Symmetry Properties of Higher-Order Bernoulli Polynomials
© Patricia J. Y. Wong 2009
Received: 11 March 2009
Accepted: 2 August 2009
Published: 26 August 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
It is easy to see that
By comparing the coefficients on both sides of (2.5) and (2.6), we obtain the following theorem.
Let and in (2.7). Then we have the following corollary.
If we take in (2.8), then we also obtain the following corollary.
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.
Let and in (2.12). Then we obtain the following Corollary 2.5.
From (2.12), we can get the well-known result due to Raabe:
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