Abstract
In this paper, we derive some interesting identities on Bernoulli and Euler polynomials by using the orthogonal property of Laguerre polynomials.
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1 Introduction
The generalized Laguerre polynomials are defined by
From (1.1), we note that
By (1.2), we see that is a polynomial with degree n. It is well known that Rodrigues’ formula for is given by
From (1.3) and a part of integration, we note that
where is a Kronecker symbol. As is well known, Bernoulli polynomials are defined by the generating function to be
with the usual convention about replacing by .
In the special case, , are called the n th Bernoulli numbers. By (1.5), we get
The Euler numbers are defined by
with the usual convention about replacing by .
In the viewpoint of (1.6), the Euler polynomials are also defined by
From (1.7) and (1.8), we note that the generating function of the Euler polynomial is given by
By (1.5) and (1.9), we get
Thus, by (1.10), we see that
By (1.7) and (1.8), we easily get
Thus, by (1.12), we see that
Throughout this paper, we assume that with . Let be the inner product space with the inner product
where . From (1.4), we note that is an orthogonal basis for .
In this paper, we give some interesting identities on Bernoulli and Euler polynomials which can be derived by an orthogonal basis for .
2 Some identities on Bernoulli and Euler polynomials
Let . Then can be generated by in to be
where
From (2.2), we note that
Let us take . Then, from (2.3), we have
Therefore, by (2.1) and (2.4), we obtain the following theorem.
Theorem 2.1 For , we have
From (1.13), we can derive the following corollary.
Corollary 2.2 For , we have
Let us take . By the same method, we get
Therefore, by (1.11), (2.1), and (2.5), we obtain the following theorem.
Theorem 2.3 For , we have
For with and with , we have
Let us take . Then can be generated by an orthogonal basis in to be
From (2.3), (2.6), and (2.7), we note that
It is easy to show that
By (2.8) and (2.9), we get
Therefore, by (2.7) and (2.10), we obtain the following theorem.
Theorem 2.4 For with and with , we have
It is easy to show that
From (2.11), we have
Let . Then by (2.12), we get
Let us take . Then can be generated by an orthogonal basis in to be
From (2.3), (2.13), and (2.14), we can determine the coefficients ’s to be
By simple calculation, we get
and
Therefore, by (2.13), (2.14), (2.15), (2.16), and (2.17), we obtain the following theorem.
Theorem 2.5 For , we get
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Kim, T., Rim, SH., Dolgy, D. et al. Some identities on Bernoulli and Euler polynomials arising from the orthogonality of Laguerre polynomials. Adv Differ Equ 2012, 201 (2012). https://doi.org/10.1186/1687-1847-2012-201
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DOI: https://doi.org/10.1186/1687-1847-2012-201