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Higher-order Frobenius-Euler and poly-Bernoulli mixed-type polynomials

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Abstract

In this paper, we consider higher-order Frobenius-Euler polynomials, associated with poly-Bernoulli polynomials, which are derived from polylogarithmic function. These polynomials are called higher-order Frobenius-Euler and poly-Bernoulli mixed-type polynomials. The purpose of this paper is to give various identities of those polynomials arising from umbral calculus.

1 Introduction

For λC with λ1, the Frobenius-Euler polynomials of order α (αR) are defined by the generating function to be

( 1 λ e t λ ) α e x t = n = 0 H n ( α ) (x|λ) t n n ! (see [1–5]).
(1.1)

When x=0, H n ( α ) (λ)= H n ( α ) (0|λ) are called the Frobenius-Euler numbers of order α. As is well known, the Bernoulli polynomials of order α are defined by the generating function to be

( t e t 1 ) α e x t = n = 0 B n ( α ) (x) t n n ! (see [6–8]).
(1.2)

When x=0, B n ( α ) = B n ( α ) (x) is called the n th Bernoulli number of order α. In the special case, α=1, B n ( 1 ) (x)= B n (x) is called the n th Bernoulli polynomial. When x=0, B n = B n (0) is called the n th ordinary Bernoulli number. Finally, we recall that the Euler polynomials of order α are given by

( 2 e t + 1 ) α e x t = n = 0 E n ( α ) (x) t n n ! (see [9–13]).
(1.3)

When x=0, E n ( α ) = E n ( α ) (0) is called the n th Euler number of order α. In the special case, α=1, E n ( 1 ) (x)= E n (x) is called the n th ordinary Euler polynomial. The classical polylogarithmic function L i k (x) is defined by

L i k (x)= n = 1 x n n k (kZ)(see [7]).
(1.4)

As is known, poly-Bernoulli polynomials are defined by the generating function to be

L i k ( 1 e t ) 1 e t e x t = n = 0 B n ( k ) (x) t n n ! (cf. [7]).
(1.5)

Let be the complex number field, and let be the set of all formal power series in the variable t over with

F= { f ( t ) = k = 0 a k k ! t k | a k C } .
(1.6)

Now, we use the notation P=C[x]. In this paper, P will be denoted by the vector space of all linear functionals on . Let us assume that L|p(x) be the action of the linear functional L on the polynomial p(x), and we remind that the vector space operations on P are defined by L+M|p(x)=L|p(x)+M|p(x), cL|p(x)=cL|p(x), where c is a complex constant in . The formal power series

f(t)= k = 0 a k k ! t k F
(1.7)

defines a linear functional on by setting

f ( t ) | x n = a n ,for all n0(see [14, 15]).
(1.8)

From (1.7) and (1.8), we note that

t k | x n =n! δ n , k (see [14, 15]),
(1.9)

where δ n , k is the Kronecker symbol.

Let us consider f L (t)= k = 0 L | x n k ! t k . Then we see that f L (t)| x n =L| x n , and so L= f L (t) as linear functionals. The map L f L (t) is a vector space isomorphism from P onto . Henceforth, 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 will be thought of as both a formal power series and a linear functional (see [14]). We shall call the umbral algebra. The umbral calculus is the study of umbral algebra. The order o(f(t)) of a nonzero power series f(t) is the smallest integer k, for which the coefficient of t k does not vanish. A series f(t) is called a delta series if o(f(t))=1, and an invertible series if o(f(t))=0. Let f(t),g(t)F. Then we have

f ( t ) g ( t ) | p ( x ) = f ( t ) | g ( t ) p ( x ) = g ( t ) | f ( t ) p ( x ) (see [14]).
(1.10)

For f(t),g(t)F with o(f(t))=1, o(g(t))=0, there exists a unique sequence S n (x) (deg S n (x)=n) such that g(t)f ( t ) k | S n (x)=n! δ n , k for 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, 15]). Let f(t)F and p(t)P. Then we have

f(t)= k = 0 f ( t ) | x k t k k ! ,p(x)= k = 0 t k | p ( x ) x k k ! .
(1.11)

From (1.11), we note that

p ( k ) (0)= t k | p ( x ) = 1 | p ( k ) ( x ) .
(1.12)

By (1.12), we get

t k p(x)= p ( k ) (x)= d k p ( x ) d x k (see [14, 15]).
(1.13)

From (1.13), we easily derive the following equation

e y t p(x)=p(x+y), e y t | p ( x ) =p(y).
(1.14)

For p(x)P, f(t)F, it is known that

f ( t ) | x p ( x ) = t f ( t ) | p ( x ) = f ( t ) | p ( x ) (see [14]).
(1.15)

Let S n (x)(g(t),f(t)). Then we have

1 g ( f ¯ ( x ) ) e y f ¯ ( t ) = n = 0 S n (y) t n n ! for all yC,
(1.16)

where f ¯ (t) is the compositional inverse of f(t) with f ¯ (f(t))=t, and

f(t) S n (x)=n S n 1 (x)(see [14, 15]).
(1.17)

The Stirling number of the second kind is defined by the generating function to be

( e t 1 ) m =m! l = m S 2 (l,m) t m m ! (m Z 0 ).
(1.18)

For S n (x)(g(t),t), it is well known that

S n + 1 (x)= ( x g ( t ) g ( t ) ) S n (x)(n0)(see [14, 15]).
(1.19)

Let S n (x)(g(t),f(t)), r n (x)(h(t),l(t)). Then we have

S n (x)= m = 0 n C n , m r m (x),
(1.20)

where

C n , m = 1 m ! h ( f ¯ ( t ) ) g ( f ¯ ( t ) ) l ( f ¯ ( t ) ) m | x n (see [14, 15]).
(1.21)

In this paper, we study higher-order Frobeniuns-Euler polynomials associated with poly-Bernoulli polynomials, which are called higher-order Frobenius-Euler and poly-Beroulli mixed-type polynomials. The purpose of this paper is to give various identities of those polynomials arising from umbral calculus.

2 Higher-order Frobenius-Euler polynomials, associated poly-Bernoulli polynomials

Let us consider the polynomials T n ( r , k ) (x|λ), called higher-order Frobenius-Euler and poly-Bernoulli mixed-type polynomials, as follows:

( 1 λ e t λ ) r L i k ( 1 e t ) 1 e t e x t = n = 0 T n ( r , k ) (x|λ) t n n ! ,
(2.1)

where λC with λ1, r,kZ.

When x=0, T n ( r , k ) (λ)= T n ( r , k ) (0|λ) is called the n th higher-order Frobenius-Euler and poly-Bernoulli mixed type number.

From (1.16) and (2.1), we note that

T n ( r , k ) (x|λ) ( g r , k ( t ) = ( e t λ 1 λ ) r 1 e t L i k ( 1 e t ) , t ) .
(2.2)

By (1.17) and (2.2), we get

t T n ( r , k ) (x|λ)=n T n 1 ( r , k ) (x|λ).
(2.3)

From (2.1), we can easily derive the following equation

T n ( r , k ) ( x | λ ) = l = 0 n ( n l ) H n l ( r ) ( λ ) B l ( k ) ( x ) = l = 0 n ( n l ) H n l ( r ) ( x | λ ) B l ( k ) .
(2.4)

By (1.16) and (2.2), we get

T n ( r , k ) (x|λ)= 1 g r , k ( t ) x n = ( 1 λ e t λ ) r L i k ( 1 e t ) 1 e t x n .
(2.5)

In [7], it is known that

L i k ( 1 e t ) 1 e t x n = m = 0 n 1 ( m + 1 ) k j = 0 m ( 1 ) j ( m j ) ( x j ) n .
(2.6)

Thus, by (2.5) and (2.6), we get

T n ( r , k ) ( x | λ ) = ( 1 λ e t λ ) r L i k ( 1 e t ) 1 e t x n = m = 0 1 ( m + 1 ) k j = 0 m ( 1 ) j ( m j ) ( 1 λ e t λ ) r ( x j ) n = m = 0 n 1 ( m + 1 ) k j = 0 m ( 1 ) j ( m j ) H n ( r ) ( x j | λ ) .
(2.7)

By (1.1), we easily see that

H n ( r ) (x|λ)= l = 0 n ( n l ) H n l ( r ) (λ) x l .
(2.8)

Therefore, by (2.7) and (2.8), we obtain the following theorem.

Theorem 2.1 For r,kZ, n0, we have

T n ( r , k ) ( x | λ ) = m = 0 n 1 ( m + 1 ) k j = 0 m ( 1 ) j ( m j ) l = 0 n ( n l ) H n l ( r ) ( λ ) ( x j ) l = l = 0 n { ( n l ) H n l ( r ) ( λ ) m = 0 1 ( m + 1 ) k j = 0 m ( 1 ) j ( m j ) } ( x j ) l .

In [7], it is known that

L i k ( 1 e t ) 1 e t x n = j = 0 n { m = 0 n j ( 1 ) n m j ( m + 1 ) k ( n j ) m ! S 2 ( n j , m ) } x j .
(2.9)

By (2.5) and (2.9), we get

T n ( r , k ) ( x | λ ) = ( 1 λ e t λ ) r L i k ( 1 e t ) 1 e t x n = j = 0 n { m = 0 n j ( 1 ) n m j ( m + 1 ) k ( n j ) m ! S 2 ( n j , m ) } ( 1 λ e t λ ) r x j = j = 0 n { m = 0 n j ( 1 ) n m j ( m + 1 ) k ( n j ) m ! S 2 ( n j , m ) } H j ( r ) ( x | λ ) .
(2.10)

Therefore, by (2.8) and (2.10), we obtain the following theorem.

Theorem 2.2 For r,kZ, n Z 0 , we have

T n ( r , k ) (x|λ)= l = 0 n { j = l n m = 0 n j ( 1 ) n m j ( n j ) ( j l ) m ! ( m + 1 ) k H j l ( r ) ( λ ) S 2 ( n j , m ) } x l .

From (1.19) and (2.2), we have

T n + 1 ( r , k ) (x|λ)= ( x g r , k ( t ) g r , k ( t ) ) T n ( r , k ) (x|λ).
(2.11)

Now, we note that

g r , k ( t ) g r , k ( t ) = ( log g r , k ( t ) ) = ( r log ( e t λ ) r log ( 1 λ ) + log ( 1 e t ) log L i k ( 1 e t ) ) = r + r λ e t λ + ( t e t 1 ) L i k ( 1 e t ) L i k 1 ( 1 e t ) t L i k ( 1 e t ) .
(2.12)

By (2.11) and (2.12), we get

T n + 1 ( r , k ) ( x | λ ) = x T n ( r , k ) ( x | λ ) r T n ( r , k ) ( x | λ ) r λ 1 λ ( 1 λ e t λ ) r + 1 L i k ( 1 e t ) 1 e t x n ( 1 λ e t λ ) r L i k ( 1 e t ) L i k 1 ( 1 e t ) t ( 1 e t ) ( t e t 1 ) x n = ( x r ) T n ( r , k ) ( x | λ ) r λ 1 λ T n ( r + 1 , k ) ( x | λ ) l = 0 n ( n l ) B n l ( 1 λ e t λ ) r L i k ( 1 e t ) L i k 1 ( 1 e t ) t ( 1 e t ) x l .
(2.13)

It is easy to show that

L i k ( 1 e t ) L i k 1 ( 1 e t ) 1 e t = 1 1 e t n = 1 { ( 1 e t ) n n k ( 1 e t ) n n k 1 } = ( 1 e t 2 k 1 e t 2 k 1 ) + = ( 1 2 k 1 2 k 1 ) t + .
(2.14)

For any delta series f(t), we have

f ( t ) t x n =f(t) 1 n + 1 x n + 1 .
(2.15)

Thus, by (2.13), (2.14) and (2.15), we get

T n + 1 ( r , k ) ( x | λ ) = ( x r ) T n ( r , k ) ( x | λ ) r λ 1 λ T n ( r + 1 , k ) ( x | λ ) l = 0 n ( n l ) B n l 1 l + 1 ( 1 λ e t λ ) r L i k ( 1 e t ) L i k 1 ( 1 e t ) 1 e t x l + 1 = ( x r ) T n ( r , k ) ( x | λ ) r λ 1 λ T n ( r + 1 , k ) ( x | λ ) l = 0 n ( n l ) l + 1 B n l { T l + 1 ( r , k ) ( x | λ ) T l + 1 ( r , k 1 ) ( x | λ ) } = ( x r ) T n ( r , k ) ( x | λ ) r λ 1 λ T n ( r + 1 , k ) ( x | λ ) 1 n + 1 l = 1 n + 1 ( n + 1 l ) B n + 1 l { T l ( r , k ) ( x | λ ) T l ( r , k 1 ) ( x | λ ) } = ( x r ) T n ( r , k ) ( x | λ ) r λ 1 λ T n ( r + 1 , k ) ( x | λ ) 1 n + 1 l = 0 n + 1 ( n + 1 l ) B n + 1 l { T l ( r , k ) ( x | λ ) T l ( r , k 1 ) ( x | λ ) } = ( x r ) T n ( r , k ) ( x | λ ) r λ 1 λ T n ( r + 1 , k ) ( x | λ ) 1 n + 1 l = 0 n + 1 ( n + 1 l ) B l { T n + 1 l ( r , k ) ( x | λ ) T n + 1 l ( r , k 1 ) ( x | λ ) } .
(2.16)

Therefore, by (2.16), we obtain the following theorem.

Theorem 2.3 For r,kZ, n Z 0 , we have

T n + 1 ( r , k ) ( x | λ ) = ( x r ) T n ( r , k ) ( x | λ ) r λ 1 λ T n ( r + 1 , k ) ( x | λ ) 1 n + 1 l = 0 n + 1 ( n + 1 l ) B l { T n + 1 l ( r , k ) ( x | λ ) T n + 1 l ( r , k 1 ) ( x | λ ) } .

Remark 1 If r=0, then we have

n = 0 B n ( k ) (x) t n n ! = L i k ( 1 e t ) ( 1 e t ) e x t = n = 0 T n ( 0 , k ) (x|λ) t n n ! .
(2.17)

Thus, by (2.17), we get B n ( k ) (x)= T n ( 0 , k ) (x|λ).

From (2.4), we have

t x T n ( r , k ) ( x | λ ) = t ( x l = 0 n ( n l ) H n l ( r ) ( λ ) B l ( k ) ( x ) ) = l = 0 n ( n l ) H n l ( r ) ( λ ) { l x B l 1 ( k ) ( x ) + B l ( k ) ( x ) } = n x l = 0 n 1 ( n 1 l ) H n 1 l ( r ) ( λ ) B l ( k ) ( x ) + l = 0 n ( n l ) H n l ( r ) ( λ ) B l ( k ) ( x ) = n x T n 1 ( r , k ) ( x | λ ) + T n ( r , k ) ( x | λ ) .
(2.18)

Applying t on both sides of Theorem 2.3, we get

( n + 1 ) T n ( r , k ) ( x | λ ) = n x T n 1 ( r , k ) ( x | λ ) + T n ( r , k ) ( x | λ ) r n T n 1 ( r , k ) ( x | λ ) r n λ 1 λ T n 1 ( r + 1 , k ) ( x | λ ) 1 n + 1 l = 0 n + 1 ( n + 1 l ) B l { ( n + 1 l ) T n l ( r , k ) ( x | λ ) ( n + 1 l ) T n l ( r , k 1 ) ( x | λ ) } .
(2.19)

Thus, by (2.19), we have

( n + 1 ) T n ( r , k ) ( x | λ ) + n ( r 1 2 x ) T n 1 ( r , k ) ( x | λ ) + l = 0 n 2 ( n l ) B n l T l ( r , k ) ( x | λ ) = r λ n 1 λ T n 1 ( r + 1 , k ) ( x | λ ) + l = 0 n ( n l ) B n l T l ( r , k 1 ) ( x | λ ) .
(2.20)

Therefore, by (2.20), we obtain the following theorem.

Theorem 2.4 For r,kZ, nZ with n2, we have

( n + 1 ) T n ( r , k ) ( x | λ ) + n ( r 1 2 x ) T n 1 ( r , k ) ( x | λ ) + l = 0 n 2 ( n l ) B n l T l ( r , k ) ( x | λ ) = r λ n 1 λ T n 1 ( r + 1 , k ) ( x | λ ) + l = 0 n ( n l ) B n l T l ( r , k 1 ) ( x | λ ) .

From (1.14) and (2.5), we note that

T n ( r , k ) ( y | λ ) = ( 1 λ e t λ ) r L i k ( 1 e t ) 1 e t e y t | x n = ( 1 λ e t λ ) r L i k ( 1 e t ) 1 e t e y t | x x n 1 .
(2.21)

By (1.15) and (2.21), we get

T n ( r , k ) ( y | λ ) = t ( ( 1 λ e t λ ) r L i k ( 1 e t ) 1 e t e y t ) | x n 1 = ( t ( 1 λ e t λ ) r ) L i k ( 1 e t ) 1 e t e y t | x n 1 + ( 1 λ e t λ ) r ( t L i k ( 1 e t ) 1 e t ) e y t | x n 1 + ( 1 λ e t λ ) r L i k ( 1 e t ) 1 e t t e y t | x n 1 .
(2.22)

Therefore, by (2.22), we obtain the following theorem.

Theorem 2.5 For r,kZ, n1, we have

T n ( r , k ) ( x | λ ) = ( x r ) T n 1 ( r , k ) ( x | λ ) r λ 1 λ T n 1 ( r + 1 , k ) ( x | λ ) + l = 0 n 1 { ( 1 ) n 1 l ( n 1 l ) m = 0 n 1 l ( 1 ) m ( m + 1 ) ! ( m + 2 ) k S 2 ( n 1 l , m ) } H l ( r ) ( x 1 | λ ) .

Now, we compute ( 1 λ e t λ ) r L i k (1 e t )| x n + 1 in two different ways.

On the one hand,

( 1 λ e t λ ) r L i k ( 1 e t ) | x n + 1 = ( 1 λ e t λ ) r L i k ( 1 e t ) 1 e t | ( 1 e t ) x n + 1 = ( 1 λ e t λ ) r L i k ( 1 e t ) 1 e t | x n + 1 ( x 1 ) n + 1 = m = 0 n ( n + 1 m ) ( 1 ) n m ( 1 λ e t λ ) r L i k ( 1 e t ) 1 e t | x m = m = 0 n ( n + 1 m ) ( 1 ) n m 1 | T m ( r , k ) ( x | λ ) = m = 0 n ( n + 1 m ) ( 1 ) n m T m ( r , k ) ( λ ) .
(2.23)

On the other hand, we get

( 1 λ e t λ ) r L i k ( 1 e t ) | x n + 1 = L i k ( 1 e t ) | ( 1 λ e t λ ) r x n + 1 = 0 t ( L i k ( 1 e s ) ) d s | H n + 1 ( r ) ( x | λ ) = 0 t e s L i k ( 1 e s ) ( 1 e s ) d s | H n + 1 ( r ) ( x | λ ) = l = 0 n ( m = 0 l ( l m ) ( 1 ) l m B m ( k 1 ) ) 1 l ! 0 t s l d s | H n + 1 ( r ) ( x | λ ) = l = 0 n m = 0 l ( l m ) ( 1 ) l m B m ( k 1 ) ( l + 1 ) ! t l + 1 | H n + 1 ( r ) ( x | λ ) = l = 0 n m = 0 l ( l m ) ( n + 1 l + 1 ) ( 1 ) l m B m ( k 1 ) H n l ( r ) ( λ ) .
(2.24)

Therefore, by (2.23) and (2.24), we obtain the following theorem.

Theorem 2.6 For r,kZ, n Z 0 , we have

m = 0 n ( n + 1 m ) ( 1 ) n m T m ( r , k ) ( λ ) = l = 0 n m = 0 l ( 1 ) l m ( l m ) ( n + 1 l + 1 ) B m ( k 1 ) H n l ( r ) ( λ ) .

Now, we consider the following two Sheffer sequences:

T n ( r , k ) ( x | λ ) ( ( e t λ 1 λ ) r 1 e t L i k ( 1 e t ) , t ) , B ( s ) ( ( e t 1 t ) s , t ) ,
(2.25)

where s Z 0 , r,kZ and λC with λ1. Let us assume that

T n ( r , k ) (x|λ)= m = 0 n C n m B m ( s ) (x).
(2.26)

By (1.21) and (2.26), we get

C n , m = 1 m ! ( e t 1 t ) s ( 1 λ e t λ ) r L i k ( 1 e t ) 1 e t t m | x n = 1 m ! ( e t 1 t ) s ( 1 λ e t λ ) r L i k ( 1 e t ) 1 e t | t m x n = ( n m ) ( e t 1 t ) s ( 1 λ e t λ ) r L i k ( 1 e t ) 1 e t | x n m = ( n m ) l = 0 n m s ! ( l + s ) ! S 2 ( l + s , s ) ( 1 λ e t λ ) r L i k ( 1 e t ) 1 e t | t l x n m = ( n m ) l = 0 n m s ! l ! ( l + s ) ! ( n m ) l l ! S 2 ( l + s , s ) 1 | T n m l ( r , k ) ( x | λ ) = ( n m ) l = 0 n m ( n m l ) ( s + l l ) S 2 ( l + s , s ) T n m l ( r , k ) ( λ ) .
(2.27)

Therefore, by (2.26) and (2.27), we obtain the following theorem.

Theorem 2.7 For r,kZ, s Z 0 , we have

T n ( r , k ) (x|λ)= m = 0 n { ( n m ) l = 0 n m ( n m l ) ( s + l l ) S 2 ( l + s , s ) T n m l ( r , k ) ( λ ) } B m ( s ) (x).

From (1.3) and (2.1), we note that

T n ( r , k ) ( x | λ ) ( ( e t λ 1 λ ) r 1 e t L i k ( 1 e t ) , t ) , E n ( r , s ) ( x ) ( ( e t + 1 2 ) s , t ) ,
(2.28)

where r,kZ, s Z 0 .

By the same method, we get

T n ( r , k ) (x|λ)= 1 2 s m = 0 n { ( n m ) j = 0 s ( s j ) T n m ( r , k ) ( j ) } E m ( s ) (x).
(2.29)

From (1.1) and (2.1), we note that

T n ( r , k ) ( x | λ ) ( ( e t λ 1 λ ) r 1 e t L i k ( 1 e t ) , t ) , H n ( s ) ( x | μ ) ( ( e t μ 1 μ ) s , t ) ,
(2.30)

where r,kZ, and λ,μC with λ1, μ1, s Z 0 .

Let us assume that

T n ( r , k ) (x|λ)= m = 0 n C n , m H m ( s ) (x|μ).
(2.31)

By (1.21) and (2.31), we get

C n , m = 1 m ! ( e t μ 1 μ ) s ( 1 λ e t λ ) r L i k ( 1 e t ) 1 e t t m | x n = ( n m ) ( 1 μ ) s ( e t μ ) s | ( 1 λ e t λ ) r L i k ( 1 e t ) 1 e t x n m = ( n m ) ( 1 μ ) s j = 0 s ( s j ) ( μ ) s j e j t | T n m ( r , k ) ( x | λ ) = ( n m ) ( 1 μ ) s j = 0 s ( s j ) ( μ ) s j T n m ( r , k ) ( j | λ ) .
(2.32)

Therefore, by (2.31) and (2.32), we obtain the following theorem.

Theorem 2.8 For r,kZ, s Z 0 , we have

T n ( r , k ) (x|λ)= 1 ( 1 μ ) s m = 0 n { ( n m ) j = 0 s ( s j ) ( μ ) s j T n m ( r , k ) ( j | λ ) } H m ( s ) (x|μ).

It is known that

T n ( r , k ) ( x | λ ) ( ( e t λ 1 λ ) r 1 e t L i k ( 1 e t ) , t ) , ( x ) n ( 1 , e t 1 ) .
(2.33)

Let

T n ( r , k ) (x|λ)= m = 0 n C n , m ( x ) m .
(2.34)

Then, by (1.21) and (2.34), we get

C n , m = 1 m ! ( 1 λ e t λ ) r L i k ( 1 e t ) 1 e t ( e t 1 ) m | x n = l = 0 S 2 ( l + m , m ) ( l + m ) ! ( 1 λ e t λ ) r L i k ( 1 e t ) 1 e t | t m + l x n = l = 0 n m S 2 ( l + m , m ) ( l + m ) ! ( n ) m + l 1 | ( 1 λ e t λ ) r L i k ( 1 e t ) 1 e t x n m l = l = 0 n m ( n l + m ) S 2 ( l + m , m ) T n m l ( r , k ) ( λ ) .
(2.35)

Therefore, by (2.34) and (2.35), we obtain the following theorem.

Theorem 2.9 For r,kZ, we have

T n ( r , k ) (x|λ)= m = 0 n { l = 0 n m ( n l + m ) S 2 ( l + m , m ) T n m l ( r , k ) ( λ ) } ( x ) m .

Finally, we consider the following two Sheffer sequences:

T n ( r , k ) ( x | λ ) ( ( e t λ 1 λ ) r 1 e t L i k ( 1 e t ) , t ) , x [ n ] ( 1 , 1 e t ) ,
(2.36)

where x [ n ] =x(x+1)(x+n1).

Let us assume that

T n ( r , k ) (x|λ)= m = 0 n C n , m x [ m ] .
(2.37)

Then, by (1.21) and (2.37), we get

C n , m = 1 m ! ( 1 λ e t λ ) r L i k ( 1 e t ) 1 e t ( 1 e t ) m | x n = l = 0 ( 1 ) l S 2 ( l + m , m ) ( l + m ) ! ( 1 λ e t λ ) r L i k ( 1 e t ) 1 e t | t m + l x n = l = 0 n m ( 1 ) l S 2 ( l + m , m ) ( l + m ) ! ( n ) m + l 1 | ( 1 λ e t λ ) r L i k ( 1 e t ) 1 e t x n m l = l = 0 n m ( 1 ) l ( n l + m ) S 2 ( l + m , m ) T n m l ( r , k ) ( λ ) .
(2.38)

Therefore, by (2.37) and (2.38), we obtain the following theorem.

Theorem 2.10 For r,kZ, n0, we have

T n ( r , k ) (x|λ)= m = 0 n { l = 0 n m ( 1 ) l ( n l + m ) S 2 ( l + m , m ) T n m l ( r , k ) ( λ ) } x [ m ] .

References

  1. 1.

    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.

  2. 2.

    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(1):239–247. 10.1216/RMJ-2011-41-1-239

  3. 3.

    Kim T: Identities involving Frobenius-Euler polynomials arising from non-linear differential equations. J. Number Theory 2012, 132(1):2854–2865.

  4. 4.

    Ryoo C: A note on the Frobenius-Euler polynomials. Proc. Jangjeon Math. Soc. 2011, 14(4):495–501.

  5. 5.

    Ryoo CS, Agarwal RP: Exploring the multiple Changhee q -Bernoulli polynomials. Int. J. Comput. Math. 2005, 82(4):483–493. 10.1080/00207160512331323362

  6. 6.

    Kim DS, Kim T, Kim YH, Lee SH: Some arithmetic properties of Bernoulli and Euler numbers. Adv. Stud. Contemp. Math. 2012, 22(4):467–480.

  7. 7.

    Kim, DS, Kim, T: Poly-Bernoulli polynomials arising from umbral calculus (communicated)

  8. 8.

    Kim T: Power series and asymptotic series associated with the q -analog of the two-variable p -adic L -function. Russ. J. Math. Phys. 2005, 12(2):186–196.

  9. 9.

    Can M, Cenkci M, Kurt V, Simsek Y: Twisted Dedekind type sums associated with Barnes’ type multiple Frobenius-Euler l -functions. Adv. Stud. Contemp. Math. 2009, 18(2):135–160.

  10. 10.

    Ding D, Yang J: Some identities related to the Apostol-Euler and Apostol-Bernoulli polynomials. Adv. Stud. Contemp. Math. 2010, 20(1):7–21.

  11. 11.

    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.

  12. 12.

    Kurt B, Simsek Y: On the generalized Apostol-type Frobenius-Euler polynomials. Adv. Differ. Equ. 2013., 2013: Article ID 1

  13. 13.

    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.

  14. 14.

    Roman S Pure and Applied Mathematics 111. In The Umbral Calculus. Academic Press, New York; 1984.

  15. 15.

    Roman S, Rota G-C: The umbral calculus. Adv. Math. 1978, 27(2):95–188. 10.1016/0001-8708(78)90087-7

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Acknowledgements

This work was supported by the National Research Foundation of Korea (NRF) grant, funded by the Korea government (MOE) (No. 2012R1A1A2003786).

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Correspondence to Taekyun Kim.

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The authors declare that they have no competing interests.

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All authors contributed equally to the manuscript and typed, read, and approved the final manuscript.

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Keywords

  • Formal Power Series
  • Linear Functional
  • Bernoulli Number
  • Bernoulli Polynomial
  • Euler Polynomial