Advances in Difference Equations

, 2012:201 | Cite as

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 

1 Introduction

The generalized Laguerre polynomials are defined by
exp ( x t 1 t ) ( 1 t ) α + 1 = n = 0 L n α ( x ) t n ( α Q  with  α > 1 ) . Open image in new window
(1.1)
From (1.1), we note that
L n α ( x ) = r = 0 n ( 1 ) r ( n + α n r ) x r r ! ( see [1–3] ) . Open image in new window
(1.2)
By (1.2), we see that L n α ( x ) Open image in new window is a polynomial with degree n. It is well known that Rodrigues’ formula for L n α ( x ) Open image in new window is given by
L n α ( x ) = x α e x n ! ( d n d x n ( e x x n + α ) ) ( see [1–3] ) . Open image in new window
(1.3)
From (1.3) and a part of integration, we note that
0 x α e x L m α ( x ) L n α ( x ) d x = Γ ( α + n + 1 ) n ! δ m , n , Open image in new window
(1.4)
where δ m , n Open image in new window is a Kronecker symbol. As is well known, Bernoulli polynomials are defined by the generating function to be
t e t 1 e x t = e B ( x ) t = n = 0 B n ( x ) t n n ! ( see [1–29] ) , Open image in new window
(1.5)

with the usual convention about replacing B n ( x ) Open image in new window by B n ( x ) Open image in new window.

In the special case, x = 0 Open image in new window, B n ( 0 ) = B n Open image in new window are called the n th Bernoulli numbers. By (1.5), we get
B n ( x ) = l = 0 n ( n l ) B n l x l ( see [1–29] ) . Open image in new window
(1.6)
The Euler numbers are defined by
E 0 = 1 , ( E + 1 ) n + E n = 2 δ 0 , n ( see [27, 28] ) , Open image in new window
(1.7)

with the usual convention about replacing E n Open image in new window by E n Open image in new window.

In the viewpoint of (1.6), the Euler polynomials are also defined by
E n ( x ) = ( E + x ) n = l = 0 n ( n l ) E n l x l ( see [11–24] ) . Open image in new window
(1.8)
From (1.7) and (1.8), we note that the generating function of the Euler polynomial is given by
2 e t + 1 e x t = e E ( x ) t = n = 0 E n ( x ) t n n ! ( see [15–29] ) . Open image in new window
(1.9)
By (1.5) and (1.9), we get
2 e t + 1 e x t = 1 t ( 2 2 2 e t + 1 ) ( t e x t e t 1 ) = 2 n = 0 ( l = 0 n E l + 1 l + 1 ( n l ) B n l ( x ) ) t n n ! . Open image in new window
(1.10)
Thus, by (1.10), we see that
E n ( x ) = 2 l = 0 n ( n l ) E l + 1 l + 1 B n l ( x ) . Open image in new window
(1.11)
By (1.7) and (1.8), we easily get
t e t 1 e x t = t 2 ( 2 e x t e t + 1 ) + ( t e t 1 ) ( 2 e x t e t + 1 ) . Open image in new window
(1.12)
Thus, by (1.12), we see that
B n ( x ) = k = 0 , k 1 n ( n k ) B k E n k ( x ) . Open image in new window
(1.13)
Throughout this paper, we assume that α Q Open image in new window with α > 1 Open image in new window. Let P n = { p ( x ) Q [ x ] | deg p ( x ) n } Open image in new window be the inner product space with the inner product
p ( x ) , q ( x ) = 0 x α e x p ( x ) q ( x ) d x , Open image in new window

where p ( x ) , q ( x ) P n Open image in new window. From (1.4), we note that { L 0 α ( x ) , L 1 α ( x ) , , L n α ( x ) } Open image in new window is an orthogonal basis for P n Open image in new window.

In this paper, we give some interesting identities on Bernoulli and Euler polynomials which can be derived by an orthogonal basis { L 0 α ( x ) , L 1 α ( x ) , , L n α ( x ) } Open image in new window for P n Open image in new window.

2 Some identities on Bernoulli and Euler polynomials

Let p ( x ) P n Open image in new window. Then p ( x ) Open image in new window can be generated by { L 0 α ( x ) , L 1 α ( x ) , , L n α ( x ) } Open image in new window in P n Open image in new window to be
p ( x ) = k = 0 n C k L k α ( x ) , Open image in new window
(2.1)
where
p ( x ) , L k α ( x ) = C k L k α ( x ) , L k α ( x ) = C k 0 x α e x L k α ( x ) L k α ( x ) d x = C k Γ ( α + k + 1 ) k ! . Open image in new window
(2.2)
From (2.2), we note that
C k = k ! Γ ( α + k + 1 ) p ( x ) , L k α ( x ) = k ! Γ ( α + k + 1 ) 1 k ! 0 ( d k d x k x k + α e x ) p ( x ) d x = 1 Γ ( α + k + 1 ) 0 ( d k d x k x k + α e x ) p ( x ) d x . Open image in new window
(2.3)
Let us take p ( x ) = m = 0 , m 1 n ( n m ) B m E n m ( x ) P n Open image in new window. Then, from (2.3), we have
C k = 1 Γ ( α + k + 1 ) 0 ( d k d x k x k + α e x ) m = 0 , m 1 n ( n m ) B n E n m ( x ) d x = ( 1 ) k Γ ( α + k + 1 ) m = 0 , m 1 n k l = k n m ( n m ) ( n m l ) B m E n m l l ! ( l k ) ! 0 x l + α e x d x = ( 1 ) k Γ ( α + k + 1 ) m = 0 , m 1 n k l = k n m ( n m ) ( n m l ) B m E n m l l ! ( l k ) ! Γ ( l + α + 1 ) = ( 1 ) k m = 0 , m 1 n k l = k n m ( n m ) ( n m l ) B m E n m l l ! ( l k ) ! ( l + α ) ( l + α 1 ) α ( α + k ) ( α + k 1 ) α = ( 1 ) k n ! m = 0 , m 1 n k l = k n m B m m ! E n m l ( n m l ) ! ( l + α l k ) . Open image in new window
(2.4)

Therefore, by (2.1) and (2.4), we obtain the following theorem.

Theorem 2.1 For n Z + Open image in new window, we have

From (1.13), we can derive the following corollary.

Corollary 2.2 For n Z + Open image in new window, we have
B n ( x ) = n ! k = 0 n ( 1 ) k ( m = 0 , m 1 n k l = k n m B m m ! E n m l ( n m l ) ! ( l + α l k ) ) L k α ( x ) . Open image in new window
Let us take p ( x ) = l = 0 n ( n l ) E l + 1 l + 1 B n l ( x ) Open image in new window. By the same method, we get
C k = 1 Γ ( α + k + 1 ) 0 ( d k d x k x k + α e x ) l = 0 n ( n l ) E l + 1 l + 1 B n l ( x ) d x = 1 Γ ( α + k + 1 ) l = 0 n k m = 0 n l ( n l ) ( n l m ) E l + 1 l + 1 B n l m 0 ( d k d x k x k + α e x ) x m d x = ( 1 ) k Γ ( α + k + 1 ) l = 0 n k m = k n l ( n l ) ( n l m ) E l + 1 l + 1 B n l m m ! ( m k ) ! Γ ( m + α + 1 ) = ( 1 ) k l = 0 n k m = k n l ( n l ) ( n l m ) m ! ( m k ) ! E l + 1 ( l + 1 ) B n l m ( α + m ) ( α + m 1 ) α ( α + k ) ( α + k 1 ) α = ( 1 ) k n ! l = 0 n k m = k n l ( α + m m k ) E l + 1 ( l + 1 ) ! B n l m ( n l m ) ! . Open image in new window
(2.5)

Therefore, by (1.11), (2.1), and (2.5), we obtain the following theorem.

Theorem 2.3 For n Z + Open image in new window, we have
E n ( x ) 2 = n ! k = 0 n ( 1 ) k ( l = 0 n k m = k n l ( α + m m k ) E m + 1 ( m + 1 ) ! B n m l ( n m l ) ! ) L k α ( x ) . Open image in new window
For n N Open image in new window with n 2 Open image in new window and m Z + Open image in new window with n m 0 Open image in new window, we have
B n m ( x ) B m ( x ) = r { ( n m 2 r ) m + ( m 2 r ) ( n m ) } B 2 r B n 2 r ( x ) n 2 r + ( 1 ) m + 1 ( n m ) ! m ! n ! B n P n ( see [8] ) . Open image in new window
(2.6)
Let us take p ( x ) = B n m ( x ) B m ( x ) P n Open image in new window. Then p ( x ) Open image in new window can be generated by an orthogonal basis { L 0 α ( x ) , L 1 α ( x ) , , L n α ( x ) } Open image in new window in P n Open image in new window to be
p ( x ) = k = 0 n C k L k α ( x ) . Open image in new window
(2.7)
From (2.3), (2.6), and (2.7), we note that
C k = 1 Γ ( α + k + 1 ) 0 ( d k d x k x k + α e x ) p ( x ) d x = 1 Γ ( α + k + 1 ) r = 0 [ n 2 ] { ( n m 2 r ) m + ( m 2 r ) ( n m ) } × B 2 r n 2 r 0 ( d k d x k x k + α e x ) B n 2 r ( x ) d x = 1 Γ ( α + k + 1 ) r = 0 [ n 2 ] { ( n m 2 r ) m + ( m 2 r ) ( n m ) } B 2 r n 2 r × l = 0 n 2 r ( n 2 r l ) B n 2 r l 0 ( d k d x k x k + α e x ) x l d x = 1 Γ ( α + k + 1 ) r = 0 [ n 2 ] l = 0 n 2 r { ( n m 2 r ) m + ( m 2 r ) ( n m ) } ( n 2 r l ) × B 2 r B n 2 r l n 2 r 0 ( d k d x k x k + α e x ) x l d x = ( 1 ) k Γ ( α + k + 1 ) r = 0 [ n k 2 ] l = k n 2 r { ( n m 2 r ) m + ( m 2 r ) ( n m ) } ( n 2 r l ) × B 2 r B n 2 r l l ! ( n 2 r ) ( l k ) ! Γ ( α + l + 1 ) . Open image in new window
(2.8)
It is easy to show that
Γ ( α + l + 1 ) Γ ( α + k + 1 ) ( l k ) ! = ( α + l ) ( α + l 1 ) α Γ ( α ) ( α + k ) ( α + k 1 ) α Γ ( α ) ( l k ) ! = ( α + l ) ( α + l 1 ) ( α + k + 1 ) ( α k ) ! = ( α + l l k ) . Open image in new window
(2.9)
By (2.8) and (2.9), we get
C k = ( 1 ) k r = 0 [ n k 2 ] l = k n 2 r { ( n m 2 r ) m + ( m 2 r ) ( n m ) } × ( n 2 r l ) ( α + l l k ) l ! B 2 r B n 2 r l ( n 2 r ) . Open image in new window
(2.10)

Therefore, by (2.7) and (2.10), we obtain the following theorem.

Theorem 2.4 For n N Open image in new window with n 2 Open image in new window and m Z + Open image in new window with n m 0 Open image in new window, we have
B n m ( x ) B m ( x ) = k = 0 n ( 1 ) k { r = 0 [ n k 2 ] l = k n 2 r ( ( n m 2 r ) m + ( m 2 r ) ( n m ) ) × ( n 2 r l ) ( α + l l k ) l ! B 2 r B n 2 r l ( n 2 r ) } L k α ( x ) . Open image in new window
It is easy to show that
t 2 e t ( x + y ) ( e t 1 ) 2 = ( x + y 1 ) t 2 e t ( x + y 1 ) e t 1 t 2 d d t ( e t ( x + y 1 ) e t 1 ) . Open image in new window
(2.11)
From (2.11), we have
k = 0 n ( n k ) B k ( x ) B n k ( y ) = ( 1 n ) B n ( x + y ) + ( x + y 1 ) n B n 1 ( x + y ) ( see [11] ) . Open image in new window
(2.12)
Let x = y Open image in new window. Then by (2.12), we get
k = 0 n ( n k ) B k ( x ) B n k ( x ) = ( 1 n ) B n ( 2 x ) + ( 2 x 1 ) B n 1 ( 2 x ) . Open image in new window
(2.13)
Let us take p ( x ) = k = 0 n ( n k ) B k ( x ) B n k ( x ) P n Open image in new window. Then p ( x ) Open image in new window can be generated by an orthogonal basis { L 0 α ( x ) , L 1 α ( x ) , , L n α ( x ) } Open image in new window in P n Open image in new window to be
p ( x ) = k = 0 n ( n k ) B k ( x ) B n k ( x ) = k = 0 n C k L k α ( x ) . Open image in new window
(2.14)
From (2.3), (2.13), and (2.14), we can determine the coefficients C k Open image in new window’s to be
C k = 1 Γ ( α + k + 1 ) 0 ( d k d x k x k + α e x ) p ( x ) d x = 1 Γ ( α + k + 1 ) { ( 1 n ) 0 ( d k d x k x k + α e x ) B n ( 2 x ) d x + n 0 ( d k d x k x k + α e x ) ( 2 x 1 ) B n 1 ( 2 x ) d x } . Open image in new window
(2.15)
By simple calculation, we get

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

Theorem 2.5 For n Z + Open image in new window, 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.MATHGoogle Scholar
  2. 2.
    Carlitz L: On the product of two Laguerre polynomials. J. Lond. Math. Soc. 1961, 36: 399–402. 10.1112/jlms/s1-36.1.399CrossRefMATHGoogle Scholar
  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.MathSciNetGoogle Scholar
  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.MathSciNetMATHGoogle Scholar
  6. 6.
    Bayad A: Modular properties of elliptic Bernoulli and Euler functions. Adv. Stud. Contemp. Math. 2010, 20(3):389–401.MathSciNetMATHGoogle Scholar
  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/s1-34.3.361CrossRefMATHGoogle Scholar
  8. 8.
    Carlitz L: Some generating functions for Laguerre polynomials. Duke Math. J. 1968, 35: 825–827. 10.1215/S0012-7094-68-03587-4MathSciNetCrossRefMATHGoogle Scholar
  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.2009.04.002MathSciNetCrossRefMATHGoogle Scholar
  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.MathSciNetMATHGoogle Scholar
  11. 11.
    Hansen ER: A Table of Series and Products. Prentice Hall, Englewood Cliffs; 1975.MATHGoogle Scholar
  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.MathSciNetMATHGoogle Scholar
  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.MathSciNetMATHGoogle Scholar
  14. 14.
    Kim T: A note on q -Bernstein polynomials. Russ. J. Math. Phys. 2011, 18(1):73–82. 10.1134/S1061920811010080MathSciNetCrossRefMATHGoogle Scholar
  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 Z p Open image in new window . Russ. J. Math. Phys. 2009, 16(4):484–491. 10.1134/S1061920809040037MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Kim T: Symmetry of power sum polynomials and multivariate fermionic p -adic invariant integral on Z p Open image in new window . Russ. J. Math. Phys. 2009, 16(1):93–96. 10.1134/S1061920809010063MathSciNetCrossRefMATHGoogle Scholar
  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 Z p Open image in new window . Adv. Stud. Contemp. Math. 2012, 22(2):239–247.MathSciNetMATHGoogle Scholar
  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.MathSciNetGoogle Scholar
  20. 20.
    Rim SH, Lee SJ: Some identities on the twisted ( h , q ) Open image in new window -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 Z p Open image in new window . 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.MathSciNetMATHGoogle Scholar
  24. 24.
    Ryoo CS: A note on the Frobenius-Euler polynomials. Proc. Jangjeon Math. Soc. 2011, 14(4):495–501.MathSciNetMATHGoogle Scholar
  25. 25.
    Ryoo CS: Some relations between twisted q -Euler numbers and Bernstein polynomials. Adv. Stud. Contemp. Math. 2011, 21(2):217–223.MathSciNetMATHGoogle Scholar
  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.MathSciNetMATHGoogle Scholar
  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.MathSciNetMATHGoogle Scholar
  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/S1061920810040114MathSciNetCrossRefMATHGoogle Scholar
  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/S1061920806030095MathSciNetCrossRefMATHGoogle Scholar

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© Kim et al.; licensee Springer 2012

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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

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