Some identities on Bernoulli and Euler polynomials arising from orthogonality of Legendre polynomials
© Kim et al.; licensee Springer 2012
Received: 26 June 2012
Accepted: 25 September 2012
Published: 9 October 2012
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 .
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 .
where is the Kronecker symbol.
with the usual convention about replacing by .
In the special case, , are called the Bernoulli numbers.
with the usual convention about replacing by .
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
Therefore, by (2.1) and (2.3), we obtain the following proposition.
Let us assume that .
where () is the gamma function.
Therefore, by (2.6) and (2.7), we obtain the following Proposition 2.2.
Therefore, by Proposition 2.1, (2.9) and (2.10), we obtain the following theorem.
Therefore, by Proposition 2.1 and (2.17), we obtain the following theorem.
where is the Hermite polynomial of degree n (see ).
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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