Open Access

Apostol-Euler polynomials arising from umbral calculus

Advances in Difference Equations20132013:301

https://doi.org/10.1186/1687-1847-2013-301

Received: 3 April 2013

Accepted: 10 October 2013

Published: 8 November 2013

Abstract

In this paper, by using the orthogonality type as defined in the umbral calculus, we derive an explicit formula for several well-known polynomials as a linear combination of the Apostol-Euler polynomials.

MSC:05A40.

Keywords

Bernoulli polynomialBessel polynomialEuler polynomialFrobenius-Euler polynomialumbral calculus

1 Introduction

Let Π n be the set of all polynomials in a single variable x over the complex field of degree at most n. Clearly, Π n is a ( n + 1 ) -dimensional vector space over . Define
H = { f ( t ) = k 0 a k t k k ! | a k C }
(1.1)
to be the algebra of formal power series in a single variable t. As is known, L | p ( x ) denotes the action of a linear functional L H on a polynomial p ( x ) , and we remind that the vector space on Π n is defined by
c L + c L | p ( x ) = c L | p ( x ) + c L | p ( x )
for any c , c C and L , L H (see [14]). The formal power series in variable t define a linear functional on Π n by setting
f ( t ) | x n = a n for all  n 0 (see [1–4]) .
(1.2)
By (1.1) and (1.2), we have
t k | x n = n ! δ n , k for all  n , k 0 (see [1–4]) ,
(1.3)

where δ n , k is the Kronecker symbol. Let f L ( t ) = k 0 L | x k t k k ! with L H . From (1.3), we have f L ( t ) | x n = L | x n . So, the map L f L ( t ) is a vector space isomorphic from Π n onto . Henceforth, is thought of as a set of both formal power series and linear functionals. We call umbral algebra. The umbral calculus is the study of umbral algebra.

Let f ( t ) H . The smallest integer k for which the coefficient of t k does not vanish is called the order of f ( t ) and is denoted by O ( f ( t ) ) (see [14]). If O ( f ( t ) ) = 1 , O ( f ( t ) ) = 0 , then f ( t ) is called a delta, an invertible series, respectively. For given two power series f ( t ) , g ( t ) H such that O ( f ( t ) ) = 1 and O ( g ( t ) ) = 0 , there exists a unique sequence S n ( x ) of polynomials with g ( t ) ( f ( t ) ) k | S n ( x ) = n ! δ n , k (this condition sometimes is called orthogonality type) for all n , k 0 . The sequence S n ( x ) is called the Sheffer sequence for ( g ( t ) , f ( t ) ) which is denoted by S n ( x ) ( g ( t ) , f ( t ) ) (see [14]).

For f ( t ) H and p ( x ) Π , we have
e y t | p ( x ) = p ( y ) , f ( t ) g ( t ) | p ( x ) = f ( t ) | g ( t ) p ( x ) ,
(1.4)
and
f ( t ) = k 0 f ( t ) | x k t k k ! , p ( x ) = k 0 t k | p ( x ) x k k !
(1.5)
(see [14]). From (1.5), we derive
t k | p ( x ) = p ( k ) ( 0 ) , 1 | p ( k ) ( x ) = p ( k ) ( 0 ) ,
(1.6)
where p ( k ) ( 0 ) denotes the k th derivative of p ( x ) with respect to x at x = 0 . Let S n ( x ) ( g ( t ) , f ( t ) ) . Then we have
1 g ( f ¯ ( t ) ) e y f ¯ ( t ) = k 0 S k ( y ) t k k ! ,
(1.7)

for all y C , where f ¯ ( t ) is the compositional inverse of f ( t ) (see [16]).

For λ C with λ 1 , the Apostol-Euler polynomials (see [710]) are defined by the generating function to be
2 λ e t + 1 e x t = k 0 E k ( x | λ ) t k k ! .
(1.8)
In particular, x = 0 , E n ( 0 | λ ) = E n ( λ ) is called the nth Apostol-Euler number. From (1.8), we can derive
E n ( x | λ ) = k = 0 n ( n k ) E n k ( λ ) x k .
(1.9)
By (1.9), we have d d x E n ( x | λ ) = n E n 1 ( x | λ ) . Also, from (1.8) we have
2 λ e t + 1 = e E ( λ ) t = n 0 E n ( λ ) t n n !
(1.10)
with the usual convention about replacing E n ( λ ) by E n ( λ ) . By (1.10), we get
2 = e E ( λ ) t ( λ e t + 1 ) = λ e ( E ( λ ) + 1 ) t + e E ( λ ) t = n 0 ( λ ( E ( λ ) + 1 ) n + E n ( λ ) ) t n n ! .
Thus, by comparing the coefficients of the both sides, we have
λ ( E ( λ ) + 1 ) n + E n ( λ ) = 2 δ n , 0 .
(1.11)
As is well known, the Bernoulli polynomial (see [1114]) is also defined by the generating function to be
t e t 1 e x t = k 0 B k ( x ) t k k ! .
(1.12)
In the special case, x = 0 , B n ( 0 ) = B n is called the nth Bernoulli number. By (1.12), we get
B n ( x ) = k = 0 n ( n k ) B n k x k .
(1.13)
From (1.12), we note that
t e t 1 = e B t = n 0 B n t n n !
(1.14)
with the usual convention about replacing B n by B n . By (1.13) and (1.14), we get
t = e B t ( e t 1 ) = e ( B + 1 ) t e B t = n 0 ( ( B + 1 ) n B n ) t n n ! ,
which implies
B n ( 1 ) B n = ( B + 1 ) n B n = δ n , 1 , B 0 = 1 .
(1.15)
Euler polynomials (see [4, 11, 13, 15]) are defined by
2 e t + 1 e x t = k 0 E k ( x ) t k k ! .
(1.16)
In the special case, x = 0 , E n ( 0 ) = E n is called the nth Euler number. By (1.16), we get
2 e t + 1 = e E t = n 0 E n t n n !
(1.17)
with the usual convention about replacing E n by E n . By (1.16) and (1.17), we get
2 = e E t ( e t + 1 ) = e ( E + 1 ) t + e E t = n 0 ( ( E + 1 ) n + E n ) t n n ! ,
which implies
E n ( 1 ) + E n = ( E + 1 ) n + E n = 2 δ n , 0 .
(1.18)
For λ C with λ 1 , the Frobenius-Euler (see [11, 1619]) polynomials are defined by
1 + λ e t + λ e x t = k 0 F k ( x | λ ) t k k ! .
(1.19)
In the special case, x = 0 , F n ( 0 | λ ) = F n ( λ ) is called the nth Frobenius-Euler number (see [17]). By (1.19), we get
1 + λ e t + λ = e F t = n 0 F n ( λ ) t n n !
(1.20)
with the usual convention about replacing F n ( λ ) by F n ( λ ) (see [17]). By (1.19) and (1.20), we get
1 + λ = e F ( λ ) t ( e t + λ ) = e ( F ( λ ) + 1 ) t + λ e F ( λ ) t = n 0 ( ( F ( λ ) + 1 ) n + λ F n ( λ ) ) t n n ! ,
which implies
λ F n ( λ ) + F n ( 1 | λ ) = λ F n ( λ ) + ( F ( λ ) + 1 ) n = ( 1 + λ ) δ n , 0 .
(1.21)

In the next section, we present our main theorem and its applications. More precisely, by using the orthogonality type, we write any polynomial in Π n as a linear combination of the Apostol-Euler polynomials. Several applications related to Bernoulli, Euler and Frobenius-Euler polynomials are derived.

2 Main results and applications

Note that the set of the polynomials E 0 ( x | λ ) , E 1 ( x | λ ) , , E n ( x | λ ) is a good basis for Π n . Thus, for p ( x ) Π n , there exist constants c 0 , c 1 , , c n such that p ( x ) = k = 0 n c k E k ( x | λ ) . Since E n ( x | λ ) ( ( 1 + λ e t ) / 2 , t ) (see (1.7) and (1.8)), we have
1 + λ e t 2 t k | E n ( x | λ ) = n ! δ n , k ,
which gives
1 + λ e t 2 t k | p ( x ) = = 0 n c 1 + λ e t 2 t k | E ( x | λ ) = = 0 n c ! δ , k = k ! c k .

Hence, we can state the following result.

Theorem 2.1 For all p ( x ) Π n , there exist constants c 0 , c 1 , , c n such that p ( x ) = k = 0 n c k E k ( x | λ ) , where
c k = 1 2 k ! ( 1 + λ e t ) t k | p ( x ) .
Now, we present several applications for our theorem. As a first application, let us take p ( x ) = x n with n 0 . By Theorem 2.1, we have x n = k = 0 n c k E k ( x | λ ) , where
c k = 1 2 k ! ( 1 + λ e t ) t k | x n = 1 2 ( n k ) 1 + λ e t | x n k = 1 2 ( n k ) ( δ n k , 0 + λ ) ,

which implies the following identity.

Corollary 2.2 For all n 0 ,
x n = 1 2 E n ( x | λ ) + λ 2 k = 0 n ( n k ) E k ( x | λ ) .
Let p ( x ) = B n ( x ) Π n , then by Theorem 2.1 we have that B n ( x ) = k = 0 n c k E k ( x | λ ) , where
c k = 1 2 k ! ( 1 + λ e t ) t k | B n ( x ) = 1 2 ( n k ) 1 + λ e t | B n k ( x ) = 1 2 ( n k ) ( B n k + λ B n k ( 1 ) ) ,

which, by (1.15), implies the following identity.

Corollary 2.3 For all n 2 ,
B n ( x ) = ( λ 1 ) n 4 E n 1 ( x | λ ) + 1 + λ 2 k = 0 , k n 1 n ( n k ) B n k E k ( x | λ ) .
Let p ( x ) = E n ( x ) , then by Theorem 2.1 we have that E n ( x ) = k = 0 n c k E k ( x | λ ) , where
c k = 1 2 k ! ( 1 + λ e t ) t k | E n ( x ) = 1 2 ( n k ) 1 + λ e t | E n k ( x ) = 1 2 ( n k ) ( E n k + λ E n k ( 1 ) ) ,

which, by (1.18), implies the following identity.

Corollary 2.4 For all n 0 ,
E n ( x ) = 1 + λ 2 k = 0 n ( n k ) E n k E k ( x | λ ) .
For another application, let p ( x ) = F n ( x | λ ) , then by Theorem 2.1 we have that F n ( x | λ ) = k = 0 n c k E k ( x | λ ) , where
c k = 1 2 k ! ( 1 + λ e t ) t k | F n ( x | λ ) = 1 2 ( n k ) 1 + λ e t | F n k ( x | λ ) = 1 2 ( n k ) ( F n k ( λ ) + λ F n k ( 1 | λ ) ) ,

which, by (1.21), implies the following identity.

Corollary 2.5 For all n 1 ,
F n ( x | λ ) = 1 + λ 2 E n ( x | λ ) + 1 λ 2 2 k = 0 n 1 ( n k ) F n k ( λ ) E k ( x | λ ) .
Again, let p ( x ) = y n ( x ) = k = 0 n ( n + k ) ! ( n k ) ! k ! x k 2 k be the n th Bessel polynomial (which is the solution of the following differential equation x 2 f ( x ) + 2 ( x + 1 ) f + n ( n + 1 ) f = 0 , where f ( x ) denotes the derivative of f ( x ) , see [3, 4]). Then, by Theorem 2.1, we can write y n ( x ) = k = 0 n c k E k ( x | λ ) , where
c k = 1 2 k ! = 0 n ( n + ) ! ( n ) ! ! 2 1 + λ e t | t k x = 1 2 = k n ( n + ) ! ( n ) ! ! 2 ( k ) 1 + λ e t | x k = 1 2 = k n ( n + ) ! ( n ) ! ! 2 ( k ) ( δ n k , 0 + λ ) = k ! 2 k + 1 ( n k ) ( n + k k ) + λ = k n k ! 2 + 1 ( k ) ( n ) ( n + ) ,

which implies the following identity.

Corollary 2.6 For all n 1 ,
y n ( x ) = k = 0 n k ! 2 k + 1 ( n k ) ( n + k k ) E k ( x | λ ) + λ k = 0 n = k n k ! 2 + 1 ( k ) ( n ) ( n + ) E k ( x | λ ) .

We end by noting that if we substitute λ = 0 in any of our corollaries, then we get the well-known value of the polynomial p ( x ) . For instance, by setting λ = 0 , the last corollary gives that y n ( x ) = k = 0 n ( n + k ) ! ( n k ) ! k ! x k 2 k , as expected.

Declarations

Acknowledgements

This paper is supported in part by the Research Grant of Kwangwoon University in 2013.

Authors’ Affiliations

(1)
Department of Mathematics, Kwangwoon University
(2)
Department of Mathematics, University of Haifa
(3)
Department of Mathematics Education, Kyungpook National University
(4)
Division of General Education, Kwangwoon University

References

  1. Kim DS, Kim T: Applications of umbral calculus associated with p -adic invariant integrals on Z p . Abstr. Appl. Anal. 2012., 2012: Article ID 865721Google Scholar
  2. Kim DS, Kim T: Some identities of Frobenius-Euler polynomials arising from umbral calculus. Adv. Differ. Equ. 2012., 2012: Article ID 196Google Scholar
  3. Roman S: More on the umbral calculus, with emphasis on the q -umbral calculus. J. Math. Anal. Appl. 1985, 107: 222-254. 10.1016/0022-247X(85)90367-1MathSciNetView ArticleMATHGoogle Scholar
  4. Roman S: The Umbral Calculus. Dover, New York; 2005.Google Scholar
  5. Kim T: Identities involving Frobenius-Euler polynomials arising from non-linear differential equations. J. Number Theory 2012, 132(12):2854-2865. 10.1016/j.jnt.2012.05.033MathSciNetView ArticleMATHGoogle Scholar
  6. Robinson TI: Formal calculus and umbral calculus. Electron. J. Comb. 2010., 17(1): Article ID R95Google Scholar
  7. Bayad, A, Kim, T: Results on values of Barnes polynomials. Rocky Mt. J. Math. Forthcoming Articles (2013)Google Scholar
  8. Kim T: Symmetry p -adic invariant integral on Z p for Bernoulli and Euler polynomials. J. Differ. Equ. Appl. 2008, 14(279):1267-1277.View ArticleMATHGoogle Scholar
  9. Tremblay R, Gaboury S, Fugére B-J: Some new classes of generalized Apostol-Euler and Apostol-Genocchi polynomials. Int. J. Math. Math. Sci. 2012., 2012: Article ID 182785Google Scholar
  10. 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: 484-491. 10.1134/S1061920809040037MathSciNetView ArticleMATHGoogle Scholar
  11. Araci S, Acikgoz M: A note on the Frobenius-Euler numbers and polynomials associated with Bernstein polynomials. Adv. Stud. Contemp. Math. 2012, 22(3):399-406.MathSciNetMATHGoogle Scholar
  12. 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
  13. Kim T: An identity of the symmetry for the Frobenius-Euler polynomials associated with the fermionic p -adic invariant q -integrals on Z p . Rocky Mt. J. Math. 2011, 41: 239-247. 10.1216/RMJ-2011-41-1-239View ArticleMATHGoogle Scholar
  14. 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
  15. 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
  16. Carlitz L: The product of two Eulerian polynomials. Math. Mag. 1959, 23: 247-260.View ArticleGoogle Scholar
  17. Carlitz L: The product of two Eulerian polynomials. Math. Mag. 1963, 36: 37-41. 10.2307/2688134MathSciNetView ArticleMATHGoogle Scholar
  18. Can M, Cenkci M, Kurt V, Simsek Y: Twisted Dedekind type sums associated with Barne’s type multiple Frobenius-Euler L -functions. Adv. Stud. Contemp. Math. 2009, 18(2):135-160.MathSciNetMATHGoogle Scholar
  19. 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

Copyright

© Kim et al.; licensee Springer. 2013

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.