Open Access

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

Advances in Difference Equations20142014:140

https://doi.org/10.1186/1687-1847-2014-140

Received: 24 February 2014

Accepted: 25 April 2014

Published: 9 May 2014

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 ) ( k Z ) 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 ) , 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 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 , 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 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 ( 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 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 ) and f ( 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 ( 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 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 ( n l , 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 ) ( x 1 | 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 n 1 , 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 n m 1 , 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 n m 1
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 ( m 1 ) ! , we obtain for n m 1
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 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 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 ×