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Theory and Modern Applications

Apostol-Euler polynomials arising from umbral calculus

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.

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 LH 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 n0(see [1–4]).
(1.2)

By (1.1) and (1.2), we have

t k | x n =n! δ n , k for all n,k0(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 LH. 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,k0. 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 yC, where f ¯ (t) is the compositional inverse of f(t) (see [16]).

For λC with λ1, 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 λC with λ1, 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 n0. 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 n0,

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 n2,

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 n0,

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 n1,

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 n1,

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.

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Acknowledgements

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

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Correspondence to Taekyun Kim.

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Kim, T., Mansour, T., Rim, SH. et al. Apostol-Euler polynomials arising from umbral calculus. Adv Differ Equ 2013, 301 (2013). https://doi.org/10.1186/1687-1847-2013-301

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