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

Advances in Difference Equations20132013:251

https://doi.org/10.1186/1687-1847-2013-251

Received: 11 July 2013

Accepted: 6 August 2013

Published: 20 August 2013

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.

Keywords

Formal Power SeriesLinear FunctionalBernoulli NumberBernoulli PolynomialEuler Polynomial

1 Introduction

For λ C with λ 1 , the Frobenius-Euler polynomials of order α ( α R ) 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 ( k Z ) ( 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 ) , c L | p ( x ) = c L | 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  n 0 ( 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 , 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, 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  y C ,
(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 ) ( n 0 ) ( 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 λ C with λ 1 , r , k Z .

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 , k Z , n 0 , 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 , k Z , 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 , k Z , 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 , k Z , n Z with n 2 , 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 , k Z , n 1 , 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 , k Z , 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 , k Z and λ C with λ 1 . 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 , k Z , 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 , k Z , 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 , k Z , and λ , μ C with λ 1 , μ 1 , 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 , k Z , 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 , k Z , 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 + n 1 ) .

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 , k Z , n 0 , 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 ] .

Declarations

Acknowledgements

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

Authors’ Affiliations

(1)
Department of Mathematics, Sogang University, Seoul, Republic of Korea
(2)
Department of Mathematics, Kwangwoon University, Seoul, Republic of Korea

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Copyright

© Kim and Kim; 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.

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