpoint of (1.5), we can show that

where are the
th Frobenius-Euler polynomials. From (1.7) and (1.8), we note that

(cf. [1–8, 11–18]). For each positive integer , let
. Then we have

The -Euler polynomials of order
, denoted
, are defined as

Then the values of at
are called the
-Euler numbers of order
. When
, the polynomials or numbers are called the
-Euler polynomials or numbers. The purpose of this paper is to investigate some properties of symmetry for the multivariate
-adic fermionic integral on
. From the properties of symmetry for the multivariate
-adic fermionic integral on
, we derive some identities of symmetry for the
-Euler polynomials of higher order. By using our identities of symmetry for the
-Euler polynomials of higher order, we can obtain many identities related to the Frobenius-Euler polynomials of higher order.
2. On the Symmetry for the
-Euler Polynomials of Higher Order
Let with
(mod 2) and
. Then we set

where

Thus, we note that this expression for is symmetry in
and
. From (2.1), we have

We can show that

By (1.4) and (1.11), we see that

Thus, we have

From (2.3), (2.4), and (2.5), we can derive

By the same method, we also see that

By comparing the coefficients on both sides of (2.7) and (2.8), we obtain the following.
Theorem 2.1.
For   with  
,
, and  
, one has

Let and
in (2.9). Then we have

From (2.10), we note that

If we take in (2.11), then we have

From (2.3), we note that

By the symmetric property of in
, we also see that

By comparing the coefficients on both sides of (2.13) and (2.14), we obtain the following theorem.
Theorem 2.2.
For    with  
 and 
, one has

Let and
, we have

From (2.16), we can derive

References
Kim T: Symmetry -adic invariant integral on for Bernoulli and Euler polynomials. Journal of Difference Equations and Applications 2008,14(12):1267–1277. 10.1080/10236190801943220
Kim T: Note on the Euler numbers and polynomials. Advanced Studies in Contemporary Mathematics 2008,17(2):131–136.
Kim T: Note on -Genocchi numbers and polynomials. Advanced Studies in Contemporary Mathematics 2008,17(1):9–15.
Kim T: The modified -Euler numbers and polynomials. Advanced Studies in Contemporary Mathematics 2008,16(2):161–170.
Kim T: On a -analogue of the -adic log gamma functions and related integrals. Journal of Number Theory 1999,76(2):320–329. 10.1006/jnth.1999.2373
Kim T: -Volkenborn integration. Russian Journal of Mathematical Physics 2002,9(3):288–299.
Kim T: -Bernoulli numbers and polynomials associated with Gaussian binomial coefficients. Russian Journal of Mathematical Physics 2008,15(1):51–57.
Kim T, Choi JY, Sug JY: Extended -Euler numbers and polynomials associated with fermionic -adic -integral on . Russian Journal of Mathematical Physics 2007,14(2):160–163. 10.1134/S1061920807020045
Kim T: Symmetry of power sum polynomials and multivariate fermionic -adic invariant integral on . Russian Journal of Mathematical Physics 2009,16(1):93–96. 10.1134/S1061920809010063
Kim T: On -adic interpolating function for -Euler numbers and its derivatives. Journal of Mathematical Analysis and Applications 2008,339(1):598–608. 10.1016/j.jmaa.2007.07.027
Agarwal RP, Ryoo CS: Numerical computations of the roots of the generalized twisted -Bernoulli polynomials. Neural, Parallel & Scientific Computations 2007,15(2):193–206.
Cenkci M, Can M, Kurt V: -adic interpolation functions and Kummer-type congruences for -twisted and -generalized twisted Euler numbers. Advanced Studies in Contemporary Mathematics 2004,9(2):203–216.
Howard FT: Applications of a recurrence for the Bernoulli numbers. Journal of Number Theory 1995,52(1):157–172. 10.1006/jnth.1995.1062
Kupershmidt BA: Reflection symmetries of -Bernoulli polynomials. Journal of Nonlinear Mathematical Physics 2005, 12: 412–422. 10.2991/jnmp.2005.12.s1.34
Ozden H, Simsek Y: Interpolation function of the -extension of twisted Euler numbers. Computers & Mathematics with Applications 2008,56(4):898–908. 10.1016/j.camwa.2008.01.020
Jang L-C: A study on the distribution of twisted -Genocchi polynomials. Advanced Studies in Contemporary Mathematics 2009,18(2):181–189.
Schork M: Ward's "calculus of sequences", -calculus and the limit . Advanced Studies in Contemporary Mathematics 2006,13(2):131–141.
Tuenter HJH: A symmetry of power sum polynomials and Bernoulli numbers. The American Mathematical Monthly 2001,108(3):258–261. 10.2307/2695389
Acknowledgment
The present research has been conducted by the research grant of the Kwangwoon University in 2009.
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Kim, T., Park, K. & Hwang, Kw. On the Identities of Symmetry for the -Euler Polynomials of Higher Order.
Adv Differ Equ 2009, 273545 (2009). https://doi.org/10.1155/2009/273545
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DOI: https://doi.org/10.1155/2009/273545