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Theory and Modern Applications

Barnes-type Daehee of the first kind and poly-Cauchy of the first kind mixed-type polynomials

Abstract

In this paper, by considering Barnes-type Daehee polynomials of the first kind as well as poly-Cauchy polynomials of the first kind, we define and investigate the mixed-type polynomials of these polynomials. From the properties of Sheffer sequences of these polynomials arising from umbral calculus, we derive new and interesting identities.

MSC:05A15, 05A40, 11B68, 11B75, 65Q05.

1 Introduction

In this paper, we consider the polynomials D n ( k ) (x| a 1 ,, a r ) called the Barnes-type Daehee of the first kind and poly-Cauchy of the first kind mixed-type polynomials, whose generating function is given by

j = 1 r ( ln ( 1 + t ) ( 1 + t ) a j 1 ) Lif k ( ln ( 1 + t ) ) ( 1 + t ) x = n = 0 D n ( k ) (x| a 1 ,, a r ) t n n ! ,
(1)

where a 1 ,, a r 0. Here, Lif k (x) (kZ) is the polyfactorial function [1] defined by

Lif k (x)= m = 0 x m m ! ( m + 1 ) k .

When x=0, D n ( k ) ( a 1 ,, a r )= D n ( k ) (0| a 1 ,, a r ) is called Barnes-type Daehee of the first kind and poly-Cauchy of the first kind mixed-type number.

Recall that the Barnes-type Daehee polynomials of the first kind, denoted by D n (x| a 1 ,, a r ), are given by the generating function

j = 1 r ( ln ( 1 + t ) ( 1 + t ) a j 1 ) ( 1 + t ) x = n = 0 D n (x| a 1 ,, a r ) t n n ! .

If a 1 == a r =1, then D n ( r ) (x)= D n (x| 1 , , 1 r ) are the Daehee polynomials of the first kind of order r. Dahee polynomials were defined by the second author [2] and have been investigated in [3, 4].

The poly-Cauchy polynomials of the first kind, denoted by c n ( k ) (x) [5, 6], are given by the generating function as

Lif k ( ln ( 1 + t ) ) ( 1 + t ) x = n = 0 c n ( k ) (x) t n n ! .

In this paper, by considering Barnes-type Daehee polynomials of the first kind as well as poly-Cauchy polynomials of the first kind, we define and investigate the mixed-type polynomials of these polynomials. From the properties of Sheffer sequences of these polynomials arising from umbral calculus, we derive new and interesting identities.

2 Umbral calculus

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

F= { f ( t ) = k = 0 a k k ! t k | a k C } .
(2)

Let P=C[x] and let P be the vector space of all linear functionals on . L|p(x) is 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).
(3)

In particular,

t k | x n =n! δ n , k (n,k0),
(4)

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 the umbral algebra. The order O(f(t)) of a power series f(t) (≠0) is the smallest integer k for which the coefficient of t k does not vanish. If O(f(t))=1, then f(t) is called a delta series; if O(f(t))=0, then f(t) is called an invertible series. For f(t),g(t)F with O(f(t))=1 and 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. Such a 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)).

For f(t),g(t)F and p(x)P, we have

f ( t ) g ( t ) | p ( x ) = f ( t ) | g ( t ) p ( x ) = g ( t ) | f ( t ) p ( x )
(5)

and

f(t)= k = 0 f ( t ) | x k t k k ! ,p(x)= k = 0 t k | p ( x ) x k k !
(6)

[7], Theorem 2.2.5]. Thus, by (6), we get

t k p(x)= p ( k ) (x)= d k p ( x ) d x k and e y t p(x)=p(x+y).
(7)

Sheffer sequences are characterized by the generating function [7], Theorem 2.3.4].

Lemma 1 The sequence s n (x) is Sheffer for (g(t),f(t)) if and only if

1 g ( f ¯ ( t ) ) e y f ¯ ( t ) = k = 0 s k ( y ) k ! t k (yC),

where f ¯ (t) is the compositional inverse of f(t).

For s n (x)(g(t),f(t)), we have the following equations [7], Theorem 2.3.7, Theorem 2.3.5, Theorem 2.3.9]:

f(t) s n (x)=n s n 1 (x)(n0),
(8)
s n (x)= j = 0 n 1 j ! g ( f ¯ ( t ) ) 1 f ¯ ( t ) j | x n x j ,
(9)
s n (x+y)= j = 0 n ( n j ) s j (x) p n j (y),
(10)

where p n (x)=g(t) s n (x).

Assume that p n (x)(1,f(t)) and q n (x)(1,g(t)). Then the transfer formula [7], Corollary 3.8.2] is given by

q n (x)=x ( f ( t ) g ( t ) ) n x 1 p n (x)(n1).

For s n (x)(g(t),f(t)) and r n (x)(h(t),l(t)), assume that

s n (x)= m = 0 n C n , m r m (x)(n0).

Then we have [7], p.132]

C n , m = 1 m ! h ( f ¯ ( t ) ) g ( f ¯ ( t ) ) l ( f ¯ ( t ) ) m | x n .
(11)

3 Main results

From the definition (1), D n ( k ) (x| a 1 ,, a r ) is the Sheffer sequence for the pair

g(t)= j = 1 r ( e a j t 1 t ) 1 Lif k ( t ) andf(t)= e t 1.

So,

D n ( k ) (x| a 1 ,, a r ) ( j = 1 r ( e a j t 1 t ) 1 Lif k ( t ) , e t 1 ) .
(12)

3.1 Explicit expressions

Recall that Barnes’ multiple Bernoulli polynomials B n (x| a 1 ,, a r ) are defined by the generating function

t r j = 1 r ( e a j t 1 ) e x t = n = 0 B n (x| a 1 ,, a r ) t n n ! ,
(13)

where a 1 ,, a r 0 [8, 9]. Let ( n ) j =n(n1)(nj+1) (j1) with ( n ) 0 =1. The (signed) Stirling numbers of the first kind S 1 (n,m) are defined by

( x ) n = m = 0 n S 1 (n,m) x m .

Theorem 1

D n ( k ) ( x | a 1 , , a r ) = m = 0 n l = 0 m S 1 ( n , m ) ( m l ) ( m l + 1 ) k B l ( x | a 1 , , a r )
(14)
= j = 0 n l = 0 n j i = 0 l ( n l ) ( l i ) S 1 (nl,j) c i ( k ) D l i ( a 1 ,, a r ) x j
(15)
= l = 0 n ( n l ) D n l ( a 1 ,, a r ) c l ( k ) (x)
(16)
= l = 0 n ( n l ) c n l ( k ) D l (x| a 1 ,, a r ).
(17)

Proof Since

j = 1 r ( e a j t 1 t ) 1 Lif k ( t ) D n ( k ) (x| a 1 ,, a r ) ( 1 , e t 1 )
(18)

and

( x ) n ( 1 , e t 1 ) ,
(19)

we have

D n ( k ) ( x | a 1 , , a r ) = j = 1 r ( t e a j t 1 ) Lif k ( t ) ( x ) n = m = 0 n S 1 ( n , m ) j = 1 r ( t e a j t 1 ) Lif k ( t ) x m = m = 0 n S 1 ( n , m ) j = 1 r ( t e a j t 1 ) l = 0 m t l l ! ( l + 1 ) k x m = m = 0 n S 1 ( n , m ) j = 1 r ( t e a j t 1 ) l = 0 m ( m ) l l ! ( l + 1 ) k x m l = m = 0 n S 1 ( n , m ) l = 0 m ( m ) l l ! ( l + 1 ) k j = 1 r ( t e a j t 1 ) x m l = m = 0 n l = 0 m S 1 ( n , m ) ( m l ) ( l + 1 ) k B m l ( x | a 1 , , a r ) = m = 0 n l = 0 m S 1 ( n , m ) ( m l ) ( m l + 1 ) k B l ( x | a 1 , , a r ) .

Thus, we get (14).

By (9) with (12), we get

g ( f ¯ ( t ) ) 1 f ¯ ( t ) j | x n = j = 1 r ( ln ( 1 + t ) ( 1 + t ) a j 1 ) Lif k ( ln ( 1 + t ) ) ( ln ( 1 + t ) ) j | x n = j = 1 r ( ln ( 1 + t ) ( 1 + t ) a j 1 ) Lif k ( ln ( 1 + t ) ) | l = 0 j ! ( l + j ) ! S 1 ( l + j , j ) t l + j x n = l = 0 n j j ! ( l + j ) ! S 1 ( l + j , j ) ( n ) l + j j = 1 r ( ln ( 1 + t ) ( 1 + t ) a j 1 ) Lif k ( ln ( 1 + t ) ) | x n l j = l = 0 n j j ! ( n l + j ) S 1 ( l + j , j ) i = 0 D i ( k ) ( a 1 , , a r ) t i i ! | x n l j = l = 0 n j j ! ( n l + j ) S 1 ( l + j , j ) D n l j ( k ) ( a 1 , , a r ) = l = 0 n j j ! ( n l ) S 1 ( n l , j ) D l ( k ) ( a 1 , , a r ) .

On the other hand,

g ( f ¯ ( t ) ) 1 f ¯ ( t ) j | x n = l = 0 n j j ! ( l + j ) ! S 1 ( l + j , j ) ( n ) l + j j = 1 r ( ln ( 1 + t ) ( 1 + t ) a j 1 ) | Lif k ( ln ( 1 + t ) ) x n l j = l = 0 n j j ! ( n l + j ) S 1 ( l + j , j ) j = 1 r ( ln ( 1 + t ) ( 1 + t ) a j 1 ) | i = 0 n l j c i ( k ) t i i ! x n l j = l = 0 n j j ! ( n l + j ) S 1 ( l + j , j ) i = 0 n l j c i ( k ) ( n l j ) i i ! j = 1 r ( ln ( 1 + t ) ( 1 + t ) a j 1 ) | x n l j i = l = 0 n j j ! ( n l + j ) S 1 ( l + j , j ) i = 0 n l j c i ( k ) ( n l j ) i i ! m = 0 D m ( a 1 , , a r ) t m m ! | x n l j i = l = 0 n j i = 0 n l j j ! ( n l + j ) ( n l j i ) S 1 ( l + j , j ) c i ( k ) D n l j i ( a 1 , , a r ) = l = 0 n j i = 0 l j ! ( n l ) ( l i ) S 1 ( n l , j ) c i ( k ) D l i ( a 1 , , a r ) .

Thus, we obtain

D n ( k ) ( x | a 1 , , a r ) = j = 0 n l = 0 n j ( n l ) S 1 ( n l , j ) D l ( k ) ( a 1 , , a r ) x j = j = 0 n l = 0 n j i = 0 l ( n l ) ( l i ) S 1 ( n l , j ) c i ( k ) D l i ( a 1 , , a r ) x j ,

which is the identity (15).

Next,

D n ( k ) ( y | a 1 , , a r ) = i = 0 D i ( k ) ( y | a 1 , , a r ) t i i ! | x n = j = 1 r ( ln ( 1 + t ) ( 1 + t ) a j 1 ) Lif k ( ln ( 1 + t ) ) ( 1 + t ) y | x n = j = 1 r ( ln ( 1 + t ) ( 1 + t ) a j 1 ) | Lif k ( ln ( 1 + t ) ) ( 1 + t ) y x n = j = 1 r ( ln ( 1 + t ) ( 1 + t ) a j 1 ) | l = 0 n c l ( k ) ( y ) t l l ! x n = l = 0 n ( n l ) c l ( k ) ( y ) j = 1 r ( ln ( 1 + t ) ( 1 + t ) a j 1 ) | x n l = l = 0 n ( n l ) c l ( k ) ( y ) i = 0 D i ( a 1 , , a r ) t i i ! | x n l = l = 0 n ( n l ) c l ( k ) ( y ) D n l ( a 1 , , a r ) .

Thus, we obtain (16).

Finally, we obtain

D n ( k ) ( y | a 1 , , a r ) = i = 0 D i ( k ) ( y | a 1 , , a r ) t i i ! | x n = j = 1 r ( ln ( 1 + t ) ( 1 + t ) a j 1 ) Lif k ( ln ( 1 + t ) ) ( 1 + t ) y | x n = Lif k ( ln ( 1 + t ) ) | j = 1 r ( ln ( 1 + t ) ( 1 + t ) a j 1 ) ( 1 + t ) y x n = Lif k ( ln ( 1 + t ) ) | l = 0 n D l ( y | a 1 , , a r ) t l l ! x n = l = 0 n D l ( y | a 1 , , a r ) ( n l ) Lif k ( ln ( 1 + t ) ) | x n l = l = 0 n D l ( y | a 1 , , a r ) ( n l ) i = 0 c i ( k ) t i i ! | x n l = l = 0 n ( n l ) D l ( y | a 1 , , a r ) c n l ( k ) .

Thus, we get the identity (17). □

3.2 Sheffer identity

Theorem 2

D n ( k ) (x+y| a 1 ,, a r )= j = 0 n ( n j ) D j ( k ) (x| a 1 ,, a r ) ( y ) n j .
(20)

Proof By (12) with

p n ( x ) = j = 1 r ( e a j t 1 t ) 1 Lif k ( t ) D n ( x | a 1 , , a r ) = ( x ) n ( 1 , e t 1 ) ,

using (10), we have (20). □

3.3 Difference relations

Theorem 3

D n ( k ) (x+1| a 1 ,, a r ) D n ( k ) (x| a 1 ,, a r )=n D n 1 ( k ) (x| a 1 ,, a r ).
(21)

Proof By (8) with (12), we get

( e t 1 ) D n ( k ) (x| a 1 ,, a r )=n D n 1 ( k ) (x| a 1 ,, a r ).

By (7), we have (21). □

3.4 Recurrence

Theorem 4

D n + 1 ( k ) ( x | a 1 , , a r ) = x D n ( k ) ( x 1 | a 1 , , a r ) m = 0 n j = 1 r l = 1 m + 1 i = 0 l ( m + 1 l ) ( l i ) ( m + 1 ) ( l i + 1 ) k S 1 ( n , m ) × ( a j ) m + 1 l B m + 1 l B i ( x 1 | a 1 , , a r ) + m = 0 n l = 0 m ( m l ) ( m + 2 l ) k S 1 ( n , m ) B l ( x 1 | a 1 , , a r ) ,
(22)

where B n is the nth ordinary Bernoulli number.

Proof By applying

s n + 1 (x)= ( x g ( t ) g ( t ) ) 1 f ( t ) s n (x)
(23)

[7], Corollary 3.7.2] with (12), we get

D n + 1 ( k ) (x| a 1 ,, a r )=x D n ( k ) (x1| a 1 ,, a r ) e t g ( t ) g ( t ) D n ( k ) (x| a 1 ,, a r ).

Now,

g ( t ) g ( t ) = ( ln g ( t ) ) = ( j = 1 r ln ( e a j t 1 ) r ln t ln Lif k ( t ) ) = j = 1 r a j e a j t e a j t 1 r t Lif k ( t ) Lif k ( t ) = j = 1 r i j ( e a i t 1 ) ( a j t e a j t e a j t + 1 ) t j = 1 r ( e a j t 1 ) Lif k ( t ) Lif k ( t ) .

Observe that

j = 1 r i j ( e a i t 1 ) ( a j t e a j t e a j t + 1 ) j = 1 r ( e a j t 1 ) = 1 2 ( j = 1 r a 1 a j 1 a j 2 a j + 1 a r ) t r + 1 + ( a 1 a r ) t r + = 1 2 ( j = 1 r a j ) t +

is a series with order ≥1. Since

D n ( k ) (x| a 1 ,, a r )= j = 1 r ( t e a j t 1 ) Lif k (t) ( x ) n = m = 0 n S 1 (n,m) j = 1 r ( t e a j t 1 ) Lif k (t) x m ,

we have

g ( t ) g ( t ) D n ( k ) ( x | a 1 , , a r ) = m = 0 n S 1 ( n , m ) g ( t ) g ( t ) ( j = 1 r t e a j t 1 ) Lif k ( t ) x m = m = 0 n S 1 ( n , m ) Lif k ( t ) ( j = 1 r t e a j t 1 ) × j = 1 r i j ( e a i t 1 ) ( a j t e a j t e a j t + 1 ) t j = 1 r ( e a j t 1 ) x m m = 0 n S 1 ( n , m ) Lif k ( t ) ( j = 1 r t e a j t 1 ) x m .
(24)

Since

j = 1 r i j ( e a i t 1 ) ( a j t e a j t e a j t + 1 ) t j = 1 r ( e a j t 1 ) x m = j = 1 r i j ( e a i t 1 ) ( a j t e a j t e a j t + 1 ) j = 1 r ( e a j t 1 ) x m + 1 m + 1 = 1 m + 1 j = 1 r ( a j t e a j t e a j t 1 1 ) x m + 1 = 1 m + 1 j = 1 r ( l = 0 ( 1 ) l B l a j l l ! t l 1 ) x m + 1 = 1 m + 1 j = 1 r ( l = 0 m + 1 ( m + 1 l ) ( a j ) l B l x m + 1 l x m + 1 ) = 1 m + 1 j = 1 r l = 1 m + 1 ( m + 1 l ) ( a j ) l B l x m + 1 l = 1 m + 1 j = 1 r l = 1 m + 1 ( m + 1 l ) ( a j ) m + 1 l B m + 1 l x l ,

the first term in (24) is

m = 0 n S 1 ( n , m ) m + 1 j = 1 r l = 1 m + 1 ( m + 1 l ) ( a j ) m + 1 l B m + 1 l Lif k ( t ) ( j = 1 r t e a j t 1 ) x l = m = 0 n S 1 ( n , m ) m + 1 j = 1 r l = 1 m + 1 ( m + 1 l ) ( a j ) m + 1 l B m + 1 l i = 0 l t i i ! ( i + 1 ) k B l ( x | a 1 , , a r ) = m = 0 n S 1 ( n , m ) m + 1 j = 1 r l = 1 m + 1 ( m + 1 l ) ( a j ) m + 1 l B m + 1 l i = 0 l ( l i ) ( i + 1 ) k B l i ( x | a 1 , , a r ) = m = 0 n j = 1 r l = 1 m + 1 i = 0 l ( m + 1 l ) ( l i ) ( m + 1 ) ( l i + 1 ) k S 1 ( n , m ) ( a j ) m + 1 l B m + 1 l B i ( x | a 1 , , a r ) .

Since

Lif k 1 (t) Lif k (t)= ( 1 2 k 1 1 2 k ) t+,
(25)

the second term in (24) is

m = 0 n S 1 ( n , m ) Lif k 1 ( t ) Lif k ( t ) t B m ( x | a 1 , , a r ) = m = 0 n S 1 ( n , m ) ( Lif k 1 ( t ) Lif k ( t ) ) B m + 1 ( x | a 1 , , a r ) m + 1 = m = 0 n S 1 ( n , m ) m + 1 ( Lif k 1 ( t ) Lif k ( t ) ) B m + 1 ( x | a 1 , , a r ) = m = 0 n S 1 ( n , m ) m + 1 ( l = 0 m + 1 t l l ! ( l + 1 ) k 1 B m + 1 ( x | a 1 , , a r ) l = 0 m + 1 t l l ! ( l + 1 ) k B m + 1 ( x | a 1 , , a r ) ) = m = 0 n S 1 ( n , m ) m + 1 ( l = 0 m + 1 ( m + 1 l ) ( l + 1 ) k 1 B m + 1 l ( x | a 1 , , a r ) l = 0 m + 1 ( m + 1 l ) ( l + 1 ) k B m + 1 l ( x | a 1 , , a r ) ) = m = 0 n S 1 ( n , m ) m + 1 l = 1 m + 1 ( m + 1 l ) l ( l + 1 ) k B m + 1 l ( x | a 1 , , a r ) = m = 0 n l = 1 m + 1 ( m l 1 ) S 1 ( n , m ) 1 ( l + 1 ) k B m + 1 l ( x | a 1 , , a r ) = m = 0 n l = 0 m ( m l ) ( m + 2 l ) k S 1 ( n , m ) B l ( x | a 1 , , a r ) .

Thus, we have

D n + 1 ( k ) ( x | a 1 , , a r ) = x D n ( k ) ( x 1 | a 1 , , a r ) m = 0 n j = 1 r l = 1 m + 1 i = 0 l ( m + 1 l ) ( l i ) ( m + 1 ) ( l i + 1 ) k S 1 ( n , m ) × ( a j ) m + 1 l B m + 1 l B i ( x 1 | a 1 , , a r ) + m = 0 n l = 0 m ( m l ) ( m + 2 l ) k S 1 ( n , m ) B l ( x 1 | a 1 , , a r ) ,

which is the identity (22). □

3.5 Differentiation

Theorem 5

d d x D n ( k ) (x| a 1 ,, a r )=n! l = 0 n 1 ( 1 ) n l 1 l ! ( n l ) D l ( k ) (x| a 1 ,, a r ).
(26)

Proof We shall use

d d x s n (x)= l = 0 n 1 ( n l ) f ¯ ( t ) | x n l s l (x)

(cf. [7], Theorem 2.3.12]). Since

f ¯ ( t ) | x n l = ln ( 1 + t ) | x n l = m = 1 ( 1 ) m 1 t m m | x n l = m = 1 n l ( 1 ) m 1 m t m | x n l = m = 1 n l ( 1 ) m 1 m ( n l ) ! δ m , n l = ( 1 ) n l 1 ( n l 1 ) ! ,

with (12), we have

d d x D n ( k ) ( x | a 1 , , a r ) = l = 0 n 1 ( n l ) ( 1 ) n l 1 ( n l 1 ) ! D l ( k ) ( x | a 1 , , a r ) = n ! l = 0 n 1 ( 1 ) n l 1 l ! ( n l ) D l ( k ) ( x | a 1 , , a r ) ,

which is the identity (26). □

3.6 One more relation

The classical Cauchy numbers c n are defined by

t ln ( 1 + t ) = n = 0 c n t n n !

(see e.g. [1, 10]).

Theorem 6

D n ( k ) ( x | a 1 , , a r ) = x D n 1 ( k ) ( x 1 | a 1 , , a r ) + 1 n l = 0 n ( n l ) c l D n l ( k 1 ) ( x 1 | a 1 , , a r ) + r 1 n l = 0 n ( n l ) c l D n l ( k ) ( x 1 | a 1 , , a r ) 1 n j = 1 r l = 0 n ( n l ) a j c l D n l ( k ) ( x + a j 1 | a 1 , , a r , a j ) .
(27)

Proof For n1, we have

D n ( k ) ( y | a 1 , , a r ) = l = 0 D l ( k ) ( y | a 1 , , a r ) t l l ! | x n = j = 1 r ( ln ( 1 + t ) ( 1 + t ) a j 1 ) Lif k ( ln ( 1 + t ) ) ( 1 + t ) y | x n = t ( j = 1 r ( ln ( 1 + t ) ( 1 + t ) a j 1 ) Lif k ( ln ( 1 + t ) ) ( 1 + t ) y ) | x n 1 = ( t j = 1 r ( ln ( 1 + t ) ( 1 + t ) a j 1 ) ) Lif k ( ln ( 1 + t ) ) ( 1 + t ) y | x n 1 + j = 1 r ( ln ( 1 + t ) ( 1 + t ) a j 1 ) ( t Lif k ( ln ( 1 + t ) ) ) ( 1 + t ) y | x n 1 + j = 1 r ( ln ( 1 + t ) ( 1 + t ) a j 1 ) Lif k ( ln ( 1 + t ) ) ( t ( 1 + t ) y ) | x n 1 .

The third term is

y j = 1 r ( ln ( 1 + t ) ( 1 + t ) a j 1 ) Lif k ( ln ( 1 + t ) ) ( 1 + t ) y 1 | x n 1 = y D n 1 ( k ) ( y 1 | a 1 , , a r ) .

By (25), the second term is

j = 1 r ( ln ( 1 + t ) ( 1 + t ) a j 1 ) Lif k 1 ( ln ( 1 + t ) ) Lif k ( ln ( 1 + t ) ) ( 1 + t ) ln ( 1 + t ) ( 1 + t ) y | x n 1 = j = 1 r ( ln ( 1 + t ) ( 1 + t ) a j 1 ) Lif k 1 ( ln ( 1 + t ) ) Lif k ( ln ( 1 + t ) ) t ( 1 + t ) y 1 | t ln ( 1 + t ) x n 1 = j = 1 r ( ln ( 1 + t ) ( 1 + t ) a j 1 ) Lif k 1 ( ln ( 1 + t ) ) Lif k ( ln ( 1 + t ) ) t ( 1 + t ) y 1 | l = 0 c l t l l ! x n 1 = l = 0 n 1 ( n 1 l ) c l × j = 1 r ( ln ( 1 + t ) ( 1 + t ) a j 1 ) ( 1 + t ) y 1 | Lif k 1 ( ln ( 1 + t ) ) Lif k ( ln ( 1 + t ) ) t x n 1 l = l = 0 n 1 ( n 1 l ) c l × j = 1 r ( ln ( 1 + t ) ( 1 + t ) a j 1 ) ( 1 + t ) y 1 | ( Lif k 1 ( ln ( 1 + t ) ) Lif k ( ln ( 1 + t ) ) ) x n l n l = 1 n l = 0 n 1 ( n l ) c l ( j = 1 r ( ln ( 1 + t ) ( 1 + t ) a j 1 ) Lif k 1 ( ln ( 1 + t ) ) ( 1 + t ) y 1 | x n l j = 1 r ( ln ( 1 + t ) ( 1 + t ) a j 1 ) Lif k ( ln ( 1 + t ) ) ( 1 + t ) y 1 | x n l ) = 1 n l = 0 n 1 ( n l ) c l ( D n l ( k 1 ) ( y 1 | a 1 , , a r ) D n l ( k ) ( y 1 | a 1 , , a r ) ) .

Since

t j = 1 r ( ln ( 1 + t ) ( 1 + t ) a j 1 ) = 1 1 + t j = 1 r ( ln ( 1 + t ) ( 1 + t ) a j 1 ) i = 1 r ( t ln ( 1 + t ) a i t ( 1 + t ) a i ( 1 + t ) a i 1 ) t ,

with

i = 1 r ( t ln ( 1 + t ) a i t ( 1 + t ) a i ( 1 + t ) a i 1 ) = 1 2 ( i = 1 r a i ) t+

a series with order (≥1), the first term is

j = 1 r ( ln ( 1 + t ) ( 1 + t ) a j 1 ) Lif k ( ln ( 1 + t ) ) ( 1 + t ) y 1 | i = 1 r ( t ln ( 1 + t ) a i t ( 1 + t ) a i ( 1 + t ) a i 1 ) t x n 1 = 1 n j = 1 r ( ln ( 1 + t ) ( 1 + t ) a j 1 ) Lif k ( ln ( 1 + t ) ) ( 1 + t ) y 1 | i = 1 r ( t ln ( 1 + t ) a i t ( 1 + t ) a i ( 1 + t ) a i 1 ) x n = r n j = 1 r ( ln ( 1 + t ) ( 1 + t ) a j 1 ) Lif k ( ln ( 1 + t ) ) ( 1 + t ) y 1 | t ln ( 1 + t ) x n 1 n i = 1 r a i ln ( 1 + t ) ( 1 + t ) a i 1 j = 1 r ( ln ( 1 + t ) ( 1 + t ) a j 1 ) Lif k ( ln ( 1 + t ) ) ( 1 + t ) y + a i 1 | t ln ( 1 + t ) x n = r n j = 1 r ( ln ( 1 + t ) ( 1 + t ) a j 1 ) Lif k ( ln ( 1 + t ) ) ( 1 + t ) y 1 | l = 0 c l t l l ! x n 1 n i = 1 r a i ln ( 1 + t ) ( 1 + t ) a i 1 j = 1 r ( ln ( 1 + t ) ( 1 + t ) a j 1 ) Lif k ( ln ( 1 + t ) ) ( 1 + t ) y + a i 1 | l = 0 c l t l l ! x n = r n l = 0 n ( n l ) c l j = 1 r ( ln ( 1 + t ) ( 1 + t ) a j 1 ) Lif k ( ln ( 1 + t ) ) ( 1 + t ) y 1 | x n l 1 n i = 1 r a i l = 0 n ( n l ) c l × ln ( 1 + t ) ( 1 + t ) a i 1 j = 1 r ( ln ( 1 + t ) ( 1 + t ) a j 1 ) Lif k ( ln ( 1 + t ) ) ( 1 + t ) y + a i 1 | x n l = r n l = 0 n ( n l ) c l D n l ( k ) ( y 1 | a 1 , , a r ) 1 n i = 1 r a i l = 0 n ( n l ) c l D n l ( k ) ( y + a i 1 | a 1 , , a r , a i ) .

Therefore, we obtain

D n ( k ) ( x | a 1 , , a r ) = x D n 1 ( k ) ( x 1 | a 1 , , a r ) + 1 n l = 0 n 1 ( n l ) c l ( D n l ( k 1 ) ( x 1 | a 1 , , a r ) D n l ( k ) ( x 1 | a 1 , , a r ) ) + r n l = 0 n ( n l ) c l D n l ( k ) ( x 1 | a 1 , , a r ) 1 n j = 1 r l = 0 n ( n l ) a j c l D n l ( k ) ( x + a j 1 | a 1 , , a r , a j ) = x D n 1 ( k ) ( x 1 | a 1 , , a r ) + 1 n l = 0 n 1 ( n l ) c l D n l ( k 1 ) ( x 1 | a 1 , , a r ) + r 1 n l = 0 n ( n l ) c l D n l ( k ) ( x 1 | a 1 , , a r ) + 1 n c n 1 n j = 1 r l = 0 n ( n l ) a j c l D n l ( k ) ( x + a j 1 | a 1 , , a r , a j ) = x D n 1 ( k ) ( x 1 | a 1 , , a r ) + 1 n l = 0 n ( n l ) c l D n l ( k 1 ) ( x 1 | a 1 , , a r ) + r 1 n l = 0 n ( n l ) c l D n l ( k ) ( x 1 | a 1 , , a r ) 1 n j = 1 r l = 0 n ( n l ) a j c l D n l ( k ) ( x + a j 1 | a 1 , , a r , a j ) ,

which is the identity (27). □

3.7 A relation including the Stirling numbers of the first kind

Theorem 7 For nm1, we have

m l = 0 n m ( n l ) S 1 ( n l , m ) D l ( k ) ( a 1 , , a r ) = m r n l = 0 n m i = 0 l ( n l ) ( l i ) S 1 ( n l , m ) c l i D i ( k ) ( 1 | a 1 , , a r ) m n l = 0 n m i = 0 l j = 1 r ( n l ) ( l i ) S 1 ( n l , m ) a j c l i D i ( k ) ( a j 1 | a 1 , , a r , a j ) + l = 0 n m ( n 1 l ) S 1 ( n l 1 , m 1 ) D l ( k 1 ) ( 1 | a 1 , , a r ) + ( m 1 ) l = 0 n m ( n 1 l ) S 1 ( n l 1 , m 1 ) D l ( k ) ( 1 | a 1 , , a r ) .
(28)

Proof We shall compute

j = 1 r ( ln ( 1 + t ) ( 1 + t ) a j 1 ) Lif k ( ln ( 1 + t ) ) ( ln ( 1 + t ) ) m | x n

in two different ways. On the one hand,

j = 1 r ( ln ( 1 + t ) ( 1 + t ) a j 1 ) Lif k ( ln ( 1 + t ) ) ( ln ( 1 + t ) ) m | x n = j = 1 r ( ln ( 1 + t ) ( 1 + t ) a j 1 ) Lif k ( ln ( 1 + t ) ) | ( ln ( 1 + t ) ) m x n = j = 1 r ( ln ( 1 + t ) ( 1 + t ) a j 1 ) Lif k ( ln ( 1 + t ) ) | l = 0 m ! ( l + m ) ! S 1 ( l + m , m ) t l + m x n = l = 0 n m m ! ( l + m ) ! S 1 ( l + m , m ) ( n ) l + m j = 1 r ( ln ( 1 + t ) ( 1 + t ) a j 1 ) Lif k ( ln ( 1 + t ) ) | x n l m = l = 0 n m m ! ( n l + m ) S 1 ( l + m , m ) D n l m ( k ) ( a 1 , , a r ) = l = 0 n m m ! ( n l ) S 1 ( n l , m ) D l ( k ) ( a 1 , , a r ) .

On the other hand,

j = 1 r ( ln ( 1 + t ) ( 1 + t ) a j 1 ) Lif k ( ln ( 1 + t ) ) ( ln ( 1 + t ) ) m | x n = t ( j = 1 r ( ln ( 1 + t ) ( 1 + t ) a j 1 ) Lif k ( ln ( 1 + t ) ) ( ln ( 1 + t ) ) m ) | x n 1 = ( t j = 1 r ( ln ( 1 + t ) ( 1 + t ) a j 1 ) ) Lif k ( ln ( 1 + t ) ) ( ln ( 1 + t ) ) m | x n 1 + j = 1 r ( ln ( 1 + t ) ( 1 + t ) a j 1 ) ( t Lif k ( ln ( 1 + t ) ) ) ( ln ( 1 + t ) ) m | x n 1 + j = 1 r ( ln ( 1 + t ) ( 1 + t ) a j 1 ) Lif k ( ln ( 1 + t ) ) ( t ( ln ( 1 + t ) ) m ) | x n 1 .
(29)

The third term of (29) is equal to

m j = 1 r ( ln ( 1 + t ) ( 1 + t ) a j 1 ) Lif k ( ln ( 1 + t ) ) ( 1 + t ) 1 | ( ln ( 1 + t ) ) m 1 x n 1 = m j = 1 r ( ln ( 1 + t ) ( 1 + t ) a j 1 ) Lif k ( ln ( 1 + t ) ) ( 1 + t ) 1 | l = 0 n m ( m 1 ) ! ( l + m 1 ) ! S 1 ( l + m 1 , m 1 ) t l + m 1 x n 1 = m l = 0 n m ( m 1 ) ! ( l + m 1 ) ! S 1 ( l + m 1 , m 1 ) ( n 1 ) l + m 1 × j = 1 r ( ln ( 1 + t ) ( 1 + t ) a j 1 ) Lif k ( ln ( 1 + t ) ) ( 1 + t ) 1 | x n l m = m ! l = 0 n m ( n 1 l + m 1 ) S 1 ( l + m 1 , m 1 ) D n l m ( k ) ( 1 | a 1 , , a r ) = m ! l = 0 n m ( n 1 l ) S 1 ( n l 1 , m 1 ) D l ( k ) ( 1 | a 1 , , a r ) .

The second term of (29) is equal to

j = 1 r ( ln ( 1 + t ) ( 1 + t ) a j 1 ) ( Lif k 1 ( ln ( 1 + t ) ) Lif k ( ln ( 1 + t ) ) ( 1 + t ) ln ( 1 + t ) ) ( ln ( 1 + t ) ) m | x n 1 = j = 1 r ( ln ( 1 + t ) ( 1 + t ) a j 1 ) Lif k 1 ( ln ( 1 + t ) ) ( 1 + t ) 1 | ( ln ( 1 + t ) ) m 1 x n 1 j = 1 r ( ln ( 1 + t ) ( 1 + t ) a j 1 ) Lif k ( ln ( 1 + t ) ) ( 1 + t ) 1 | ( ln ( 1 + t ) ) m 1 x n 1 = ( m 1 ) ! l = 0 n m ( n 1 l ) S 1 ( n l 1 , m 1 ) D l ( k 1 ) ( 1 | a 1 , , a r ) ( m 1 ) ! l = 0 n m ( n 1 l ) S 1 ( n l 1 , m 1 ) D l ( k ) ( 1 | a 1 , , a r ) .

The first term of (29) is equal to

1 1 + t i = 1 r ( ln ( 1 + t ) ( 1 + t ) a i 1 ) j = 1 r ( t ln ( 1 + t ) a j t ( 1 + t ) a j ( 1 + t ) a j 1 ) t Lif k ( ln ( 1 + t ) ) ( ln ( 1 + t ) ) m | x n 1 = i = 1 r ( ln ( 1 + t ) ( 1 + t ) a i 1 ) Lif k ( ln ( 1 + t ) ) ( 1 + t ) 1 ( ln ( 1 + t ) ) m | j = 1 r ( t ln ( 1 + t ) a j t ( 1 + t ) a j ( 1 + t ) a j 1 ) t x n 1 = 1 n i = 1 r ( ln ( 1 + t ) ( 1 + t ) a i 1 ) Lif k ( ln ( 1 + t ) ) ( 1 + t ) 1 ( ln ( 1 + t ) ) m | j = 1 r ( t ln ( 1 + t ) a j t ( 1 + t ) a j ( 1 + t ) a j 1 ) x n = 1 n i = 1 r ( ln ( 1 + t ) ( 1 + t ) a i 1 ) Lif k ( ln ( 1 + t ) ) ( 1 + t ) 1 × j = 1 r ( t ln ( 1 + t ) a j t ( 1 + t ) a j ( 1 + t ) a j 1 ) | ( ln ( 1 + t ) ) m x n = 1 n i = 1 r ( ln ( 1 + t ) ( 1 + t ) a i 1 ) Lif k ( ln ( 1 + t ) ) ( 1 + t ) 1 × j = 1 r ( t ln ( 1 + t ) a j t ( 1 + t ) a j ( 1 + t ) a j 1 ) | l = 0 m ! ( l + m ) ! S 1 ( l + m , m ) t l + m x n = 1 n l = 0 n m m ! ( l + m ) ! S 1 ( l + m , m ) ( n ) l + m i = 1 r ( ln ( 1 + t ) ( 1 + t ) a i 1 ) Lif k ( ln ( 1 + t ) ) ( 1 + t ) 1 × j = 1 r ( t ln ( 1 + t ) a j t ( 1 + t ) a j ( 1 + t ) a j 1 ) | x n l m = m ! n l = 0 n m ( n l + m ) S 1 ( l + m , m ) × ( r i = 1 r ( ln ( 1 + t ) ( 1 + t ) a i 1 ) Lif k ( ln ( 1 + t ) ) ( 1 + t ) 1 | t ln ( 1 + t ) x n l m j = 1 r a j ln ( 1 + t ) ( 1 + t ) a j 1 i = 1 r ( ln ( 1 + t ) ( 1 + t ) a i 1 ) Lif k ( ln ( 1 + t ) ) ( 1 + t ) a j 1 | t ln ( 1 + t ) x n l m ) = m ! n l = 0 n m ( n l + m ) S 1 ( l + m , m ) × ( r i = 1 r ( ln ( 1 + t ) ( 1 + t ) a i 1 ) Lif k ( ln ( 1 + t ) ) ( 1 + t ) 1 | ν = 0 c ν t ν ν ! x n l m j = 1 r a j ln ( 1 + t ) ( 1 + t ) a j 1 i = 1 r ( ln ( 1 + t ) ( 1 + t ) a i 1 ) Lif k ( ln ( 1 + t ) ) ( 1 + t ) a j 1 | ν = 0 c ν t ν ν ! x n l m ) = m ! n l = 0 n m ( n l + m ) S 1 ( l + m , m ) × ( r ν = 0 n l m ( n l m ν ) c ν i = 1 r ( ln ( 1 + t ) ( 1 + t ) a i 1 ) Lif k ( ln ( 1 + t ) ) ( 1 + t ) 1 | x n l m ν j = 1 r a j ν = 0 n l m ( n l m ν ) c ν × ln ( 1 + t ) ( 1 + t ) a j 1 i = 1 r ( ln ( 1 + t ) ( 1 + t ) a i 1 ) Lif k ( ln ( 1 + t ) ) ( 1 + t ) a j 1 | x n l m ν ) = m ! n l = 0 n m ( n l + m ) S 1 ( l + m , m ) × ( r ν = 0 n l m ( n l m ν ) c ν D n l m ν ( k ) ( 1 | a 1 , , a r ) j = 1 r ν = 0 n l m ( n l m ν ) a j c ν D n l m ν ( k ) ( a j 1 | a 1 , , a r , a j ) ) = m ! n l = 0 n m ( n l ) S 1 ( n l , m ) × ( r i = 0 l ( l i ) c i D n i ( k ) ( 1 | a 1 , , a r ) j = 1 r i = 0 l ( l i ) a j c i D l i ( k ) ( a j 1 | a 1 , , a r , a j ) ) .

Therefore, we get for nm1

m ! l = 0 n m ( n l ) S 1 ( n l , m ) D l ( k ) ( a 1 , , a r ) = m ! r n l = 0 n m i = 0 l ( n l ) ( l i ) S 1 ( n l , m ) c i D l i ( k ) ( 1 | a 1 , , a r ) m ! 1 n l = 0 n m i = 0 l j = 1 r ( n l ) ( l i ) S 1 ( n l , m ) a j c i D l i ( k ) ( a j 1 | a 1 , , a r , a j ) + ( m 1 ) ! l = 0 n m ( n 1 l ) S 1 ( n l 1 , m 1 ) D l ( k 1 ) ( 1 | a 1 , , a r ) ( m 1 ) ! l = 0 n m ( n 1 l ) S 1 ( n l 1 , m 1 ) D l ( k ) ( 1 | a 1 , , a r ) + m ! l = 0 n m ( n 1 l ) S 1 ( n l 1 , m 1 ) D l ( k ) ( 1 | a 1 , , a r ) .

Dividing both sides by (m1)!, we obtain for nm1

m l = 0 n m ( n l ) S 1 ( n l , m ) D l ( k ) ( a 1 , , a r ) = m r n l = 0 n m i = 0 l ( n l ) ( l i ) S 1 ( n l , m ) c l i D i ( k ) ( 1 | a 1 , , a r ) m n l = 0 n m i = 0 l j = 1 r ( n l ) ( l i ) S 1 ( n l , m ) a j c l i D i ( k ) ( a j 1 | a 1 , , a r , a j ) + l = 0 n m ( n 1 l ) S 1 ( n l 1 , m 1 ) D l ( k 1 ) ( 1 | a 1 , , a r ) + ( m 1 ) l = 0 n m ( n 1 l ) S 1 ( n l 1 , m 1 ) D l ( k ) ( 1 | a 1 , , a r ) .

Thus, we get (28). □

3.8 A relation with the falling factorials

Theorem 8

D n ( k ) (x| a 1 ,, a r )= m = 0 n ( n m ) D n m ( k ) ( a 1 ,, a r ) ( x ) m .
(30)

Proof For (12) and (19), assume that D n ( k ) (x| a 1 ,, a r )= m = 0 n C n , m ( x ) m . By (11), we have

C n , m = 1 m ! 1 j = 1 r ( e a j ln ( 1 + t ) 1 ln ( 1 + t ) ) 1 Lif k ( ln ( 1 + t ) ) t m | x n = 1 m ! j = 1 r ( ln ( 1 + t ) ( 1 + t ) a j 1 ) Lif k ( ln ( 1 + t ) ) | t m x n = ( n m ) j = 1 r ( ln ( 1 + t ) ( 1 + t ) a j 1 ) Lif k ( ln ( 1 + t ) ) | x n m = ( n m ) D n m ( k ) ( a 1 , , a r ) .

Thus, we get the identity (30). □

3.9 A relation with higher-order Frobenius-Euler polynomials

For λC with λ1, the Frobenius-Euler polynomials of order r, H n ( r ) (x|λ) are defined by the generating function

( 1 λ e t λ ) r e x t = n = 0 H n ( r ) (x|λ) t n n !

(see e.g. [11]).

Theorem 9

D n ( k ) ( x | a 1 , , a r ) = m = 0 n ( j = 0 n m l = 0 n m j ( s j ) ( n j l ) ( n ) j × ( 1 λ ) j S 1 ( n j l , m ) D l ( k ) ( a 1 , , a r ) ) H m ( s ) ( x | λ ) .
(31)

Proof For (12) and

H n ( s ) (x|λ) ( ( e t λ 1 λ ) s , t ) ,
(32)

assume that D n ( k ) (x| a 1 ,, a r )= m = 0 n C n , m H m ( s ) (x|λ). By (11), similarly to the proof of (28), we have

C n , m = 1 m ! ( e ln ( 1 + t ) λ 1 λ ) s j = 1 r ( e a j ln ( 1 + t ) 1 ln ( 1 + t ) ) 1 Lif k ( ln ( 1 + t ) ) ( ln ( 1 + t ) ) m | x n = 1 m ! ( 1 λ ) s j = 1 r ( ln ( 1 + t ) ( 1 + t ) a j 1 ) Lif k ( ln ( 1 + t ) ) ( ln ( 1 + t ) ) m ( 1 λ + t ) s | x n = 1 m ! ( 1 λ ) s × j = 1 r ( ln ( 1 + t ) ( 1 + t ) a j 1 ) Lif k ( ln ( 1 + t ) ) ( ln ( 1 + t ) ) m | i = 0 min { s , n } ( s i ) ( 1 λ ) s i t i x n = 1 m ! ( 1 λ ) s i = 0 n m ( s i ) ( 1 λ ) s i ( n ) i × j = 1 r ( ln ( 1 + t ) ( 1 + t ) a j 1 ) Lif k ( ln ( 1 + t ) ) ( ln ( 1 + t ) ) m | x n i = 1 m ! ( 1 λ ) s i = 0 n m ( s i ) ( 1 λ ) s i ( n ) i l = 0 n m i m ! ( n i l ) S 1 ( n i l , m ) D l ( k ) ( a 1 , , a r ) = i = 0 n m l = 0 n m i ( s i ) ( n i l ) ( n ) i ( 1 λ ) i S 1 ( n i l , m ) D l ( k ) ( a 1 , , a r ) .

Thus, we get the identity (31). □

3.10 A relation with higher-order Bernoulli polynomials

Bernoulli polynomials B n ( r ) (x) of order r are defined by

( t e t 1 ) r e x t = n = 0 B n ( r ) ( x ) n ! t n

(see e.g. [7], Section 2.2]). In addition, Cauchy numbers of the first kind C n ( r ) of order r are defined by

( t ln ( 1 + t ) ) r = n = 0 C n ( r ) n ! t n

(see e.g. [12], (2.1)], [13], (6)]).

Theorem 10

D n ( k ) ( x | a 1 , , a r ) = m = 0 n ( i = 0 n m l = 0 n m i ( n i ) ( n i l ) C i ( s ) S 1 ( n i l , m ) D l ( k ) ( a 1 , , a r ) ) B m ( s ) ( x ) .
(33)

Proof For (12) and

B n ( s ) (x) ( ( e t 1 t ) s , t ) ,
(34)

assume that D n ( k ) (x| a 1 ,, a r )= m = 0 n C n , m B m ( s ) (x). By (11), similarly to the proof of (28), we have

C n , m = 1 m ! ( e ln ( 1 + t ) 1 ln ( 1 + t ) ) s j = 1 r ( e a j ln ( 1 + t ) 1 ln ( 1 + t ) ) 1 Lif k ( ln ( 1 + t ) ) ( ln ( 1 + t ) ) m | x n = 1 m ! j = 1 r ( ln ( 1 + t ) ( 1 + t ) a j 1 ) Lif k ( ln ( 1 + t ) ) ( ln ( 1 + t ) ) m | ( t ln ( 1 + t ) ) s x n = 1 m ! j = 1 r ( ln ( 1 + t ) ( 1 + t ) a j 1 ) Lif k ( ln ( 1 + t ) ) ( ln ( 1 + t ) ) m | i = 0 C i ( s ) t i i ! x n = 1 m ! i = 0 n m C i ( s ) ( n i ) j = 1 r ( ln ( 1 + t ) ( 1 + t ) a j 1 ) Lif k ( ln ( 1 + t ) ) ( ln ( 1 + t ) ) m | x n i = 1 m ! i = 0 n m C i ( s ) ( n i ) l = 0 n m i m ! ( n i l ) S 1 ( n i l , m ) D l ( k ) ( a 1 , , a r ) = i = 0 n m l = 0 n m i ( n i ) ( n i l ) C i ( s ) S 1 ( n i l , m ) D l ( k ) ( a 1 , , a r ) .

Thus, we get the identity (33). □

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Acknowledgements

This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (MOE) (No. 2012R1A1A2003786). The second author was supported by Kwangwoon University in 2014.

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Kim, D.S., Kim, T., Komatsu, T. et al. Barnes-type Daehee of the first kind and poly-Cauchy of the first kind mixed-type polynomials. Adv Differ Equ 2014, 140 (2014). https://doi.org/10.1186/1687-1847-2014-140

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