Open Access

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

Journal of Inequalities and Applications20142014:214

https://doi.org/10.1186/1029-242X-2014-214

Received: 14 March 2014

Accepted: 22 May 2014

Published: 29 May 2014

Abstract

In this paper, we introduce the mixed-type polynomials: Barnes-type Daehee polynomials of the second kind and poly-Cauchy polynomials of the second kind. From the properties of Sheffer sequences of these polynomials arising from umbral calculus, we derive new and interesting identities.

MSC:05A19, 05A40, 11B68, 11B75.

1 Introduction

In this paper, we consider the polynomials D ˆ n ( k ) ( x | a 1 , , a r ) called the Barnes-type Daehee of the second kind and poly-Cauchy of the second kind mixed-type polynomials, whose generating function is given by
j = 1 r ( ln ( 1 + t ) ( 1 + t ) a j ( 1 + t ) a j 1 ) Lif k ( ln ( 1 + t ) ) ( 1 + t ) x = n = 0 D ˆ n ( k ) ( x | a 1 , , a r ) t n n ! ,
(1)
where a 1 , , a r 0 . Here, Lif k ( x ) ( k Z ) is the polyfactorial function [1] defined by
Lif k ( x ) = m = 0 x m m ! ( m + 1 ) k .

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

Recall that the Barnes-type Daehee polynomials of the second kind, denoted by D ˆ n ( x | a 1 , , a r ) , are given by the generating function to be
j = 1 r ( ln ( 1 + t ) ( 1 + t ) a j ( 1 + t ) a j 1 ) ( 1 + t ) x = n = 0 D ˆ n ( x | a 1 , , a r ) t n n ! .

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

The poly-Cauchy polynomials of the second kind, denoted by c ˆ n ( k ) ( x ) [5, 6], are given by the generating function as follows:
Lif k ( ln ( 1 + t ) ) ( 1 + t ) x = n = 0 c ˆ n ( k ) ( x ) t n n ! .

In this paper, we introduce the mixed-type polynomials: Barnes-type Daehee polynomials of the second kind and poly-Cauchy polynomials of the second kind. 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 ) , c L | p ( x ) = c L | 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 ( n 0 ) .
(3)
In particular,
t k | x n = n ! δ n , k ( n , k 0 ) ,
(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 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 , k 0 . 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 in 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 ( y C ) ,

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 ) ( n 0 ) ,
(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 ) ( n 1 ) .
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 ) ( n 0 ) .
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 definition (1), D ˆ 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 e a j t ) 1 Lif k ( t ) and f ( t ) = e t 1 .
So,
D ˆ n ( k ) ( x | a 1 , , a r ) ( j = 1 r ( e a j t 1 t e a j 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 as follows:
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 ( n 1 ) ( n j + 1 ) ( j 1 ) 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 ˆ n ( k ) ( x | a 1 , , a r ) = m = 0 n l = 0 m S 1 ( n , m ) ( 1 ) m l ( m l ) ( m l + 1 ) k B l ( x + a 1 + + a r | a 1 , , a r )
(14)
= j = 0 n l = 0 n j i = 0 l ( n l ) ( l i ) S 1 ( n l , j ) c ˆ i ( k ) D ˆ l i ( a 1 , , a r ) x j
(15)
= l = 0 n ( n l ) D ˆ n l ( a 1 , , a r ) c ˆ l ( k ) ( x )
(16)
= l = 0 n ( n l ) c ˆ n l ( k ) D ˆ l ( x | a 1 , , a r ) .
(17)
Proof Since
j = 1 r ( e a j t 1 t e a j t ) 1 Lif k ( t ) D ˆ n ( k ) ( x | a 1 , , a r ) ( 1 , e t 1 )
(18)
and
( x ) n ( 1 , e t 1 ) ,
(19)
we have
D ˆ n ( k ) ( x | a 1 , , a r ) = j = 1 r ( t e a j 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 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 e a j t 1 ) l = 0 m ( 1 ) l t l l ! ( l + 1 ) k x m = m = 0 n S 1 ( n , m ) j = 1 r ( t e a j t e a j t 1 ) l = 0 m ( 1 ) l ( m ) l l ! ( l + 1 ) k x m l = m = 0 n S 1 ( n , m ) l = 0 m ( 1 ) l ( m ) l l ! ( l + 1 ) k j = 1 r ( t e a j t e a j t 1 ) x m l = m = 0 n l = 0 m S 1 ( n , m ) ( 1 ) l ( m l ) ( l + 1 ) k B m l ( x + a 1 + + a r | a 1 , , a r ) = m = 0 n l = 0 m S 1 ( n , m ) ( 1 ) m l ( m l ) ( m l + 1 ) k B l ( x + a 1 + + a r | a 1 , , a r ) .

So, 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 + 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 + 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 + 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 + t ) a j 1 ) | i = 0 n l j c ˆ 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 ˆ i ( k ) ( n l j ) i i ! j = 1 r ( ln ( 1 + t ) ( 1 + t ) a j ( 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 ˆ i ( k ) ( n l j ) i i ! m = 0 D ˆ 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 ˆ i ( k ) D ˆ 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 ˆ i ( k ) D ˆ l i ( a 1 , , a r ) .
Thus, we obtain
D ˆ n ( k ) ( x | a 1 , , a r ) = j = 0 n l = 0 n j i = 0 l ( n l ) ( l i ) S 1 ( n l , j ) c ˆ i ( k ) D ˆ l i ( a 1 , , a r ) x j ,

which is identity (15).

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

Thus, we obtain (16).

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

Thus, we get identity (17). □

3.2 Sheffer identity

Theorem 2
D ˆ n ( k ) ( x + y | a 1 , , a r ) = j = 0 n ( n j ) D ˆ 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 e a j t ) 1 Lif k ( t ) D ˆ n ( k ) ( x | a 1 , , a r ) = ( x ) n ( 1 , e t 1 ) ,

using (10), we have (20). □

3.3 Difference relations

Theorem 3
D ˆ n ( k ) ( x + 1 | a 1 , , a r ) D ˆ n ( k ) ( x | a 1 , , a r ) = n D ˆ n 1 ( k ) ( x | a 1 , , a r ) .
(21)
Proof By (8) with (12), we get
( e t 1 ) D ˆ n ( k ) ( x | a 1 , , a r ) = n D ˆ n 1 ( k ) ( x | a 1 , , a r ) .

By (7), we have (21). □

3.4 Recurrence

Theorem 4
D ˆ n + 1 ( k ) ( x | a 1 , , a r ) = x D ˆ n ( k ) ( x 1 | a 1 , , a r ) m = 0 n j = 1 r l = 0 m i = 0 l ( 1 ) l i ( 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 1 | a 1 , , a r ) + j = 1 r a j m = 0 n i = 0 m ( 1 ) m i ( m i ) ( m i + 1 ) k S 1 ( n , m ) × B i ( x + a 1 + + a r 1 | a 1 , , a r ) m = 0 n l = 0 m + 1 ( 1 ) m + 1 l ( m l ) ( m + 2 l ) k S 1 ( n , m ) × B l ( x + a 1 + + a r 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 ˆ n + 1 ( k ) ( x | a 1 , , a r ) = x D ˆ n ( k ) ( x 1 | a 1 , , a r ) e t g ( t ) g ( t ) D ˆ 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 ( j = 1 r a j ) t ln Lif k ( t ) ) = j = 1 r a j e a j t e a j t 1 r t j = 1 r a j + 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 ) j = 1 r a j + 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 ˆ n ( k ) ( x | a 1 , , a r ) = j = 1 r ( t e a j 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 e a j t 1 ) Lif k ( t ) x m ,
we have
g ( t ) g ( t ) D ˆ n ( k ) ( x | a 1 , , a r ) = m = 0 n S 1 ( n , m ) Lif k ( t ) ( j = 1 r t e a j t e a j t 1 ) g ( t ) g ( t ) x m = m = 0 n S 1 ( n , m ) Lif k ( t ) ( j = 1 r t e a j 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 j = 1 r a j m = 0 n S 1 ( n , m ) Lif k ( t ) ( j = 1 r t e a j t e a j t 1 ) x m + m = 0 n S 1 ( n , m ) ( j = 1 r t e a j t e a j t 1 ) Lif k ( t ) 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 ( a j ) l B l l ! t l 1 ) x m + 1 = 1 m + 1 j = 1 r l = 0 m ( 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 = 0 m ( m + 1 l ) ( a j ) m + 1 l B m + 1 l Lif k ( t ) ( j = 1 r t e a j t e a j t 1 ) x l = m = 0 n S 1 ( n , m ) m + 1 j = 1 r l = 0 m ( m + 1 l ) ( a j ) m + 1 l B m + 1 l × i = 0 l ( 1 ) i t i i ! ( i + 1 ) k B l ( x + a 1 + + a r | a 1 , , a r ) = m = 0 n j = 1 r l = 0 m i = 0 l ( 1 ) l i ( 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 | a 1 , , a r ) .
Since
Lif k 1 ( t ) Lif k ( t ) = ( 1 2 k 1 2 k 1 ) t + ,
(25)
the second term in (24) is
j = 1 r a j m = 0 n S 1 ( n , m ) Lif k ( t ) B m ( x + a 1 + + a r | a 1 , , a r ) = j = 1 r a j m = 0 n S 1 ( n , m ) i = 0 m ( 1 ) i t i i ! ( i + 1 ) k B m ( x + a 1 + + a r | a 1 , , a r ) = j = 1 r a j m = 0 n i = 0 m ( 1 ) m i ( m i ) ( m i + 1 ) k S 1 ( n , m ) B i ( x + a 1 + + a r | a 1 , , a r ) .
The third 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 | 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 | a 1 , , a r ) m + 1 = m = 0 n S 1 ( n , m ) m + 1 ( l = 0 m + 1 ( 1 ) l t l l ! ( l + 1 ) k 1 B m + 1 ( x + a 1 + + a r | a 1 , , a r ) l = 0 m + 1 ( 1 ) l t l l ! ( l + 1 ) k B m + 1 ( x + a 1 + + a r | a 1 , , a r ) ) = m = 0 n l = 0 m + 1 ( 1 ) m + 1 l ( m l ) ( m + 2 l ) k S 1 ( n , m ) B l ( x + a 1 + + a r | a 1 , , a r ) .

Thus we have identity (22). □

3.5 Differentiation

Theorem 5
d d x D ˆ n ( k ) ( x | a 1 , , a r ) = n ! l = 0 n 1 ( 1 ) n l 1 l ! ( n l ) D ˆ 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 = ( 1 ) n l 1 ( n l 1 ) ! ,
with (12), we have
d d x D ˆ n ( k ) ( x | a 1 , , a r ) = l = 0 n 1 ( n l ) ( 1 ) n l 1 ( n l 1 ) ! D ˆ l ( k ) ( x | a 1 , , a r ) = n ! l = 0 n 1 ( 1 ) n l 1 l ! ( n l ) D ˆ l ( k ) ( x | a 1 , , a r ) ,

which is identity (26). □

3.6 More relations

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 ˆ n ( k ) ( x | a 1 , , a r ) = ( x + j = 1 r a j ) D ˆ n 1 ( k ) ( x 1 | a 1 , , a r ) + 1 n l = 0 n ( n l ) c l D ˆ n l ( k 1 ) ( x 1 | a 1 , , a r ) + r 1 n l = 0 n ( n l ) c l D ˆ n l ( k ) ( x 1 | a 1 , , a r ) 1 n j = 1 r l = 0 n ( n l ) a j c l D ˆ n l ( k ) ( x 1 | a 1 , , a r , a j ) .
(27)
Proof For n 1 , we have
D ˆ n ( k ) ( y | a 1 , , a r ) = l = 0 D ˆ l ( k ) ( y | a 1 , , a r ) t l l ! | x n = j = 1 r ( ln ( 1 + t ) ( 1 + t ) a j ( 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 + 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 + 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 ) 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 + 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 + 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 + 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 + 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 = 1 n l = 0 n 1 ( n l ) c l ( j = 1 r ( ln ( 1 + t ) ( 1 + t ) a j ( 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 + 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 ˆ n l ( k 1 ) ( y 1 | a 1 , , a r ) D ˆ n l ( k ) ( y 1 | a 1 , , a r ) ) .
Observe that
t j = 1 r ( ln ( 1 + t ) ( 1 + t ) a j ( 1 + t ) a j 1 ) = 1 1 + t j = 1 r ( ln ( 1 + t ) ( 1 + t ) a j ( 1 + t ) a j 1 ) j = 1 r ( t ln ( 1 + t ) a j t ( 1 + t ) a j ( 1 + t ) a j 1 ) t + ( j = 1 r a j ) 1 1 + t j = 1 r ( ln ( 1 + t ) ( 1 + t ) a j ( 1 + t ) a j 1 ) ,
with
j = 1 r ( t ln ( 1 + t ) a j t ( 1 + t ) a j ( 1 + t ) a j 1 ) = 1 2 ( j = 1 r a j ) t +

a series with order ≥1.

Now, the first term is
j = 1 r ( ln ( 1 + t ) ( 1 + t ) a j ( 1 + t ) a j 1 ) Lif k ( ln ( 1 + t ) ) ( 1 + t ) y 1 | j = 1 r ( t ln ( 1 + t ) a j t ( 1 + t ) a j ( 1 + t ) a j 1 ) t x n 1 + j = 1 r a j j = 1 r ( ln ( 1 + t ) ( 1 + t ) a j ( 1 + t ) a j 1 ) Lif k ( ln ( 1 + t ) ) ( 1 + t ) y 1 | x n 1 = ( j = 1 r a j ) D ˆ n 1 ( k ) ( y 1 | a 1 , , a r ) + r n j = 1 r ( ln ( 1 + t ) ( 1 + t ) a j ( 1 + t ) a j 1 ) Lif k ( ln ( 1 + t ) ) ( 1 + t ) y 1 | t ln ( 1 + t ) x n 1 n j = 1 r a j ln ( 1 + t ) ( 1 + t ) a j ( 1 + t ) a j 1 j = 1 r ( ln ( 1 + t ) ( 1 + t ) a j ( 1 + t ) a j 1 ) Lif k ( ln ( 1 + t ) ) × ( 1 + t ) y 1 | t ln ( 1 + t ) x n = ( j = 1 r a j ) D ˆ n 1 ( k ) ( y 1 | a 1 , , a r ) + r n l = 0 n ( n l ) c l D ˆ n l ( k ) ( y 1 | a 1 , , a r ) 1 n i = 1 r a j l = 0 n ( n l ) c l D ˆ n l ( k ) ( y 1 | a 1 , , a r , a j ) .
Altogether, we obtain
D ˆ n ( k ) ( x | a 1 , , a r ) = ( x + j = 1 r a j ) D ˆ n 1 ( k ) ( x 1 | a 1 , , a r ) + 1 n l = 0 n 1 ( n l ) c l ( D ˆ n l ( k 1 ) ( x 1 | a 1 , , a r ) D ˆ n l ( k ) ( x 1 | a 1 , , a r ) ) + r n l = 0 n ( n l ) c l D ˆ n l ( k ) ( x 1 | a 1 , , a r ) 1 n j = 1 r l = 0 n ( n l ) a j c l D ˆ n l ( k ) ( x 1 | a 1 , , a r , a j ) ,

from which identity (27) follows. □

3.7 A relation including the Stirling numbers of the first kind

Theorem 7 For n 1 m 1 , we have
m l = 0 n m ( n l ) S 1 ( n l , m ) D ˆ l ( k ) ( a 1 , , a r ) = m j = 1 r a j l = 0 n m 1 ( n 1 l ) S 1 ( n l 1 , m ) D ˆ l ( k ) ( 1 | 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 ˆ 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 ˆ i ( k ) ( 1 | a 1 , , a r , a j ) + l = 0 n m ( n 1 l ) S 1 ( n l 1 , m 1 ) D ˆ 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 ˆ l ( k ) ( 1 | a 1 , , a r ) .
(28)
Proof We shall compute
j = 1 r ( ln ( 1 + t ) ( 1 + t ) a j ( 1 + t ) a j 1 ) Lif k ( ln ( 1 + t ) ) ( ln ( 1 + t ) ) m | x n
in two different ways. On the one hand, it is
j = 1 r ( ln ( 1 + t ) ( 1 + t ) a j ( 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 + 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 ! ( n l ) S 1 ( n l , m ) D ˆ l ( k ) ( a 1 , , a r ) .
On the other hand, it is
t ( j = 1 r ( ln ( 1 + t ) ( 1 + t ) a j ( 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 + 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 ) 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 + 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 + 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 + 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 ( n 1 l ) S 1 ( n l 1 , m 1 ) D ˆ 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 + t ) a j 1 ) ( Lif k 1 ( ln ( 1 + t ) ) Lif k 1 ( 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 + 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 + 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 ˆ 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 ˆ l ( k ) ( 1 | a 1 , , a r ) .
The first term of (29) is equal to
1 n j = 1 r ( ln ( 1 + t ) ( 1 + t ) a j ( 1 + t ) a j 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 + j = 1 r a j j = 1 r ( ln ( 1 + t ) ( 1 + t ) a j ( 1 + t ) a j 1 ) Lif k ( ln ( 1 + t ) ) ( 1 + t ) 1 | ( ln ( 1 + t ) ) m x n 1 = m ! j = 1 r a j l = 0 n m 1 ( n 1 l + m ) S 1 ( l + m , m ) D ˆ n l m 1 ( k ) ( 1 | a 1 , , a r ) + m ! n l = 0 n m ( n l + m ) S 1 ( l + m , m ) × ( r j = 1 r ( ln ( 1 + t ) ( 1 + t ) a j ( 1 + t ) a j 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 + t ) a j 1 j = 1 r ( ln ( 1 + t ) ( 1 + t ) a j ( 1 + t ) a j 1 ) × Lif k ( ln ( 1 + t ) ) ( 1 + t ) 1 | t ln ( 1 + t ) x n l m ) = m ! j = 1 r a j l = 0 n m 1 ( n 1 l ) S 1 ( n l 1 , m ) D ˆ l ( k ) ( 1 | a 1 , , a r ) + m ! n l = 0 n m ( n l ) S 1 ( n l , m ) ( r i = 0 l ( l i ) c i D ˆ l i ( k ) ( 1 | a 1 , , a r ) j = 1 r i = 0 l ( l i ) a j c i D ˆ l i ( k ) ( 1 | a 1 , , a r , a j ) ) .
Therefore, we get, for n 1 m 1 ,
m ! l = 0 n m ( n l ) S 1 ( n l , m ) D ˆ l ( k ) ( a 1 , , a r ) = m ! j = 1 r a j l = 0 n m 1 ( n 1 l ) S 1 ( n l 1 , m ) D ˆ l ( k ) ( 1 | 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 ˆ 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 ˆ l i ( k ) ( 1 | a 1 , , a r , a j ) + ( m 1 ) ! l = 0 n m ( n 1 l ) S 1 ( n l 1 , m 1 ) D ˆ 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 ˆ l ( k ) ( 1 | a 1 , , a r ) + m ! l = 0 n m ( n 1 l ) S 1 ( n l 1 , m 1 ) D ˆ l ( k ) ( 1 | a 1 , , a r ) .

Dividing both sides by ( m 1 ) ! , we get (28). □

3.8 A relation with the falling factorials

Theorem 8
D ˆ n ( k ) ( x | a 1 , , a r ) = m = 0 n ( n m ) D ˆ n m ( k ) ( a 1 , , a r ) ( x ) m .
(30)
Proof For (12) and (19), assume that D ˆ n ( k ) ( x | a 1 , , a r ) = m = 0 n C n , m ( x ) m . By (11), we have
C n , m = 1 m ! j = 1 r ( ln ( 1 + t ) ( 1 + t ) a j ( 1 + t ) a j 1 ) Lif k ( ln ( 1 + t ) ) | t m x n = ( n m ) D ˆ n m ( k ) ( a 1 , , a r ) .

Thus, we get identity (30). □

3.9 A relation with higher-order Frobenius-Euler polynomials

For λ C with λ 1 , 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 ˆ 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 ˆ 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 ˆ 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 ! ( 1 λ ) s j = 1 r ( ln ( 1 + t ) ( 1 + t ) a j ( 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 + 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 + 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 ˆ 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 ˆ l ( k ) ( a 1 , , a r ) .

Thus, we get 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, the 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 ˆ 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 ˆ 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 ˆ 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 ! j = 1 r ( ln ( 1 + t ) ( 1 + t ) a j ( 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 + 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 + 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 ˆ 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 ˆ l ( k ) ( a 1 , , a r ) .

Thus, we get identity (33). □

Declarations

Acknowledgements

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

Authors’ Affiliations

(1)
Department of Mathematics, Sogang University
(2)
Department of Mathematics, Kwangwoon University

References

  1. Komatsu T: Poly-Cauchy numbers. Kyushu J. Math. 2013, 67: 143–153. 10.2206/kyushujm.67.143MathSciNetView ArticleMATHGoogle Scholar
  2. Kim T: An invariant p -adic integral associated with Daehee numbers. Integral Transforms Spec. Funct. 2002, 13: 65–69. 10.1080/10652460212889MathSciNetView ArticleMATHGoogle Scholar
  3. Kim DS, Kim T, Rim S-H: On the associated sequence of special polynomials. Adv. Stud. Contemp. Math. (Kyungshang) 2013, 23: 355–366.MathSciNetMATHGoogle Scholar
  4. Ozden H, Cangul IN, Simsek Y: Remarks on q -Bernoulli numbers associated with Daehee numbers. Adv. Stud. Contemp. Math. (Kyungshang) 2009, 18: 41–48.MathSciNetMATHGoogle Scholar
  5. Kim T: On Euler-Barnes multiple zeta functions. Russ. J. Math. Phys. 2003, 10: 261–267.MathSciNetMATHGoogle Scholar
  6. Simsek Y, Rim S-H, Jang LC, Kang D-G, Seo J-J: A note on q -Daehee sums. J. Anal. Comput. 2005, 1: 151–160.MathSciNetGoogle Scholar
  7. Roman S: The Umbral Calculus. Dover, New York; 2005.MATHGoogle Scholar
  8. Dolgy, DV, Kim, DS, Kim, T, Komatsu, T, Lee, S-H: Barnes’ multiple Bernoulli and poly-Bernoulli mixed-type polynomials. J. Comput. Anal. Appl. (to appear)Google Scholar
  9. Kim DS, Kim T: Daehee numbers and polynomials. Appl. Math. Sci. 2013, 7: 5969–5976.MathSciNetGoogle Scholar
  10. Comtet L: Advanced Combinatorics. Reidel, Dordrecht; 1974.View ArticleMATHGoogle Scholar
  11. Kim DS, Kim T: Some identities of Frobenius-Euler polynomials arising from umbral calculus. Adv. Differ. Equ. 2012., 2012: Article ID 196Google Scholar
  12. Carlitz L: A note on Bernoulli and Euler polynomials of the second kind. Scr. Math. 1961, 25: 323–330.MathSciNetMATHGoogle Scholar
  13. Liang H, Wuyungaowa : Identities involving generalized harmonic numbers and other special combinatorial sequences. J. Integer Seq. 2012., 15: Article ID 12.9.6Google Scholar

Copyright

© Kim and Kim; licensee Springer. 2014

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/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.