Skip to main content

Advertisement

Hermite and poly-Bernoulli mixed-type polynomials

Article metrics

Abstract

In this paper, we consider Hermite and poly-Bernoulli mixed-type polynomials and investigate the properties of those polynomials which are derived from umbral calculus. Finally, we give various identities associated with Stirling numbers, Bernoulli and Frobenius-Euler polynomials of higher order.

1 Introduction

For r Z 0 , as is well known, the Bernoulli polynomials of order r are defined by the generating function to be

n = 0 B n ( r ) ( x ) n ! t n = ( t e t 1 ) r e x t (see [1–16]).
(1.1)

For kZ, the polylogarithm is defined by

Li k (x)= n = 1 x n n k .
(1.2)

Note that Li 1 (x)=log(1x).

The poly-Bernoulli polynomials are defined by the generating function to be

Li k ( 1 e t ) 1 e t e x t = n = 0 B n ( k ) (x) t n n ! (see [5, 8]).
(1.3)

When x=0, B n ( k ) = B n ( k ) (0) are called the poly-Bernoulli numbers (of index k).

For ν(0)R, the Hermite polynomials of order ν are given by the generating function to be

e ν t 2 2 e x t = n = 0 H n ( ν ) (x) t n n ! (see [6, 12, 13]).
(1.4)

When x=0, H n ( ν ) = H n ( ν ) (0) are called the Hermite numbers of order ν.

In this paper, we consider the Hermite and poly-Bernoulli mixed-type polynomials H B n ( ν , k ) (x) which are defined by the generating function to be

e ν t 2 2 Li k ( 1 e t ) 1 e t e x t = n = 0 H B n ( ν , k ) (x) t n n ! ,
(1.5)

where kZ and ν(0)R.

When x=0, H B n ( ν , k ) =H B n ( ν , k ) (0) are called the Hermite and poly-Bernoulli mixed-type numbers.

Let be the set of all formal power series in the variable t over as follows:

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

Let P=C[x] and P denote the vector space of all linear functionals on .

L|p(x) denotes the action of the linear functional L on the polynomial p(x), and we recall 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 . For f(t)F, let us define the linear functional on by setting

f ( t ) | x n = a n (n0).
(1.7)

Then, by (1.6) and (1.7), we get

t k | x n =n! δ n , k (n,k0),
(1.8)

where δ n , k is the Kronecker symbol.

For f L (t)= k = 0 L | x k k ! t k , we have f L (t)| x n =L| x n . That is, L= f L (t). The map L f L (t) is a vector space isomorphism from P onto . Henceforth, denotes 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. We call the umbral algebra and the umbral calculus is the study of umbral algebra. The order O(f) of the power series f(t)0 is the smallest integer for which a k does not vanish. If O(f)=0, then f(t) is called an invertible series. If O(f)=1, then f(t) is called a delta series. For f(t),g(t)F, we have

f ( t ) g ( t ) | p ( x ) = f ( t ) | g ( t ) p ( x ) = g ( t ) | f ( t ) p ( x ) .
(1.9)

Let f(t)F and p(x)P. Then we have

f(t)= k = 0 f ( t ) | x k k ! t k ,p(x)= k = 0 t k | p ( x ) k ! x k (see [8, 9, 11, 13, 14]).
(1.10)

By (1.10), we get

p ( k ) (0)= t k | p ( x ) = 1 | p ( k ) ( x ) ,
(1.11)

where p ( k ) (0)= d k p ( x ) d x k | x = 0 .

From (1.11), we have

t k p(x)= p ( k ) (x)= d k p ( x ) d x k (see [8, 9, 13]).
(1.12)

By (1.12), we easily get

e y t p(x)=p(x+y), e y t | p ( x ) =p(y).
(1.13)

For O(f(t))=1, O(g(t))=0, there exists a unique sequence s n (x) of polynomials such that g(t)f ( t ) k | x n =n! δ n , k (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)).

Let p(x)P, f(t)F. Then we see that

f ( t ) | x p ( x ) = t f ( t ) | p ( x ) = d f ( t ) d t | p ( x ) .
(1.14)

For s n (x)(g(t),f(t)), we have the following equations:

h(t)= k = 0 h ( t ) | s k ( x ) k ! g(t)f ( t ) k ,p(x)= k = 0 g ( t ) f ( t ) k | p ( x ) k ! s k (x),
(1.15)

where h(t)F, p(x)P,

1 g ( f ¯ ( t ) ) e y f ¯ ( t ) = n = 0 s n (y) t n n ! ,
(1.16)

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

s n (x+y)= k = 0 n ( n k ) s k (y) p n k (x),where  p n (x)=g(t) s n (x),
(1.17)
f(t) s n (x)=n s n 1 (x), s n + 1 (x)= ( x g ( t ) g ( t ) ) 1 f ( t ) s n (x),
(1.18)

and the conjugate representation is given by

s n (x)= j = 0 n 1 j ! g ( f ¯ ( t ) ) 1 f ¯ ( t ) j | x n x j .
(1.19)

For s n (x)(g(t),f(t)), r n (x)(h(t),l(t)), 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 [8, 9, 13]).
(1.21)

In this paper, we consider Hermite and poly-Bernoulli mixed-type polynomials and investigate the properties of those polynomials which are derived from umbral calculus. Finally, we give various identities associated with Bernoulli and Frobenius-Euler polynomials of higher order.

2 Hermite and poly-Bernoulli mixed-type polynomials

From (1.5) and (1.16), we note that

H B n ( ν , k ) (x) ( e ν t 2 2 1 e t Li k ( 1 e t ) , t ) ,
(2.1)

and, by (1.3), (1.4) and (1.16), we get

B n ( k ) (x) ( 1 e t Li k ( 1 e t ) , t ) ,
(2.2)
H n ( ν ) (x) ( e ν t 2 2 , t ) ,where n0.
(2.3)

From (1.18), (2.1), (2.2) and (2.3), we have

t B n ( k ) (x)=n B n 1 ( k ) (x),t H n ( ν ) (x)=n H n 1 ( ν ) (x),tH B n ( ν , k ) (x)=nH B n 1 ( ν , k ) (x).
(2.4)

By (1.5), (1.8) and (2.1), we get

H B n ( ν , k ) ( x ) = e ν t 2 2 Li k ( 1 e t ) 1 e t x n = e ν t 2 2 B n ( k ) ( x ) = m = 0 [ n 2 ] 1 m ! ( ν 2 ) m ( n ) 2 m B n 2 m ( k ) ( x ) = m = 0 [ n 2 ] ( n 2 m ) ( 2 m ) ! m ! ( ν 2 ) m B n 2 m ( k ) ( x ) .
(2.5)

Therefore, by (2.5), we obtain the following proposition.

Proposition 1 For n0, we have

H B n ( ν , k ) (x)= m = 0 [ n 2 ] ( n 2 m ) ( 2 m ) ! m ! ( ν 2 ) m B n 2 m ( k ) (x).

From (1.5), we can also derive

H B n ( ν , k ) ( x ) = Li k ( 1 e t ) 1 e t e ν t 2 2 x n = Li k ( 1 e t ) 1 e t H n ( ν ) ( x ) = m = 0 ( 1 e t ) m ( m + 1 ) k H n ( ν ) ( x ) = m = 0 n 1 ( m + 1 ) k j = 0 m ( m j ) ( 1 ) j e j t H n ( ν ) ( x ) = m = 0 n 1 ( m + 1 ) k j = 0 m ( m j ) ( 1 ) j H n ( ν ) ( x j ) .
(2.6)

Therefore, by (2.6), we obtain the following theorem.

Theorem 2 For n0, we have

H B n ( ν , k ) (x)= m = 0 n 1 ( m + 1 ) k j = 0 m ( m j ) ( 1 ) j H n ( ν ) (xj).

By (1.5), we get

H B n ( ν , k ) ( x ) = e ν t 2 2 B n ( k ) ( x ) = l = 0 1 l ! ( ν 2 ) l t 2 l B n ( k ) ( x ) = l = 0 [ n 2 ] 1 l ! ( ν 2 ) l m = 0 n 1 ( m + 1 ) k j = 0 m ( 1 ) j ( m j ) t 2 l ( x j ) n = l = 0 [ n 2 ] j = 0 n { m = j n ( n 2 l ) ( 2 l ) ! l ! ( ν 2 ) l ( 1 ) j ( m j ) ( m + 1 ) k } ( x j ) n 2 l .
(2.7)

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

Theorem 3 For n0, we have

H B n ( ν , k ) (x)= l = 0 [ n 2 ] j = 0 n { m = j n ( n 2 l ) ( 2 l ) ! l ! ( ν 2 ) l ( 1 ) j ( m j ) ( m + 1 ) k } ( x j ) n 2 l .

By (2.6), we get

H B n ( ν , k ) ( x ) = m = 0 n ( 1 e t ) m ( m + 1 ) k H n ( ν ) ( x ) = m = 0 n 1 ( m + 1 ) k a = 0 n m m ! ( a + m ) ! ( 1 ) a S 2 ( a + m , m ) ( n ) a + m H n a m ( ν ) ( x ) = m = 0 n a = 0 n m ( 1 ) n a m m ! ( m + 1 ) k ( n n a ) S 2 ( n a , m ) H a ( ν ) ( x ) = ( 1 ) n a = 0 n { m = 0 n a ( 1 ) m + a m ! ( m + 1 ) k ( n a ) S 2 ( n a , m ) } H a ( ν ) ( x ) ,
(2.8)

where S 2 (n,m) is the Stirling number of the second kind.

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

Theorem 4 For n0, we have

H B n ( ν , k ) (x)= ( 1 ) n a = 0 n { m = 0 n a ( 1 ) a + m m ! ( m + 1 ) k ( n a ) S 2 ( n a , m ) } H a ( ν ) (x).

From (1.19) and (2.1), we have

H B n ( ν , k ) ( x ) = j = 0 n ( n j ) e ν t 2 2 Li k ( 1 e t ) 1 e t | x n j x j = j = 0 n ( n j ) e ν t 2 2 | B n j ( k ) ( x ) x j = j = 0 n ( n j ) l = 0 [ n j 2 ] ( ν 2 ) l l ! ( n j ) 2 l 1 | B n j 2 l ( k ) ( x ) x j = j = 0 n ( n j ) l = 0 [ n j 2 ] 1 l ! ( ν 2 ) l ( n j ) 2 l B n j 2 l ( k ) x j = j = 0 n { l = 0 [ n j 2 ] ( n j ) ( n j 2 l ) ( 2 l ) ! l ! ( ν 2 ) l B n j 2 l ( k ) } x j .
(2.9)

Therefore, by (2.9), we obtain the following theorem.

Theorem 5 For n0, we have

H B n ( ν , k ) (x)= j = 0 n { l = 0 [ n j 2 ] ( n j ) ( n j 2 l ) ( 2 l ) ! l ! ( ν 2 ) l B n j 2 l ( k ) } x j .

Remark By (1.17) and (2.1), we easily get

H B n ( ν , k ) (x+y)= j = 0 n ( n j ) H B j ( ν , k ) (x) y n j .
(2.10)

We note that

H B n ( ν , k ) (x) ( g ( t ) = e ν t 2 2 1 e t Li k ( 1 e t ) , f ( t ) = t ) .
(2.11)

From (1.18) and (2.11), we have

H B n + 1 ( ν , k ) (x)= ( x g ( t ) g ( t ) ) H B n ( ν , k ) (x).
(2.12)

Now, we observe that

g ( t ) g ( t ) = ( log ( g ( t ) ) ) = ( log e ν t 2 2 + log ( 1 e t ) log ( Li k ( 1 e t ) ) ) = ν t + e t 1 e t ( 1 Li k 1 ( 1 e t ) Li k ( 1 e t ) ) .
(2.13)

By (2.12) and (2.13), we get

H B n + 1 ( ν , k ) ( x ) = x H B n ( ν , k ) ( x ) g ( t ) g ( t ) H B n ( ν , k ) ( x ) = x H B n ( ν , k ) ( x ) ν n H B n 1 ( ν , k ) ( x ) e ν t 2 2 t e t 1 Li k ( 1 e t ) Li k 1 ( 1 e t ) t ( 1 e t ) x n .
(2.14)

It is easy to show that

Li k ( 1 e t ) Li k 1 ( 1 e t ) 1 e t = m = 2 ( 1 m k 1 m k 1 ) ( 1 e t ) m 1 = ( 1 2 k 1 2 k 1 ) t + .
(2.15)

Thus, by (2.15), we get

Li k ( 1 e t ) Li k 1 ( 1 e t ) t ( 1 e t ) x n = Li k ( 1 e t ) Li k 1 ( 1 e t ) 1 e t x n + 1 n + 1 .
(2.16)

From (2.16), we can derive

e ν t 2 2 t e t 1 Li k ( 1 e t ) Li k 1 ( 1 e t ) t ( 1 e t ) x n = 1 n + 1 ( l = 0 B l l ! t l ) ( H B n + 1 ( ν , k ) ( x ) H B n + 1 ( ν , k 1 ) ( x ) ) = 1 n + 1 l = 0 n + 1 B l l ! t l ( H B n + 1 ( ν , k ) ( x ) H B n + 1 ( ν , k 1 ) ( x ) ) = 1 n + 1 l = 0 n + 1 ( n + 1 l ) B l ( H B n + 1 l ( ν , k ) ( x ) H B n + 1 l ( ν , k 1 ) ( x ) ) .
(2.17)

Therefore, by (2.14) and (2.17), we obtain the following theorem.

Theorem 6 For n0, we have

H B n + 1 ( ν , k ) ( x ) = x H B n ( ν , k ) ( x ) ν n H B n 1 ( ν , k ) ( x ) 1 n + 1 l = 0 n + 1 ( n + 1 l ) B l { H B n + 1 l ( ν , k ) ( x ) H B n + 1 l ( ν , k 1 ) ( x ) } .
(2.18)

Let us take t on the both sides of (2.18). Then we have

( n + 1 ) H B n ( ν , k ) ( x ) = ( x t + 1 ) H B n ( ν , k ) ( x ) ν n ( n 1 ) H B n 2 ( ν , k ) ( x ) 1 n + 1 l = 0 n + 1 ( n + 1 l ) ( n + 1 l ) B l { H B n l ( ν , k ) ( x ) H B n l ( ν , k 1 ) ( x ) } = n x H B n 1 ( ν , k ) ( x ) + H B n ( ν , k ) ( x ) ν n ( n 1 ) H B n 2 ( ν , k ) ( x ) l = 0 n ( n l ) B l ( H B n l ( ν , k ) ( x ) H B n l ( ν , k 1 ) ( x ) ) ,
(2.19)

where n3.

Thus, by (2.19), we obtain the following theorem.

Theorem 7 For n3, we have

l = 0 n ( n l ) B l H B n l ( ν , k 1 ) ( x ) = ( n + 1 ) H B n ( ν , k ) ( x ) n ( x + 1 2 ) H B n 1 ( ν , k ) ( x ) + n ( n 1 ) ( ν + 1 12 ) H B n 2 ( ν , k ) ( x ) + l = 0 n 3 ( n l ) B n l H B l ( ν , k ) ( x ) .

By (1.5) and (1.8), we get

H B n ( ν , k ) ( y ) = e ν t 2 2 Li k ( 1 e t ) 1 e t e y t | x n = t ( e ν t 2 2 Li k ( 1 e t ) 1 e t e y t ) | x n 1 = ( t e ν t 2 2 ) Li k ( 1 e t ) 1 e t e y t | x n 1 + e ν t 2 2 ( t Li k ( 1 e t ) 1 e t ) e y t | x n 1 + e ν t 2 2 Li k ( 1 e t ) 1 e t ( t e y t ) | x n 1 = ν ( n 1 ) e ν t 2 2 Li k ( 1 e t ) 1 e t e y t | x n 2 + y e ν t 2 2 Li k ( 1 e t ) 1 e t e y t | x n 1 + e ν t 2 2 ( t Li k ( 1 e t ) 1 e t ) e y t | x n 1 = ν ( n 1 ) H B n 2 ( ν , k ) ( y ) + y H B n 1 ( ν , k ) ( y ) + e ν t 2 2 ( t Li k ( 1 e t ) 1 e t ) e y t | x n 1 .
(2.20)

Now, we observe that

t ( Li k ( 1 e t ) 1 e t ) = Li k 1 ( 1 e t ) Li k ( 1 e t ) ( 1 e t ) 2 e t .
(2.21)

From (2.21), we have

e ν t 2 2 ( t Li k ( 1 e t ) 1 e t ) e y t | x n 1 = e ν t 2 2 ( Li k 1 ( 1 e t ) Li k ( 1 e t ) ( 1 e t ) 2 ) e t e y t | 1 n t x n = 1 n e ν t 2 2 Li k 1 ( 1 e t ) Li k ( 1 e t ) 1 e t e y t | t e t 1 x n = 1 n e ν t 2 2 Li k 1 ( 1 e t ) Li k ( 1 e t ) 1 e t e y t | B n ( x ) = 1 n l = 0 n ( n l ) B l e ν t 2 2 Li k 1 ( 1 e t ) Li k ( 1 e t ) 1 e t e y t | x n l = 1 n l = 0 n ( n l ) B l { H B n l ( ν , k 1 ) ( y ) H B n l ( ν , k ) ( y ) } ,
(2.22)

where B n are the ordinary Bernoulli numbers which are defined by the generating function to be

t e t 1 = n = 0 B n n ! t n .

Therefore, by (2.20) and (2.22), we obtain the following theorem.

Theorem 8 For n2, we have

H B n ( ν , k ) ( x ) = ν ( n 1 ) H B n 2 ( ν , k ) ( x ) + x H B n 1 ( ν , k ) ( x ) + 1 n l = 0 n ( n l ) B l ( H B n l ( ν , k 1 ) ( x ) H B n l ( ν , k ) ( x ) ) .

Now, we compute

e ν t 2 2 Li k ( 1 e t ) | x n + 1

in two different ways.

On the one hand,

e ν t 2 2 Li k ( 1 e t ) | x n + 1 = e ν t 2 2 Li k ( 1 e t ) 1 e t ( 1 e t ) | x n + 1 = e ν t 2 2 Li k ( 1 e t ) 1 e t | ( 1 e t ) x n + 1 = e ν t 2 2 Li k ( 1 e t ) 1 e t | x n + 1 ( x 1 ) n + 1 = m = 0 n ( 1 ) n m ( n + 1 m ) e ν t 2 2 Li k ( 1 e t ) 1 e t | x m = m = 0 n ( 1 ) n m ( n + 1 m ) H B m ( ν , k ) .
(2.23)

On the other hand,

e ν t 2 2 Li k ( 1 e t ) | x n + 1 = Li k ( 1 e t ) | e ν t 2 2 x n + 1 = 0 t ( Li k ( 1 e s ) ) d s | e ν t 2 2 x n + 1 = 0 t e s Li k 1 ( 1 e s ) 1 e s d s | e ν t 2 2 x n + 1 = l = 0 ( m = 0 l ( 1 ) l m ( l m ) B m ( k 1 ) t l + 1 ( l + 1 ) ! ) | H n + 1 ( ν ) ( x ) = l = 0 n m = 0 l ( 1 ) l m ( l m ) B m ( k 1 ) 1 ( l + 1 ) ! t l + 1 | H n + 1 ( ν ) ( x ) = l = 0 n m = 0 l ( 1 ) l m ( l m ) ( n + 1 l + 1 ) B m ( k 1 ) H n l ( ν ) .
(2.24)

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

Theorem 9 For n0, we have

m = 0 n ( 1 ) n m ( n + 1 m ) H B m ( ν , k ) = m = 0 n l = m n ( 1 ) l m ( l m ) ( n + 1 l + 1 ) B m ( k 1 ) H n l ( ν ) .

Let us consider the following two Sheffer sequences:

H B n ( ν , k ) (x) ( e ν t 2 2 1 e t Li k ( 1 e t ) , t )
(2.25)

and

B n ( r ) (x) ( ( e t 1 t ) r , t ) (r Z 0 ).
(2.26)

Let us assume that

H B n ( ν , k ) (x)= m = 0 n C n , m B m ( r ) (x).
(2.27)

Then, by (1.20) and (1.21), we get

C n , m = 1 m ! ( e t 1 t ) r t m | e ν t 2 2 Li k ( 1 e t ) 1 e t x n = 1 m ! ( e t 1 t ) r | t m H B n ( ν , k ) ( x ) = 1 m ! ( n ) m ( e t 1 t ) r | H B n m ( ν , k ) ( x ) = ( n m ) l = 0 r ! ( l + r ) ! S 2 ( l + r , r ) t l | H B n m ( ν , k ) ( x ) = ( n m ) l = 0 n m ( n m ) l r ! ( l + r ) ! S 2 ( l + r , r ) H B n m l ( ν , k ) = ( n m ) l = 0 n m ( n m l ) ( l + r r ) S 2 ( l + r , r ) H B n m l ( ν , k ) .
(2.28)

Therefore, by (2.27) and (2.28), we obtain the following theorem.

Theorem 10 For n,r Z 0 , we have

H B n ( ν , k ) (x)= m = 0 n { ( n m ) l = 0 n m ( n m l ) ( l + r r ) S 2 ( l + r , r ) H B n m l ( ν , k ) } B m ( r ) (x).

For λ(1)C, r Z 0 , the Frobenius-Euler polynomials of order r are defined by the generating function to be

( 1 λ e t λ ) r e x t = n = 0 H n ( r ) (x|λ) t n n ! (see [1, 4, 7, 9, 10]).
(2.29)

From (1.16) and (2.29), we note that

H n ( r ) (x|λ) ( ( e t λ 1 λ ) r , t ) .
(2.30)

Let us assume that

H B n ( ν , k ) (x)= m = 0 n C n , m H m ( r ) (x|λ).
(2.31)

By (1.21), we get

C n , m = 1 m ! ( e t λ 1 λ ) r t m | e ν t 2 2 Li k ( 1 e t ) 1 e t x n = ( n ) m m ! ( 1 λ ) r l = 0 r ( r l ) ( λ ) r l e l t | H B n m ( ν , k ) ( x ) = ( n m ) 1 ( 1 λ ) r l = 0 r ( r l ) ( λ ) r l 1 | e l t H B n m ( ν , k ) ( x ) = ( n m ) ( 1 λ ) r l = 0 r ( r l ) ( λ ) r l H B n m ( ν , k ) ( l ) .
(2.32)

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

Theorem 11 For n,r Z 0 , we have

H B n ( ν , k ) (x)= 1 ( 1 λ ) r m = 0 n ( n m ) { l = 0 r ( r l ) ( λ ) r l H B n m ( ν , k ) ( l ) } H m ( r ) (x|λ).

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.

    Dere R, Simsek Y: Application of umbral algebra to some special polynomials. Adv. Stud. Contemp. Math. 2012, 22(3):433-438.

  3. 3.

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

  4. 4.

    Ozden H, Cangul IN, Simsek Y: Remarks on q -Bernoulli numbers associated with Daehee numbers. Adv. Stud. Contemp. Math. 2009, 18(1):41-48.

  5. 5.

    Kaneko M: Poly-Bernoulli numbers. J. Théor. Nr. Bordx. 1997, 9(1):221-228. 10.5802/jtnb.197

  6. 6.

    Kim DS, Kim T, Dolgy DV, Rim SH: Some new identities of Bernoulli, Euler and Hermite polynomials arising from umbral calculus. Adv. Differ. Equ. 2013., 2013(2013): Article ID 73

  7. 7.

    Kim DS, Kim T: Some identities of Frobenius-Euler polynomials arising from umbral calculus. Adv. Differ. Equ. 2012., 2012: Article ID 196

  8. 8.

    Kim DS, Kim T, Lee SH: A note on poly-Bernoulli polynomials arising from umbral calculus. Adv. Stud. Theor. Phys. 2013, 7(15):731-744.

  9. 9.

    Kim DS, Kim T, Lee SH: Poly-Cauchy numbers and polynomials with umbral calculus viewpoint. Int. J. Math. Anal. 2013, 7: 2235-2253.

  10. 10.

    Kim DS, Kim T, Lee SH: Higher-order Cauchy of the first kind and poly-Cauchy of the first kind mixed type polynomials. Adv. Stud. Contemp. Math. 2013, 23: 543-554.

  11. 11.

    Kim DS, Kim T: Some identities of Bernoulli and Euler polynomials arising from umbral calculus. Adv. Stud. Contemp. Math. 2013, 23(1):159-171.

  12. 12.

    Kurt B, Simsek Y: On Hermite based Genocchi polynomials. Adv. Stud. Contemp. Math. 2013, 23(1):13-17.

  13. 13.

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

  14. 14.

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

  15. 15.

    Rim SH, Jeong J: On the modified q -Euler numbers of higher order with weight. Adv. Stud. Contemp. Math. 2012, 22(1):93-98.

  16. 16.

    Simsek Y: Generating functions of the twisted Bernoulli numbers and polynomials associated with their interpolation functions. Adv. Stud. Contemp. Math. 2008, 16(2):251-278.

Download references

Acknowledgements

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

Author information

Correspondence to Taekyun Kim.

Additional information

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

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

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article

Keywords

  • Vector Space
  • Formal Power Series
  • Linear Functional
  • Hermite Polynomial
  • Bernoulli Number