, 2012:201

# Some identities on Bernoulli and Euler polynomials arising from the orthogonality of Laguerre polynomials

• Taekyun Kim
• Seog-Hoon Rim
• DV Dolgy
• Sang-Hun Lee
Open Access
Research

## Abstract

In this paper, we derive some interesting identities on Bernoulli and Euler polynomials by using the orthogonal property of Laguerre polynomials.

## Keywords

Differential Equation Generate Function Partial Differential Equation Ordinary Differential Equation Functional Analysis
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## 1 Introduction

The generalized Laguerre polynomials are defined by
(1.1)
From (1.1), we note that
(1.2)
By (1.2), we see that is a polynomial with degree n. It is well known that Rodrigues’ formula for is given by
(1.3)
From (1.3) and a part of integration, we note that
(1.4)
where is a Kronecker symbol. As is well known, Bernoulli polynomials are defined by the generating function to be
(1.5)

with the usual convention about replacing by .

In the special case, , are called the n th Bernoulli numbers. By (1.5), we get
(1.6)
The Euler numbers are defined by
(1.7)

with the usual convention about replacing by .

In the viewpoint of (1.6), the Euler polynomials are also defined by
(1.8)
From (1.7) and (1.8), we note that the generating function of the Euler polynomial is given by
(1.9)
By (1.5) and (1.9), we get
(1.10)
Thus, by (1.10), we see that
(1.11)
By (1.7) and (1.8), we easily get
(1.12)
Thus, by (1.12), we see that
(1.13)
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
(2.1)
where
(2.2)
From (2.2), we note that
(2.3)
Let us take . Then, from (2.3), we have
(2.4)

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
(2.5)

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
(2.6)
Let us take . Then can be generated by an orthogonal basis in to be
(2.7)
From (2.3), (2.6), and (2.7), we note that
(2.8)
It is easy to show that
(2.9)
By (2.8) and (2.9), we get
(2.10)

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
(2.11)
From (2.11), we have
(2.12)
Let . Then by (2.12), we get
(2.13)
Let us take . Then can be generated by an orthogonal basis in to be
(2.14)
From (2.3), (2.13), and (2.14), we can determine the coefficients ’s to be
(2.15)
By simple calculation, we get
(2.16)
and
(2.17)

Therefore, by (2.13), (2.14), (2.15), (2.16), and (2.17), we obtain the following theorem.

Theorem 2.5 For , we get

## Notes

### Acknowledgements

The authors would like to express their deep gratitude to the referees for their valuable suggestions and comments.

## Supplementary material

13662_2012_299_MOESM1_ESM.pdf (207 kb)
Authors’ original file for figure 1

## References

1. 1.
Carlitz L: An integral for the product of two Laguerre polynomials. Boll. Unione Mat. Ital. 1962, 17(17):25–28.
2. 2.
Carlitz L: On the product of two Laguerre polynomials. J. Lond. Math. Soc. 1961, 36: 399–402. 10.1112/jlms/s1-36.1.399
3. 3.
Cangul N, Kurt V, Ozden H, Simsek Y: On the higher-order w - q -Genocchi numbers. Adv. Stud. Contemp. Math. 2009, 19(1):39–57.
4. 4.
Akemann G, Kieburg M, Phillips MJ: Skew-orthogonal Laguerre polynomials for chiral real asymmetric random matrices. J. Phys. A 2010., 43(37): Article ID 375207Google Scholar
5. 5.
Bayad A, Kim T: Identities for the Bernoulli, the Euler and the Genocchi numbers and polynomials. Adv. Stud. Contemp. Math. 2010, 20(2):247–253.
6. 6.
Bayad A: Modular properties of elliptic Bernoulli and Euler functions. Adv. Stud. Contemp. Math. 2010, 20(3):389–401.
7. 7.
Carlitz L: Note on the integral of the product of several Bernoulli polynomials. J. Lond. Math. Soc. 1959, 34: 361–363. 10.1112/jlms/s1-34.3.361
8. 8.
Carlitz L: Some generating functions for Laguerre polynomials. Duke Math. J. 1968, 35: 825–827. 10.1215/S0012-7094-68-03587-4
9. 9.
Costin RD: Orthogonality of Jacobi and Laguerre polynomials for general parameters via the Hadamard finite part. J. Approx. Theory 2010, 162(1):141–152. 10.1016/j.jat.2009.04.002
10. 10.
Ding D, Yang J: Some identities related to the Apostol-Euler and Apostol-Bernoulli polynomials. Adv. Stud. Contemp. Math. 2010, 20(1):7–21.
11. 11.
Hansen ER: A Table of Series and Products. Prentice Hall, Englewood Cliffs; 1975.
12. 12.
Kudo A: A congruence of generalized Bernoulli number for the character of the first kind. Adv. Stud. Contemp. Math. 2000, 2: 1–8.
13. 13.
Kim T, Choi J, Kim YH, Ryoo CS: On q -Bernstein and q -Hermite polynomials. Proc. Jangjeon Math. Soc. 2011, 14(2):215–221.
14. 14.
Kim T: A note on q -Bernstein polynomials. Russ. J. Math. Phys. 2011, 18(1):73–82. 10.1134/S1061920811010080
15. 15.
Kim T: Some identities on the q -Euler polynomials of higher order and q -Stirling numbers by the fermionic p -adic integral on . Russ. J. Math. Phys. 2009, 16(4):484–491. 10.1134/S1061920809040037
16. 16.
Kim T: Symmetry of power sum polynomials and multivariate fermionic p -adic invariant integral on . Russ. J. Math. Phys. 2009, 16(1):93–96. 10.1134/S1061920809010063
17. 17.
Ozden H, Cangul IN, Simsek Y: Multivariate interpolation functions of higher-order q -Euler numbers and their applications. Abstr. Appl. Anal. 2008., 2008: Article ID 390857Google Scholar
18. 18.
Choi J, Kim DS, Kim T, Kim YH: Some arithmetic identities on Bernoulli and Euler numbers arising from the p -adic integrals on . Adv. Stud. Contemp. Math. 2012, 22(2):239–247.
19. 19.
Ozden H, Cangul IN, Simsek Y: Remarks on q -Bernoulli numbers associated with Daehee numbers. Adv. Stud. Contemp. Math. 2009, 18(1):41–48.
20. 20.
Rim SH, Lee SJ: Some identities on the twisted -Genocchi numbers and polynomials associated with q -Bernstein polynomials. Int. J. Math. Math. Sci. 2011., 2011: Article ID 482840Google Scholar
21. 21.
Rim SH, Bayad A, Moon EJ, Jin JH, Lee SJ: A new construction on the q -Bernoulli polynomials. Adv. Differ. Equ. 2011., 2011: Article ID 34Google Scholar
22. 22.
Rim SH, Jin JH, Moon EJ, Lee SJ: Some identities on the q -Genocchi polynomials of higher-order and q -Stirling numbers by the fermionic p -adic integral on . Int. J. Math. Math. Sci. 2010., 2010: Article ID 860280Google Scholar
23. 23.
Ryoo CS: On the generalized Barnes type multiple q -Euler polynomials twisted by ramified roots of unity. Proc. Jangjeon Math. Soc. 2010, 13(2):255–263.
24. 24.
Ryoo CS: A note on the Frobenius-Euler polynomials. Proc. Jangjeon Math. Soc. 2011, 14(4):495–501.
25. 25.
Ryoo CS: Some relations between twisted q -Euler numbers and Bernstein polynomials. Adv. Stud. Contemp. Math. 2011, 21(2):217–223.
26. 26.
Ryoo CS: Some identities of the twisted q -Euler numbers and polynomials associated with q -Bernstein polynomials. Proc. Jangjeon Math. Soc. 2011, 14(2):239–248.
27. 27.
Simsek Y: Generating functions of the twisted Bernoulli numbers and polynomials associated with their interpolation functions. Adv. Stud. Contemp. Math. (Kyungshang) 2008, 16(2):251–278.
28. 28.
Simsek Y: Special functions related to Dedekind-type DC-sums and their applications. Russ. J. Math. Phys. 2010, 17(4):495–508. 10.1134/S1061920810040114
29. 29.
Simsek Y: On p -adic twisted q - L -functions related to generalized twisted Bernoulli numbers. Russ. J. Math. Phys. 2006, 13(3):340–348. 10.1134/S1061920806030095

## Authors and Affiliations

• Taekyun Kim
• 1
• Seog-Hoon Rim
• 2
• DV Dolgy
• 3
• Sang-Hun Lee
• 4
1. 1.Department of MathematicsKwangwoon UniversitySeoulRepublic of Korea
2. 2.Department of Mathematics EducationKyungpook National UniversityTaeguRepublic of Korea
3. 3.HanrimwonKwangwoon UniversitySeoulRepublic of Korea
4. 4.Division of General EducationKwangwoon UniversitySeoulRepublic of Korea