Open Access

Some identities on Bernoulli and Euler polynomials arising from orthogonality of Legendre polynomials

Journal of Inequalities and Applications20122012:227

https://doi.org/10.1186/1029-242X-2012-227

Received: 26 June 2012

Accepted: 25 September 2012

Published: 9 October 2012

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 P n .

1 Introduction

As is well known, the Legendre polynomial P n ( x ) is a solutions of the following differential equation:
( 1 x 2 ) u 2 x u + n ( n + 1 ) u = 0 ( see [1–7] ) ,

where n = 0 , 1 , 2 ,  .

It is a polynomial of degree n. If n is even or odd, then P n ( x ) is accordingly even or odd. They are determined up to constant and normalized so that P n ( 1 ) = 1 .

Rodrigues’ formula is given by
P n ( x ) = 1 2 n n ! { ( d d x ) n ( x 2 1 ) n } , n Z + .
(1.1)
Integrating by parts, we can derive
1 1 P m ( x ) P n ( x ) d x = 2 2 n + 1 δ m , n ( see [1–7] ) ,
(1.2)

where δ m , n is the Kronecker symbol.

By (1.1), we get
P n ( x ) = 1 2 n k = 0 [ n 2 ] ( 1 ) k ( n k ) ( 2 n 2 k n ) x n 2 k .
(1.3)
The generating function is given by
( 1 2 x t + t 2 ) 1 2 = n = 0 P n ( x ) t n .
(1.4)
The Bernoulli polynomial is defined by a generating function to be
t e t 1 e x t = e B ( x ) t = n = 0 B n ( x ) t n n ! ( see [8–13] )
(1.5)

with the usual convention about replacing B n ( x ) by B n ( x ) .

In the special case, x = 0 , B n ( 0 ) = B n are called the Bernoulli numbers.

From (1.5), we have
B n ( x ) = l = 0 n ( n l ) B n l x l ( see [10–26] ) .
(1.6)
As is well known, the Euler numbers are defined by
E 0 = 1 , ( E + 1 ) n + E n = 2 δ 0 , n ( see [10–13] )
(1.7)

with the usual convention about replacing E n by E n .

The Euler polynomials are defined as
E n ( x ) = l = 0 n ( n l ) E n l x l ( see [27–31] ) .
(1.8)
Let P n = { p ( x ) O [ x ] | deg p ( x ) n } . Then P n is an inner product space with respect to the inner product , with
q 1 ( x ) , q 2 ( x ) = 1 1 q 1 ( x ) q 2 ( x ) d x ,

where q 1 ( x ) , q 2 ( x ) P n .

From (1.2), we can show that { P 0 ( x ) , P 1 ( x ) , , P n ( x ) } is an orthogonal basis for P n . In this paper, we derive some interesting identities on the Bernoulli and Euler polynomials from the orthogonality of Legendre polynomials in P n .

2 Some identities on the Bernoulli and Euler polynomials

For q ( x ) P n , let
q ( x ) = k = 0 n C k P k ( x ) .
(2.1)
Then, from (1.2), we have
q ( x ) , P k ( x ) = C k P k ( x ) , P k ( x ) = C k 1 1 { P k ( x ) } 2 d x = 2 2 k + 1 C k .
(2.2)
By (2.2), we get
C k = 2 k + 1 2 q ( x ) , P k ( x ) = 2 k + 1 2 1 1 P k ( x ) q ( x ) d x = ( 2 k + 1 2 ) 1 2 k k ! 1 1 ( d k d x k ( x 2 1 ) k ) q ( x ) d x = ( 2 k + 1 2 k + 1 k ! ) 1 1 ( d k d x k ( x 2 1 ) k ) q ( x ) d x .
(2.3)

Therefore, by (2.1) and (2.3), we obtain the following proposition.

Proposition 2.1 For q ( x ) P n , let
q ( x ) = k = 0 n C k P k ( x ) .
Then
C k = 2 k + 1 2 k + 1 k ! 1 1 ( d k d x k ( x 2 1 ) k ) q ( x ) d x .

Let us assume that q ( x ) = x n P n .

From Proposition 2.1, we have
C k = 2 k + 1 2 k + 1 k ! 1 1 ( d k d x k ( x 2 1 ) k ) x n d x = 2 k + 1 2 k + 1 ( 1 ) k ( n k ) 1 1 ( x 2 1 ) k x n k d x = 2 k + 1 2 k + 1 ( n k ) ( 1 + ( 1 ) n k ) 0 1 ( 1 x 2 ) k x n k d x .
(2.4)
For n k 0 ( mod 2 ) , by (2.4), we get
C k = 2 k + 1 2 k + 1 ( n k ) 0 1 ( 1 y ) k y n k 1 2 d y = 2 k + 1 2 k + 1 ( n k ) B ( k + 1 , n k + 1 2 ) = 2 k + 1 2 k + 1 ( n k ) Γ ( k + 1 ) Γ ( n k + 1 2 ) Γ ( n + k + 1 2 + 1 ) = 2 k + 1 2 k + 1 ( n k ) k ! Γ ( n k + 1 2 ) ( n + k + 1 2 ) ( n + k 1 2 ) ( n k + 1 2 ) Γ ( n k + 1 2 ) = 2 k + 1 2 k + 1 ( n k ) k ! 2 k + 1 ( n k ) ! ( n + k + 2 ) ( n + k ) ( n k + 2 ) ( n + k + 2 ) ! = ( 2 k + 1 ) 2 k + 1 ( n + k + 2 ) ! × n ! ( n + k + 2 2 ) ! ( n k 2 ) ! .
(2.5)
Here the beta function B ( x , y ) is defined by
B ( x , y ) = 0 1 t x 1 ( 1 t ) y 1 d t ( Re ( x ) , Re ( y ) > 0 ) ,
and it is well known that
B ( x , y ) = Γ ( x ) Γ ( y ) Γ ( x + y ) ,

where Γ ( s ) = 0 t s 1 e t d t ( Re ( s ) > 0 ) is the gamma function.

By Proposition 2.1 and (2.5), we get
x n = 0 k n , n k 0 ( mod 2 ) ( 2 k + 1 ) n ! 2 k + 1 ( n + k + 2 2 ) ! ( n + k + 2 ) ! ( n k 2 ) ! P k ( x ) .
(2.6)
From (1.5), we can easily derive the following equation (2.7):
x n = 1 n + 1 l = 0 n ( n + 1 l ) B l ( x ) ( n Z + ) .
(2.7)

Therefore, by (2.6) and (2.7), we obtain the following Proposition 2.2.

Proposition 2.2 For n Z + , we have
l = 0 n B l ( x ) ( n + 1 l ) ! l ! = 0 k n , n k 0 ( mod 2 ) ( 2 k + 1 ) 2 k + 1 ( n + k + 2 2 ) ! ( n + k + 2 ) ! ( n k 2 ) ! P k ( x ) .
Let us take q ( x ) = B n ( x ) P n . By Proposition 2.1, we get
C k = 2 k + 1 2 k + 1 k ! 1 1 ( d k d x k ( x 2 1 ) k ) B n ( x ) d x = ( 1 ) k ( 2 k + 1 ) 2 k + 1 ( n k ) 1 1 ( x 2 1 ) k B n k ( x ) d x = ( 1 ) k ( 2 k + 1 ) 2 k + 1 ( n k ) l = 0 n k ( n k l ) B n k l 1 1 ( x 2 1 ) k x l d x = 2 k + 1 2 k + 1 ( n k ) l = 0 n k ( n k l ) B n k l ( 1 + ( 1 ) l ) 0 1 ( 1 x 2 ) k x l d x .
(2.8)
For l Z + with l 0 ( mod 2 ) , we have
C k = 2 k + 1 2 k + 1 ( n k ) 0 l n k , l is even ( n k l ) B n k l 0 1 ( 1 y ) k y l 1 2 d y = 2 k + 1 2 k + 1 ( n k ) 0 l n k , l is even ( n k l ) B n k l Γ ( k + 1 ) Γ ( l + 1 2 ) Γ ( 2 k + l + 1 2 + 1 ) = ( 2 k + 1 ) 2 k + 1 n ! 0 l n k , l 0 ( mod 2 ) B n k l ( n k l ) ! × ( 2 k + l + 2 2 ) ! ( 2 k + l + 2 ) ! ( l 2 ) ! .
(2.9)
In [14], we showed that
B n ( x ) = k = 0 n 2 ( n k ) B n k E k ( x ) + E n ( x ) = k = 0 , k n 1 n ( n k ) B n k E k ( x ) .
(2.10)

Therefore, by Proposition 2.1, (2.9) and (2.10), we obtain the following theorem.

Theorem 2.3 For n Z + , we have
1 n ! k = 0 , k n 1 n ( n k ) B n k E k ( x ) = k = 0 n ( 0 l n k , l 0 ( mod 2 ) ( 2 k + 1 ) 2 k + 1 ( l + 2 k + 2 2 ) ! B n k l ( n k l ) ! ( l + 2 k + 2 ) ! ( l 2 ) ! ) P k ( x ) .
By the same method of Theorem 2.3, we easily see that
E n ( x ) n ! = k = 0 n ( 0 l n k , l 0 ( mod 2 ) ( 2 k + 1 ) 2 k + 1 ( l + 2 k + 2 2 ) ! B n k l ( n k l ) ! ( l + 2 k + 2 ) ! ( l 2 ) ! ) P k ( x ) .
(2.11)
Let us take q ( x ) = k = 0 n B k ( x ) B n k ( x ) P n . Then we see that
(2.12)

The equation (2.12) was proved in [14].

By (2.12) and Proposition 2.2, we have
C k = 2 k + 1 2 k + 1 k ! { ( n + 1 ) l = 0 n ( n l ) n l + 1 ( m = l n B m l B n m + B n 1 l ) × 1 1 E l ( x ) ( d k d x k ( x 2 1 ) k ) d x + ( n 2 1 ) n 12 1 1 ( d k d x k ( x 2 1 ) k ) E n 2 ( x ) d x } .
(2.13)
Integrating by parts, we get
(2.14)
Then we see that
(2.15)
It is easy to show that
(2.16)
Therefore, by (2.13), (2.14), (2.15) and (2.16), we get
C k = ( 2 k + 1 ) 2 k + 1 { ( n + 1 ) l = k n ( n k ) n l + 1 ( m = l n B m l B n m + B n 1 l ) × k j l , j k 0 ( mod 2 ) ( l j ) E l j j ! ( j + k + 2 ) ! × ( j + k + 2 2 ) ! ( j k 2 ) ! + ( n 2 1 ) n 12 k j n 2 , j k 0 ( mod 2 ) ( n 2 j ) E n 2 j j ! ( j + k + 2 ) ! × ( j + k + 2 2 ) ! ( j k 2 ) ! } .
(2.17)

Therefore, by Proposition 2.1 and (2.17), we obtain the following theorem.

Theorem 2.4 For n Z + , we have
k = 0 n B k ( x ) B n k ( x ) = k = 0 n ( 2 k + 1 ) 2 k + 1 { ( n + 1 ) l = k n ( n k ) n l + 1 ( m = l n B m l B n m + B n 1 l ) × k j l , j k 0 ( mod 2 ) ( l j ) E l j j ! ( j + k + 2 ) ! × ( j + k + 2 2 ) ! ( j k 2 ) ! + ( n 2 1 ) n 12 k j n 2 , j k 0 ( mod 2 ) ( n 2 j ) E n 2 j j ! ( j + k + 2 ) ! × ( j + k + 2 2 ) ! ( j k 2 ) ! } P k ( x ) .
Remark 2.5 The extended Laguerre polynomials are given by
L n α ( x ) = r = 0 n ( 1 ) r r ! ( n + α n r ) x r ( α > 1 ) .
By the same method, we get
L n α ( x ) = k = 0 n 0 l n k , l 0 ( mod 2 ) ( 1 ) k + l ( 2 k + 1 ) 2 k + 1 ( n + α n k l ) ( l + 2 k + 2 2 ) ! ( l + 2 k + 2 ) ! ( l 2 ) ! P k ( x )
and
H n ( x ) = k = 0 n 0 l n k , l 0 ( mod 2 ) ( 2 k + 1 ) 2 2 k + l + 1 n ! ( l + 2 k + 2 2 ) ! H n k l ( n k l ) ! ( l + 2 k + 2 ) ! ( l 2 ) ! P k ( x ) ,

where H n ( x ) is the Hermite polynomial of degree n (see [7]).

Declarations

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.

Authors’ Affiliations

(1)
Department of Mathematics, Sogang University, Seoul, Republic of Korea
(2)
Department of Mathematics Education, Kyungpook National University, Taegu, Republic of Korea
(3)
Department of Mathematics, Kwangwoon University, Seoul, Republic of Korea

References

  1. Carlitz L: Some integrals containing products of Legendre polynomials. Arch. Math. 1961, 12: 334–340. 10.1007/BF01650571MathSciNetView ArticleMATHGoogle Scholar
  2. Carlitz L: Some congruence properties of the Legendre polynomials. Math. Mag. 1960/1961, 34: 387–390.MathSciNetView ArticleGoogle Scholar
  3. Carlitz L: Some arithmetic properties of the Legendre polynomials. Acta Arith. 1958, 4: 99–107.MathSciNetMATHGoogle Scholar
  4. Al-Salam WA, Carlitz L: Finite summation formulas and congruences for Legendre and Jacobi polynomials. Monatshefte Math. 1958, 62: 108–118. 10.1007/BF01301283MathSciNetView ArticleMATHGoogle Scholar
  5. Carlitz L: Some arithmetic properties of the Legendre polynomials. Proc. Camb. Philos. Soc. 1957, 53: 265–268. 10.1017/S0305004100032278View ArticleMathSciNetMATHGoogle Scholar
  6. Carlitz L: Note on Legendre polynomials. Bull. Calcutta Math. Soc. 1954, 46: 93–95.MathSciNetMATHGoogle Scholar
  7. Carlitz L: Congruence properties of the polynomials of Hermite, Laguerre and Legendre. Math. Z. 1954, 59: 474–483.MathSciNetView ArticleMATHGoogle Scholar
  8. 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
  9. Kim G, Kim B, Choi J: The DC algorithm for computing sums of powers of consecutive integers and Bernoulli numbers. Adv. Stud. Contemp. Math. 2008, 17(2):137–145.MathSciNetMATHGoogle Scholar
  10. Kim T: A note on q -Bernstein polynomials. Russ. J. Math. Phys. 2011, 18(1):73–82. 10.1134/S1061920811010080MathSciNetView ArticleMATHGoogle Scholar
  11. 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 . Russ. J. Math. Phys. 2009, 16(4):484–491. 10.1134/S1061920809040037MathSciNetView ArticleMATHGoogle Scholar
  12. Kim T: Symmetry of power sum polynomials and multivariate fermionic p -adic invariant integral on Z p . Russ. J. Math. Phys. 2009, 16(1):93–96. 10.1134/S1061920809010063MathSciNetView ArticleMATHGoogle Scholar
  13. Kim T: q -Bernoulli numbers and polynomials associated with Gaussian binomial coefficients. Russ. J. Math. Phys. 2008, 15(1):51–57.MathSciNetView ArticleMATHGoogle Scholar
  14. Kim, T, Kim, DS, Dolgy, DV, Rim, SH: Some identities on the Euler numbers arising from Euler basis polynomials. ARS Comb. 109 (2013, in press)Google Scholar
  15. 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
  16. Leyendekkers JV, Shannon AG, Wong GCK: Integer structure analysis of the product of adjacent integers and Euler’s extension of Fermat’s last theorem. Adv. Stud. Contemp. Math. 2008, 17(2):221–229.MathSciNetMATHGoogle Scholar
  17. Ozden H, Cangul IN, Simsek Y: Remarks on q -Bernoulli numbers associated with Daehee numbers. Adv. Stud. Contemp. Math. 2009, 18(1):41–48.MathSciNetMATHGoogle Scholar
  18. Rim S-H, Lee SJ: Some identities on the twisted ( h , q ) -Genocchi numbers and polynomials associated with q -Bernstein polynomials. Int. J. Math. Math. Sci. 2011., 2011: Article ID 482840Google Scholar
  19. 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 . Int. J. Math. Math. Sci. 2010., 2010: Article ID 860280Google Scholar
  20. Ryoo CS: A note on the Frobenius-Euler polynomials. Proc. Jangjeon Math. Soc. 2011, 14(4):495–501.MathSciNetMATHGoogle Scholar
  21. 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
  22. Ryoo CS: Some relations between twisted q -Euler numbers and Bernstein polynomials. Adv. Stud. Contemp. Math. 2011, 21(2):217–223.MathSciNetMATHGoogle Scholar
  23. Simsek Y, Acikgoz M:A new generating function of ( q ) Bernstein-type polynomials and their interpolation function. Abstr. Appl. Anal. 2010., 2010: Article ID 769095Google Scholar
  24. Simsek Y: Generating functions of the twisted Bernoulli numbers and polynomials associated with their interpolation functions. Adv. Stud. Contemp. Math. 2008, 16(2):251–278.MathSciNetMATHGoogle Scholar
  25. Simsek Y:Complete sum of products of ( h , q ) -extension of Euler polynomials and numbers. J. Differ. Equ. Appl. 2010, 16(11):1331–1348. 10.1080/10236190902813967MathSciNetView ArticleMATHGoogle Scholar
  26. Zhang Z, Yang H: Some closed formulas for generalized Bernoulli-Euler numbers and polynomials. Proc. Jangjeon Math. Soc. 2008, 11(2):191–198.MathSciNetMATHGoogle Scholar
  27. Acikgoz M, Erdal D, Araci S: A new approach to q -Bernoulli numbers and q -Bernoulli polynomials related to q -Bernstein polynomials. Adv. Differ. Equ. 2010., 2010: Article ID 951764Google Scholar
  28. Araci S, Erdal D, Seo J: A study on the fermionic p -adic q -integral representation on Z p associated with weighted q -Bernstein and q -Genocchi polynomials. Abstr. Appl. Anal. 2011., 2011: Article ID 649248Google Scholar
  29. Bayad A: Modular properties of elliptic Bernoulli and Euler functions. Adv. Stud. Contemp. Math. 2010, 20(3):389–401.MathSciNetMATHGoogle Scholar
  30. 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
  31. Bayad A, Kim T: Identities involving values of Bernstein, q -Bernoulli, and q -Euler polynomials. Russ. J. Math. Phys. 2011, 18(2):133–143. 10.1134/S1061920811020014MathSciNetView ArticleMATHGoogle Scholar

Copyright

© 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.