Abstract
The purpose of this paper is to investigate some interesting identities on the Bernoulli and Euler polynomials arising from the orthogonality of Legendre polynomials in the inner product space .
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1 Introduction
As is well known, the Legendre polynomial is a solutions of the following differential equation:
where .
It is a polynomial of degree n. If n is even or odd, then is accordingly even or odd. They are determined up to constant and normalized so that .
Rodrigues’ formula is given by
Integrating by parts, we can derive
where is the Kronecker symbol.
By (1.1), we get
The generating function is given by
The Bernoulli polynomial is defined by a generating function to be
with the usual convention about replacing by .
In the special case, , are called the Bernoulli numbers.
From (1.5), we have
As is well known, the Euler numbers are defined by
with the usual convention about replacing by .
The Euler polynomials are defined as
Let . Then is an inner product space with respect to the inner product with
where .
From (1.2), we can show that is an orthogonal basis for . In this paper, we derive some interesting identities on the Bernoulli and Euler polynomials from the orthogonality of Legendre polynomials in .
2 Some identities on the Bernoulli and Euler polynomials
For , let
Then, from (1.2), we have
By (2.2), we get
Therefore, by (2.1) and (2.3), we obtain the following proposition.
Proposition 2.1 For, let
Then
Let us assume that .
From Proposition 2.1, we have
For , by (2.4), we get
Here the beta function is defined by
and it is well known that
where () is the gamma function.
By Proposition 2.1 and (2.5), we get
From (1.5), we can easily derive the following equation (2.7):
Therefore, by (2.6) and (2.7), we obtain the following Proposition 2.2.
Proposition 2.2 For, we have
Let us take . By Proposition 2.1, we get
For with , we have
In [14], we showed that
Therefore, by Proposition 2.1, (2.9) and (2.10), we obtain the following theorem.
Theorem 2.3 For, we have
By the same method of Theorem 2.3, we easily see that
Let us take . Then we see that
The equation (2.12) was proved in [14].
By (2.12) and Proposition 2.2, we have
Integrating by parts, we get
Then we see that
It is easy to show that
Therefore, by (2.13), (2.14), (2.15) and (2.16), we get
Therefore, by Proposition 2.1 and (2.17), we obtain the following theorem.
Theorem 2.4 For, we have
Remark 2.5 The extended Laguerre polynomials are given by
By the same method, we get
and
where is the Hermite polynomial of degree n (see [7]).
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Acknowledgements
The authors would like to express their sincere gratitude to referee for his/her valuable comments and information. This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science, and Technology 2012R1A1A2003786.
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Kim, D.S., Rim, SH. & Kim, T. Some identities on Bernoulli and Euler polynomials arising from orthogonality of Legendre polynomials. J Inequal Appl 2012, 227 (2012). https://doi.org/10.1186/1029-242X-2012-227
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DOI: https://doi.org/10.1186/1029-242X-2012-227