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# Symmetry Properties of Higher-Order Bernoulli Polynomials

*Advances in Difference Equations*
**volume 2009**, Article number: 318639 (2009)

## Abstract

We investigate properties of identities and some interesting identities of symmetry for the Bernoulli polynomials of higher order using the multivariate -adic invariant integral on .

## 1. Introduction

Let be a fixed prime number. Throughout this paper and will, respectively, denote the ring of -adic rational integers, the field of -adic rational numbers, and the completion of algebraic closure of . For , we use the notation . Let be the space of uniformly differentiable functions on and let be the normalized exponential valuation of with . For with , the -Volkenborn integral on is defined as

(see [1, 2]). The ordinary -adic invariant integral on is given by

(see [1–15]). Let . Then we easily see that

From (1.3), we can derive

(see [2, 8–10]), where are the th Bernoulli numbers.

By (1.2) and (1.3), we easily see that

where for

It is known that the Bernoulli polynomials are defined by

where are called the th Bernoulli polynomials. The Bernoulli polynomials of order , denoted , are defined as

(see [3–6]). Then the values of at are called the Bernoulli numbers of order . When , the polynomials or numbers are called the Bernoulli polynomials or numbers. The purpose of this paper is to investigate some interesting properties of symmetry for the multivariate -adic invariant integral on . From the properties of symmetry for the multivariate -adic invariant integral on , we derive some interesting identities of symmetry for the Bernoulli polynomials of higher order.

## 2. Symmetry Properties of Higher-Order Bernoulli Polynomials

Let . Then we define

From (2.1), we note that

where

In (2.1), we note that is symmetric in . By (2.1), we see that

It is easy to see that

From (2.1), (2.3), and the above formula, we can derive

By the symmetry of in and , we see that

By comparing the coefficients on both sides of (2.5) and (2.6), we obtain the following theorem.

Theorem 2.1.

For , one has

Let and in (2.7). Then we have the following corollary.

Corollary 2.2.

For , one has

If we take in (2.8), then we also obtain the following corollary.

Corollary 2.3.

For one has

By the definition of , we easily see that

From the symmetric property of in , we note that

By comparing the coefficients on both sides of (2.10) and (2.11), we obtain the following theorem.

Theorem 2.4.

For , one has

Let and in (2.12). Then we obtain the following Corollary 2.5.

Corollary 2.5.

For , one has

From (2.12), we can get the well-known result due to Raabe:

## References

- 1.
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 - 2.
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 - 3.
Abramowitz M, Stegun IA:

*Handbook of Mathematical Functions*. National Bureau of Standards; 1964. - 4.
Jordan Ch:

*Calculus of Finite Differences*. 2nd edition. Chelsea, New York, NY, USA; 1950. - 5.
Milne-Thomson LM:

*The Calculus of Finite Differences*. Macmillan, London, UK; 1933. - 6.
Nörlund NE:

*Vorlesungen über Differenzenrechnung*. Springer, Berlin, Germany; 1924. - 7.
Kim YH:

**On the -adic interpolation functions of the generalized twisted -Euler numbers.***International Journal of Mathematical Analysis*2009,**3:**897–904. - 8.
Kim T:

**-Volkenborn integration.***Russian Journal of Mathematical Physics*2002,**9**(3):288–299. - 9.
Kim T:

**-Bernoulli numbers and polynomials associated with Gaussian binomial coefficients.***Russian Journal of Mathematical Physics*2008,**15**(1):51–57. - 10.
Kim T:

**Analytic continuation of multiple -zeta functions and their values at negative integers.***Russian Journal of Mathematical Physics*2004,**11**(1):71–76. - 11.
Kim T:

**Non-Archimedean -integrals associated with multiple Changhee -Bernoulli polynomials.***Russian Journal of Mathematical Physics*2003,**10**(1):91–98. - 12.
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 - 13.
Ozden H, Simsek Y:

**A new extension of -Euler numbers and polynomials related to their interpolation functions.***Applied Mathematics Letters*2008,**21**(9):934–939. 10.1016/j.aml.2007.10.005 - 14.
Simsek Y:

**On -adic twisted --functions related to generalized twisted Bernoulli numbers.***Russian Journal of Mathematical Physics*2006,**13**(3):340–348. 10.1134/S1061920806030095 - 15.
Kim Y-H, Hwang K-W:

**A symmetry of power sum and twisted Bernoulli polynomials.***Advanced Studies in Contemporary Mathematics*2009,**18**(2):127–133.

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Kim, T., Hwang, KW. & Kim, YH. Symmetry Properties of Higher-Order Bernoulli Polynomials.
*Adv Differ Equ* **2009, **318639 (2009). https://doi.org/10.1155/2009/318639

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### Keywords

- Differential Equation
- Partial Differential Equation
- Ordinary Differential Equation
- Functional Analysis
- Functional Equation