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Symmetry Properties of Higher-Order Bernoulli Polynomials
Advances in Difference Equations volume 2009, Article number: 318639 (2009)
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 .
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
From (1.3), we can derive
By (1.2) and (1.3), we easily see that
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
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.
For , one has
Let and in (2.7). Then we have the following corollary.
For , one has
If we take in (2.8), then we also obtain the following corollary.
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.
For , one has
Let and in (2.12). Then we obtain the following Corollary 2.5.
For , one has
From (2.12), we can get the well-known result due to Raabe:
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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
- Differential Equation
- Partial Differential Equation
- Ordinary Differential Equation
- Functional Analysis
- Functional Equation