Open Access

Some identities of Frobenius-Euler polynomials arising from umbral calculus

Advances in Difference Equations20122012:196

https://doi.org/10.1186/1687-1847-2012-196

Received: 18 October 2012

Accepted: 5 November 2012

Published: 14 November 2012

Abstract

In this paper, we study some interesting identities of Frobenius-Euler polynomials arising from umbral calculus.

1 Introduction

Let C be the complex number field, and let F be the set of all formal power series in the variable t over C with
F = { f ( t ) = k = 0 a k k ! t k | a k C } .

We use notation P = C [ x ] and P denotes the vector space of all linear functional on .

Also, L | p ( x ) denotes the action of the linear functional L on the polynomial p ( x ) , and we remind that the vector space operations on P is defined by
L + M | p ( x ) = L | p ( x ) + M | p ( x ) , c L | p ( x ) = c L | p ( x ) ( see [1] ) ,

where c is any constant in C.

The formal power series
f ( t ) = k = 0 a k k ! t k F ( see [1, 2] ) ,
(1)
defines a linear functional on by setting
f ( t ) | x n = a n , for all  n 0 .
(2)
In particular,
t k | x n = n ! δ n , k ,
(3)

where δ n , k is the Kronecker symbol. If f L ( t ) = k = 0 L | x k k ! t k , then we get f L ( t ) | x n = L | x n and so as linear functionals L = f L ( t ) (see [1, 2]).

In addition, the map L f L ( t ) is a vector space isomorphism from P onto F (see [1, 2]). Henceforth, F will denote both the algebra of formal power series in t and the vector space of all linear functionals on , and so an element f ( t ) of F will be thought of as both a formal power series and a linear functional. We shall call F the umbral algebra (see [1, 2]).

Let us give an example. For y in C the evaluation functional is defined to be the power series e y t . From (2), we have e y t | x n = y n and so e y t | p ( x ) = p ( y ) (see [1, 2]). Notice that for all f ( t ) in F,
f ( t ) = k = 0 f ( t ) | x t k ! t k
(4)
and for all polynomial p ( x )
p ( x ) = k 0 t k | p ( x ) k ! x k ( see [1, 2] ) .
(5)
For f 1 ( t ) , f 2 ( t ) , , f m ( t ) F , we have
f 1 ( t ) f 2 ( t ) f m ( t ) | x n = ( n i 1 , , i m ) f 1 ( t ) | x i 1 f n ( t ) | x i m ,

where the sum is over all nonnegative integers i 1 , i 2 , , i m such that i 1 + + i m = n (see [1, 2]). The order o ( f ( t ) ) of the power series f ( t ) 0 is the smallest integer k for which a k does not vanish. We define o ( f ( t ) ) = if f ( t ) = 0 . We see that o ( f ( t ) g ( t ) ) = o ( f ( t ) ) + o ( g ( t ) ) and o ( f ( t ) + g ( t ) ) min { o ( f ( t ) ) , o ( g ( t ) ) } . The series f ( t ) has a multiplicative inverse, denoted by f ( t ) 1 or 1 f ( t ) , if and only if o ( f ( t ) ) = 0 . Such series is called an invertible series. A series f ( t ) for which o ( f ( t ) ) = 1 is called a delta series (see [1, 2]). For f ( t ) , g ( t ) F , we have f ( t ) g ( t ) | p ( x ) = f ( t ) | g ( t ) p ( x ) .

A delta series f ( t ) has a compositional inverse f ¯ ( t ) such that f ( f ¯ ( t ) ) = f ¯ ( f ( t ) ) = t .

For f ( t ) , g ( t ) F , we have f ( t ) g ( t ) | p ( x ) = f ( t ) | g ( t ) p ( x ) .

From (5), we have
p ( k ) ( x ) = d k p ( x ) d x k = l = k t l | p ( x ) l ! l ( l 1 ) ( l k + 1 ) x l k .
Thus, we see that
p ( k ) ( 0 ) = t k | p ( x ) = 1 | p ( k ) ( x ) .
(6)
By (6), we get
t k p ( x ) = p ( k ) ( x ) = d k ( p ( x ) ) d x k ( see [1, 2] ) .
(7)
By (7), we have
e y t p ( x ) = p ( x + y ) ( see [1, 2] ) .
(8)

Let S n ( x ) be a polynomial with deg S n ( x ) = n .

Let f ( t ) be a delta series, and let g ( t ) be an invertible series. Then there exists a unique sequence S n ( x ) of polynomials such that g ( t ) f ( t ) k | S n ( x ) = n ! δ n , k for all n , k 0 . The sequence S n ( x ) is called the Sheffer sequence for ( g ( t ) , f ( t ) ) or that S n ( t ) is Sheffer for ( g ( t ) , f ( t ) ) .

The Sheffer sequence for ( 1 , f ( t ) ) is called the associated sequence for f ( t ) or S n ( x ) is associated to f ( t ) . The Sheffer sequence for ( g ( t ) , t ) is called the Appell sequence for g ( t ) or S n ( x ) is Appell for g ( t ) (see [1, 2]). The umbral calculus is the study of umbral algebra and the modern classical umbral calculus can be described as a systemic study of the class of Sheffer sequences. Let p ( x ) P . Then we have
(9)
(10)
and
e y t 1 | p ( x ) = p ( y ) p ( 0 ) ( see [1, 2] ) .
(11)
Let S n ( x ) be Sheffer for ( g ( t ) , f ( t ) ) . Then
(12)
(13)
(14)
(15)
For λ ( 1 ) C , we recall that the Frobenius-Euler polynomials are defined by the generating function to be
1 λ e t λ e x t = e H ( x | λ ) t = n = 0 H n ( x | λ ) t n n ! ,
(16)
with the usual convention about replacing H n ( x | λ ) by H n ( x | λ ) (see [3]). In the special case, x = 0 , H n ( 0 | λ ) = H n ( λ ) are called the n th Frobenius-Euler numbers. By (16), we get
H n ( x | λ ) = ( H ( λ ) + x ) n = l = 0 n ( n l ) H n l ( λ ) x l ,
(17)
and
( H ( λ ) + 1 ) n λ H n ( λ ) = ( 1 λ ) δ 0 , n ( see [1, 4–13] ) .
(18)

From (17), we note that the leading coefficient of H n ( x | λ ) is H 0 ( λ ) = 1 . So, H n ( x | λ ) is a monic polynomial of degree n with coefficients in Q ( λ ) .

In this paper, we derive some new identities of Frobenius-Euler polynomials arising from umbral calculus.

2 Applications of umbral calculus to Frobenius-Euler polynomials

Let S n ( x ) be an Appell sequence for g ( t ) . From (14), we have
1 g ( t ) x n = S n ( x ) if and only if x n = g ( t ) S n ( x ) ( n 0 ) .
(19)

For λ ( 1 ) C , let us take g λ ( t ) = e t λ 1 λ F .

Then we see that g λ ( t ) is an invertible series.

From (16), we have
k = 0 H k ( x | λ ) k ! t k = 1 g λ ( t ) e x t .
(20)
By (20), we get
1 g λ ( t ) x n = H n ( x | λ ) ( λ ( 1 ) C , n 0 ) ,
(21)
and by (17), we get
t H n ( x | λ ) = H n ( x | λ ) = n H n 1 ( x | λ ) .
(22)

Therefore, by (21) and (22), we obtain the following proposition.

Proposition 1 For λ ( 1 ) C , n 0 , we see that H n ( x | λ ) is the Appell sequence for g λ ( t ) = e t λ 1 λ .

From (20), we have
k = 1 H k ( x | λ ) k ! k t k 1 = x g λ ( t ) e x t g λ ( t ) e x t g λ ( t ) 2 = k = 0 { x 1 g λ ( t ) x k g λ ( t ) g λ ( t ) 1 g λ ( t ) x k } t k k ! .
(23)
By (21) and (23), we get
H k + 1 ( x | λ ) = x H k ( x | λ ) g λ ( t ) g λ ( t ) H k ( x | λ ) .
(24)

Therefore, by (24) we obtain the following theorem.

Theorem 2 Let g λ ( t ) = e t λ 1 λ F . Then we have
H k + 1 ( x | λ ) = ( x g λ ( t ) g λ ( t ) ) H k ( x | λ ) ( k 0 ) .
From (16), we have
n = 0 ( H n ( x + 1 | λ ) λ H n ( x | λ ) ) t n n ! = 1 λ e t λ e ( x + 1 ) t λ 1 λ e t λ e x t = ( 1 λ ) e x t .
(25)
By (25), we get
H n ( x + 1 | λ ) λ H n ( x | λ ) = ( 1 λ ) x n .
(26)
From Theorem 2, we can derive the following equation (27):
g λ ( t ) H k + 1 ( x | λ ) = ( g λ ( t ) x g λ ( t ) ) H k ( x | λ ) .
(27)
By (27), we get
( e t λ 1 λ ) H k + 1 ( x | λ ) = e t λ 1 λ x H k ( x | λ ) e t 1 λ H k ( x | λ ) .
(28)
From (8) and (28), we have
H k + 1 ( x + 1 | λ ) λ H k + 1 ( x | λ ) = ( x + 1 ) H k ( x + 1 | λ ) λ x H k ( x | λ ) H k ( x + 1 | λ ) = x H k ( x + 1 | λ ) λ x H k ( x | λ ) .

Therefore, by (26), we obtain the following theorem.

Theorem 3 For k 0 , we have
H k + 1 ( x + 1 | λ ) = λ H k + 1 ( x | λ ) + ( 1 λ ) x k + 1 .
From (16), (17), and (18), we note that
x x + y H n ( u | λ ) d u = 1 n + 1 { H n + 1 ( x + y | λ ) H n + 1 ( x | λ ) } = 1 n + 1 k = 1 ( n + 1 k ) H n + 1 k ( x | λ ) y k = k = 1 n ( n 1 ) ( n k + 2 ) k ! H n + 1 k ( x | λ ) y k = k = 1 y k k ! t k 1 H n ( x | λ ) = 1 t ( k = 0 y k k ! t k 1 ) H n ( x | λ ) = e y t 1 t H n ( x | λ ) .
(29)

Therefore, by (29), we obtain the following theorem.

Theorem 4 For λ ( 1 ) C , n 0 , we have
x x + y H n ( u | λ ) d u = e y t 1 t H n ( x | λ ) .
By (15) and Proposition 1, we get
t { 1 n + 1 H n + 1 ( x | λ ) } = H n ( x | λ ) .
(30)
From (30), we can derive equation (31):
e y t 1 | H n + 1 ( x | λ ) n + 1 = e y t 1 t | t { H n + 1 ( x | λ ) n + 1 } = e y t 1 t | H n ( x | λ ) .
(31)
By (11) and (31), we get
e y t 1 t | H n ( x | λ ) = e y t 1 | H n + 1 ( x | λ ) n + 1 = 1 n + 1 { H n + 1 ( y | λ ) H n + 1 ( λ ) } = 0 y H n ( u | λ ) d u .
(32)

Therefore, by (32), we obtain the following corollary.

Corollary 5 For n 0 , we have
e y t 1 t | H n ( x | λ ) = 0 y H n ( u | λ ) d u .

Let P ( λ ) = { p ( x ) Q ( λ ) [ x ] | deg p ( x ) n } be a vector space over Q ( λ ) .

For p ( x ) P n ( λ ) , let us take
p ( x ) = k = 0 n b k H k ( x | λ ) .
(33)
By Proposition 1, H n ( x | λ ) is an Appell sequence for g λ ( t ) = e t λ 1 λ where λ ( 1 ) C . Thus, we have
e t λ 1 λ t k | H n ( x | λ ) = n ! δ n , k .
(34)
From (33) and (34), we can derive
e t λ 1 λ t k | p ( x ) = l = 0 n b l e t λ 1 λ t k | H l ( x | λ ) = l = 0 n b l l ! δ l , k = k ! b k .
(35)
Thus, by (35), we get
b k = 1 k ! e t λ 1 λ t k | p ( x ) = 1 k ! ( 1 λ ) ( e t λ ) t k | p ( x ) = 1 k ! ( 1 λ ) e t λ | p ( k ) ( x ) .
(36)
From (11) and (36), we have
b k = 1 k ! ( 1 λ ) { p ( k ) ( 1 ) λ p ( k ) ( 0 ) } ,
(37)

where p ( k ) ( x ) = d k p ( x ) d x k .

Therefore, by (37), we obtain the following theorem.

Theorem 6 For p ( x ) P n ( λ ) , let us assume that p ( x ) = k = 0 n b k H k ( x | λ ) . Then we have
b k = 1 k ! ( 1 λ ) { p ( k ) ( 1 ) λ p ( k ) ( 0 ) } ,

where p ( k ) ( 1 ) = d k p ( x ) d x k | x = 1 .

The higher-order Frobenius-Euler polynomials are defined by
( 1 λ e t λ ) r e x t = n = 0 H n ( r ) ( x | λ ) t n n ! ,
(38)

where λ ( 1 ) C and r N (see [4, 11]).

In the special case, x = 0 , H n ( r ) ( 0 | λ ) = H n ( r ) ( λ ) are called the n th Frobenius-Euler numbers of order r. From (38), we have
H n ( r ) ( x ) = l = 0 n ( n l ) H n l ( r ) ( λ ) x l = n 1 + + n r = n ( n n 1 , , n r ) H n 1 ( x | λ ) H n r ( x | λ ) .
(39)

Note that H n ( r ) ( x | λ ) is a monic polynomial of degree n with coefficients in Q ( λ ) .

For r N , λ ( 1 ) C , let g λ r ( t ) = ( e t λ 1 λ ) r . Then we easily see that g λ r ( t ) is an invertible series.

From (38) and (39), we have
1 g λ r ( t ) e x t = n = 0 H n ( r ) ( x | λ ) t n n ! ,
(40)
and
t H n ( r ) ( x | λ ) = n H n 1 ( r ) ( x | λ ) .
(41)
By (40), we get
1 g λ r ( t ) x n = H n ( r ) ( x | λ ) ( n Z + , r N ) .
(42)

Therefore, by (41) and (42), we obtain the following proposition.

Proposition 7 For n Z + , H n ( r ) ( x | λ ) is an Appell sequence for
g λ r ( t ) = ( e t λ 1 λ ) r .
Moreover,
1 g λ r ( t ) x n = H n ( r ) ( x | λ ) and t H n ( r ) ( x | λ ) = n H n 1 ( r ) ( x | λ ) .
Remark Note that
1 λ e t λ | x n = H n ( λ ) .
(43)
From (43), we have
(44)
(45)
By (43), (44), and (45), we get
n = i 1 + + i r ( n i 1 , , i r ) H i 1 ( λ ) H i r ( λ ) = H n ( r ) ( λ ) .
Let us take p ( x ) P n ( λ ) with
p ( x ) = k = 0 n C k ( r ) H k ( r ) ( x | λ ) .
(46)
From the definition of Appell sequences, we have
( e t λ 1 λ ) r | H n ( r ) ( x | λ ) = n ! δ n , k .
(47)
By (46) and (47), we get
( e t λ 1 λ ) r t k | p ( x ) = l = 0 n C l ( r ) ( e t λ 1 λ ) r t k | H l ( x | λ ) = l = 0 n C l ( r ) l ! δ l , k = k ! C k ( r ) .
(48)
Thus, from (48), we have
C k ( r ) = 1 k ! ( e t λ 1 λ ) r t k | p ( x ) = 1 k ! ( 1 λ ) r ( e t λ ) r t k | p ( x ) = 1 k ! ( 1 λ ) r l = 0 r ( r l ) ( λ ) r l e l t | p ( k ) ( x ) = 1 k ! ( 1 λ ) r l = 0 r ( r l ) ( λ ) r l p ( k ) ( l ) .
(49)

Therefore, by (46) and (49), we obtain the following theorem.

Theorem 8 For p ( x ) P n ( λ ) , let
p ( x ) = k = 0 n C k ( r ) H k ( r ) ( x | λ ) .
Then we have
C k ( r ) = 1 k ! ( 1 λ ) r l = 0 r ( r l ) ( λ ) r l p ( k ) ( l ) ,

where r N and p ( k ) ( l ) = d k p ( x ) d x k | x = l .

Remark Let S n ( x ) be a Sheffer sequence for ( g ( t ) , f ( t ) ) . Then Sheffer identity is given by
S n ( x + y ) = k = 0 n ( n k ) P k ( y ) S n k ( x ) = k = 0 n ( n k ) P k ( x ) S n k ( y ) ,
(50)

where P k ( y ) = g ( t ) S k ( y ) is associated to f ( t ) (see [1, 2]).

From (21), Proposition 1, and (50), we have
H n ( x + y | λ ) = k = 0 n ( n k ) P k ( y ) S n k ( x ) = k = 0 n ( n k ) H n k ( y | λ ) x k .
By Proposition 7 and (50), we get
H n ( r ) ( x + y | λ ) = k = 0 n ( n k ) H n k ( r ) ( y | λ ) x k .
Let α ( 0 ) C . Then we have
H n ( α x | λ ) = α n g λ ( t ) g λ ( t α ) H n ( x | λ ) .

Declarations

Acknowledgements

This research was supported by the 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
(2)
Department of Mathematics, Kwangwoon University

References

  1. Roman S: The Umbral Calculus. Dover, New York; 2005.Google Scholar
  2. Dere R, Simsek Y: Applications of umbral algebra to some special polynomials. Adv. Stud. Contemp. Math. 2012, 22(3):433–438.MathSciNetGoogle Scholar
  3. 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.MathSciNetGoogle Scholar
  4. 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 ArticleGoogle Scholar
  5. Kim T, Choi J: A note on the product of Frobenius-Euler polynomials arising from the p -adic integral on Z p . Adv. Stud. Contemp. Math. 2012, 22(2):215–223.MathSciNetGoogle Scholar
  6. 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 ArticleGoogle Scholar
  7. 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.MathSciNetGoogle Scholar
  8. Rim S-H, Lee J: Some identities on the twisted ( h , q ) -Geonocchi numbers and polynomials associated with q -Bernstein polynomials. Int. J. Math. Math. Sci. 2011., 2011: Article ID 482840Google Scholar
  9. Simsek Y, Yurekli O, Kurt V: On interpolation functions of the twisted generalized Frobenius-Euler numbers. Adv. Stud. Contemp. Math. 2007, 15(2):187–194.MathSciNetGoogle Scholar
  10. Ryoo CS: A note on the Frobenius-Euler polynomials. Proc. Jangjeon Math. Soc. 2011, 14(4):495–501.MathSciNetGoogle Scholar
  11. Simsek Y, Bayad A, Lokesha V: q -Bernstein polynomials related to q -Frobenius-Euler polynomials, l -functions, and q -Stirling numbers. Math. Methods Appl. Sci. 2012, 35(8):877–884. 10.1002/mma.1580MathSciNetView ArticleGoogle Scholar
  12. Shiratani K: On the Euler numbers. Mem. Fac. Sci., Kyushu Univ., Ser. A, Math. 1973, 27: 1–5. 10.2206/kyushumfs.27.1MathSciNetGoogle Scholar
  13. Shiratani K, Yamamoto S: On a p -adic interpolation function for the Euler numbers and its derivatives. Mem. Fac. Sci., Kyushu Univ., Ser. A, Math. 1985, 39(1):113–125. 10.2206/kyushumfs.39.113MathSciNetGoogle Scholar

Copyright

© Kim and Kim; 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.