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Some identities of Genocchi polynomials arising from Genocchi basis

  • 1Email author,
  • 2,
  • 3 and
  • 4
Journal of Inequalities and Applications20132013:43

https://doi.org/10.1186/1029-242X-2013-43

  • Received: 21 December 2012
  • Accepted: 13 January 2013
  • Published:

Abstract

In this paper, we give some interesting identities which are derived from the basis of Genocchi. From our methods which are treated in this paper, we can derive some new identities associated with Bernoulli and Euler polynomials.

MSC:11B68, 11S80.

Keywords

  • Vector Space
  • Dimensional Vector
  • Good Basis
  • Natural Basis
  • Euler Number

1 Introduction

As is well known, the Genocchi polynomials are defined by the generating function to be
2 t e t + 1 e x t = e G ( x ) t = n = 0 G n ( x ) t n n ! ( see [1–9] )
(1)

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

In the special case x = 0 , G n ( 0 ) = G n are called the n th Genocchi numbers. From (1), we note that
G 0 = 0 , G n ( 1 ) + G n = 2 δ n , 1 ( see [10–16] ) ,
(2)
where δ n , k is the Kronecker symbol.
G n ( x ) = ( G + x ) n = l = 0 n ( n l ) G l x n l ( see [6–8, 17] ) .
(3)
Thus, by (2) and (3), we see that
d d x G n ( x ) = n G n 1 ( x ) , deg G n ( x ) = n 1 .
(4)
The n th Bernoulli polynomials are also 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 [14–16] )
(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 n th Bernoulli numbers. By (5), we get
B 0 = 1 , B n ( 1 ) B n = δ 1 , n ( see [8, 9, 17] )
(6)
and
B n ( x ) = l = 0 n ( n l ) B l x n l = l = 0 n ( n l ) B n l x l .
(7)
The Euler numbers are defined by
E 0 = 1 , ( E + 1 ) n + E n = 2 δ 0 , n .
(8)
The Euler polynomials are defined by
E n ( x ) = ( E + x ) n = l = 0 n ( n l ) E n l x l ( see [7–13, 17] ) .
(9)

Let P n = { p ( x ) Q [ x ] deg p ( x ) n } be the ( n + 1 ) -dimensional vector space over . Probably, { 1 , x , , x n } is the most natural basis for P n . But { G 1 ( x ) , G 2 ( x ) , , G n + 1 ( x ) } is also a good basis for the space P n for our purpose of arithmetical applications of Genocchi polynomials. Let p ( x ) P n . Then p ( x ) can be expressed by p ( x ) = 1 k n + 1 b k G k ( x ) .

In this paper, we develop some new methods to obtain some new identities and properties of Genocchi polynomials which are derived from the Genocchi basis.

2 Genocchi basis and some identities of Genocchi polynomials

Let us take p ( x ) P n . Then p ( x ) can be expressed as a -linear combination of G 1 ( x ) , G 2 ( x ) , , G n + 1 ( x ) as follows:
p ( x ) = 1 k n + 1 b k G k ( x ) = b 1 G 1 ( x ) + b 2 G 2 ( x ) + + b n + 1 G n + 1 ( x ) .
(10)
Now, let us define the operator ˜ by
˜ p ( x ) = p ( x + 1 ) + p ( x ) .
(11)
Then, by (10) and (11), we set
g ( x ) = ˜ p ( x ) = 1 k n + 1 b k ( G k ( x + 1 ) + G k ( x ) ) .
(12)
From (1), we note that
n = 0 { G n ( x + 1 ) + G n ( x ) } t n n ! = 2 t e t + 1 e ( x + 1 ) t + 2 t e t + 1 e x t .
(13)
By (2), (3) and (13), we get
G n + 1 ( x + 1 ) + G n + 1 ( x ) n + 1 = 2 x n .
(14)
From (12) and (14), we get
g ( x ) = ˜ p ( x ) = 2 1 k n + 1 k b k x k 1 .
(15)
For r N , let us take the r th derivative of g ( x ) in (15) as follows:
g ( r ) ( x ) = d r g ( x ) d x r = 2 1 k n + 1 k ( k 1 ) ( k 1 r + 1 ) b k x k 1 r .
(16)
Thus, by (16), we get
g ( r ) ( 0 ) = d r g ( x ) d x r | x = 0 = 2 ( r + 1 ) ! b r + 1 .
(17)
From (11) and (17), we have
b r + 1 = 1 2 ( r + 1 ) ! { p ( r ) ( 1 ) + p ( r ) ( 0 ) } ,
(18)

where p ( r ) ( a ) = d r g ( x ) d x r | x = a .

Therefore, by (10) and (18), we obtain the following theorem.

Theorem 1 For n N , let p ( x ) P n with p ( x ) = 1 k n + 1 b k G k ( x ) .

Then we have
b k = 1 2 k ! ( p ( k 1 ) ( 1 ) + p ( k 1 ) ( 0 ) ) .
Let us assume that p ( x ) = B n ( x ) . Then by Theorem 1, we get
B n ( x ) = 1 k n + 1 b k G k ( x ) ,
(19)
where
b k = 1 2 k ! { p ( k 1 ) ( 1 ) + p ( k 1 ) ( 0 ) } = 1 2 k ! ( n ) k 1 { B n k + 1 ( 1 ) + B n k + 1 } .
(20)
From (6) and (20), we have
b k = 1 2 ( n + 1 ) ( n + 1 k ) { δ n , k + 2 B n k + 1 } .
(21)
By (19) and (21), we get
B n ( x ) = 1 n + 1 1 k n 1 ( n + 1 k ) B n k + 1 G k ( x ) + 1 2 ( 1 + 2 B 1 ) G n ( x ) + 1 2 ( n + 1 ) 2 G n + 1 ( x ) = 1 n + 1 1 k n 1 ( n + 1 k ) B n k + 1 G k ( x ) + 1 n + 1 G n + 1 ( x ) .
(22)

Therefore, by (22), we obtain the following theorem.

Theorem 2 For n N , we have
B n ( x ) = 1 n + 1 1 k n 1 ( n + 1 k ) B n k + 1 G k ( x ) + 1 n + 1 G n + 1 ( x ) .
In particular, if we take p ( x ) = E n ( x ) P n , then we have
E n ( x ) = 1 k n + 1 b k G k ( x ) ,
(23)
where
b k = 1 2 k ! { p ( k 1 ) ( 1 ) + p ( k 1 ) ( 0 ) } = 1 2 k ! ( n ) k 1 { E n k + 1 ( 1 ) + E n k + 1 } .
(24)
By (8) and (24), we get
b k = 1 2 ( n + 1 ) ( n + 1 k ) { 2 δ n k + 1 , 0 E n k + 1 + E n k + 1 } = 1 n + 1 ( n + 1 k ) δ n + 1 , k .
(25)
From (23) and (25), we have
E n ( x ) = 1 n + 1 G n + 1 ( x ) .
Let us take p ( x ) P n with
p ( x ) = 0 k n B k ( x ) B n k ( x ) .
(26)
Then we have
Continuing this process, we get
d k p ( x ) d x k = p ( k ) ( x ) = ( n + 1 ) n ( n + 1 k + 1 ) l = k n B l k ( x ) B n l ( x ) = ( n + 1 ) ! ( n + 1 k ) ! l = k n B l k ( x ) B n l ( x ) .
(27)
From (27), we have
p ( k 1 ) ( 1 ) = ( n + 1 ) ! ( n + 2 k ) ! l = k 1 n B l + 1 k ( 1 ) B n l ( 1 ) .
(28)
By (6), we get
B l + 1 k ( 1 ) B n l ( 1 ) = ( δ l + 1 k , 1 + B l + 1 k ) ( δ n l , 1 + B n l ) = { δ k , n 1 + B n k + B n k + B l + 1 k B n l } .
(29)
From (28) and (29), we have
p ( k 1 ) ( 1 ) = ( n + 1 ) ! ( n + 2 k ) ! { δ k , n 1 + 2 B n k + k 1 l n B l + 1 k B n l } .
(30)
By Theorem 1, p ( x ) = 0 k n B k ( x ) B n k ( x ) can be expressed by
p ( x ) = 1 k n + 1 b k ( x ) G k ( x ) ,
(31)
where
b k = 1 2 k ! { p ( k 1 ) ( 1 ) + p ( k 1 ) ( 0 ) } = ( n + 1 ) ! 2 k ! ( n + 2 k ) ! { δ k , n 1 + 2 B n k + 2 l = k 1 B l + 1 k B n l } .
(32)
Thus, by (31) and (32), we get
p ( x ) = n ( n + 1 ) 12 G n 1 ( x ) + 1 k n + 1 1 k ( n + 1 k 1 ) B n k G k ( x ) + 1 k n + 1 1 k ( n + 1 k 1 ) l = k 1 n B l + 1 k B n l G k ( x ) .
(33)

Therefore, by (31) and (33), we obtain the following theorem.

Theorem 3 For n N , we have
k = 0 n B k ( x ) B n k ( x ) = n ( n + 1 ) 12 G n 1 ( x ) + 1 k n + 1 1 k ( n + 1 k 1 ) B n k G k ( x ) + 1 k n + 1 ( k 1 l n 1 k ( n + 1 k 1 ) B l + 1 k B n l ) G k ( x ) .

Declarations

Acknowledgements

Dedicated to Professor Hari M Srivastava.

The authors would like to express their gratitude for the valuable comments and suggestions of referees. This research was supported by Kwangwoon University in 2013.

Authors’ Affiliations

(1)
Department of Mathematics, Kwangwoon University, Seoul, 139-701, South Korea
(2)
Department of Mathematics Education, Kyungpook National University, Daegu, 702-701, South Korea
(3)
Hanrimwon, Kwangwoon University, Seoul, 139-701, South Korea
(4)
Division of General Education, Kwangwoon University, Seoul, 139-701, South Korea

References

  1. Araci S, Acikgöz M, Jolany H, Seo JJ: A unified generating function of the q -Genocchi polynomials with their interpolation functions. Proc. Jangjeon Math. Soc. 2012, 15(2):227–233.MATHMathSciNetGoogle Scholar
  2. Araci S, Erdal D, Seo JJ: 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
  3. 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.MATHMathSciNetGoogle Scholar
  4. Bayad A: Modular properties of elliptic Bernoulli and Euler functions. Adv. Stud. Contemp. Math. 2010, 20(3):389–401.MATHMathSciNetGoogle Scholar
  5. Cangul IN, 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
  6. Ding D, Yang J: Some identities related to the Apostol-Euler and Apostol-Bernoulli polynomials. Adv. Stud. Contemp. Math. 2010, 20(1):7–21.MATHMathSciNetGoogle Scholar
  7. Dolgy DV, Kim T, Lee B, Ryoo CS: On the q -analogue of Euler measure with weight. Adv. Stud. Contemp. Math. 2011, 21(4):429–435.MATHMathSciNetGoogle Scholar
  8. Kim DS, Lee N, Na J, Park KH: Identities of symmetry for higher-order Euler polynomials in three variables (I). Adv. Stud. Contemp. Math. 2012, 22(1):51–74.MATHMathSciNetGoogle Scholar
  9. Kim T: Some identities for the Bernoulli, the Euler and the Genocchi numbers and polynomials. Adv. Stud. Contemp. Math. 2010, 20(1):23–28.MATHMathSciNetGoogle Scholar
  10. Kim T: On the multiple q -Genocchi and Euler numbers. Russ. J. Math. Phys. 2008, 15(4):481–486. 10.1134/S1061920808040055MATHMathSciNetView ArticleGoogle 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/S1061920809040037MATHMathSciNetView ArticleGoogle Scholar
  12. 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
  13. Rim S-H, Jeong J: On the modified q -Euler numbers of higher order with weight. Adv. Stud. Contemp. Math. 2012, 22(1):93–98.MATHMathSciNetGoogle Scholar
  14. Ryoo CS: Calculating zeros of the twisted Genocchi polynomials. Adv. Stud. Contemp. Math. 2008, 17(2):147–159.MATHMathSciNetGoogle Scholar
  15. 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.MATHMathSciNetGoogle Scholar
  16. Simsek Y: Theorems on twisted L-function and twisted Bernoulli numbers. Adv. Stud. Contemp. Math. 2005, 11(2):205–218.MATHMathSciNetGoogle Scholar
  17. Kim DS, Kim T: Some identities of higher order Euler polynomials arising from Euler basis. Integral Transforms Spec. Funct. 2012. doi:10.1080/10652469.2012.754756Google Scholar

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