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Barnes-type Narumi of the second kind and Barnes-type Peters of the second kind hybrid polynomials

Journal of Inequalities and Applications20142014:376

https://doi.org/10.1186/1029-242X-2014-376

Received: 23 July 2014

Accepted: 10 September 2014

Published: 29 September 2014

Abstract

In this paper, by considering Barnes-type Narumi polynomials of the second kind and Barnes-type Peters polynomials of the second kind, we define and investigate the hybrid 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, 11B83.

1 Introduction

In this paper, we consider the polynomials
NS ˆ n ( x ) = NS ˆ n ( x | a ; λ ; μ ) = NS ˆ n ( x | a 1 , , a r ; λ 1 , , λ s ; μ 1 , , μ s )
called Barnes-type Narumi of the second kind and Barnes-type Peters of the second kind hybrid polynomials, whose generating function is given by
i = 1 r ( ( 1 + t ) a i 1 ( 1 + t ) a i ln ( 1 + t ) ) j = 1 s ( ( 1 + t ) λ j 1 + ( 1 + t ) λ j ) μ j ( 1 + t ) x = n = 0 NS ˆ n ( x | a 1 , , a r ; λ 1 , , λ s ; μ 1 , , μ s ) t n n ! ,
(1)

where a 1 , , a r , λ 1 , , λ s , μ 1 , , μ s C with a 1 , , a r , λ 1 , , λ s 0 . When x = 0 , NS ˆ n = NS ˆ n ( 0 ) = NS ˆ n ( 0 | a ; λ ; μ ) = NS ˆ n ( 0 | a 1 , , a r ; λ 1 , , λ s ; μ 1 , , μ s ) are called Barnes-type Narumi of the second kind and Barnes-type Peters of the second kind hybrid numbers.

Recall that the Barnes-type Narumi polynomials of the second kind, denoted by N ˆ n ( x | a 1 , , a r ) , are given by the generating function as
i = 1 r ( ( 1 + t ) a i 1 ( 1 + t ) a i ln ( 1 + t ) ) ( 1 + t ) x = n = 0 N ˆ n ( x | a 1 , , a r ) t n n ! .
If x = 0 , we write N ˆ n ( a 1 , , a r ) = N ˆ n ( 0 | a 1 , , a r ) . Narumi polynomials were mentioned in [[1], p.127]. In addition, the Barnes-type Peters polynomials of the second kind, denoted by s ˆ n ( x | λ 1 , , λ s ; μ 1 , , μ s ) , are given by the generating function as
j = 1 s ( ( 1 + t ) λ j 1 + ( 1 + t ) λ j ) μ j ( 1 + t ) x = n = 0 s ˆ n ( x | λ 1 , , λ s ; μ 1 , , μ s ) t n n ! .

If s = 1 , then s ˆ n ( x | λ ; μ ) are the Peters polynomials of the second kind. Peters polynomials were mentioned in [[1], p.128] and have been investigated in e.g. [2].

In this paper, by considering Barnes-type Narumi polynomials of the second kind and Barnes-type Peters polynomials of the second kind, we define and investigate the hybrid 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 Kronecker’s 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)
[[1], 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 of [[1], 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 [[1], 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 [[1], 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 [[1], p.132]
C n , m = 1 m ! h ( f ¯ ( t ) ) g ( f ¯ ( t ) ) l ( f ¯ ( t ) ) m | x n .
(11)

3 Main results

For convenience, we introduce an Appell sequence of polynomials F n ( a 1 , , a r ) ( a 1 , , a r 0 ), defined by
i = 1 r ( e a i t 1 t ) e x t = n = 0 F n ( x | a 1 , , a r ) t n n ! .
If x = 0 , we write F n ( a 1 , , a r ) = F n ( 0 | a 1 , , a r ) . By [[1], Theorem 2.5.8] with y = 0 , we have
F n ( x | a 1 , , a r ) = m = 0 n ( n m ) F n m ( a 1 , , a r ) x m .
More precisely, one can show that
F n ( x | a 1 , , a r ) = i = 0 n l 1 + + l r = i i ! ( i + r ) ! ( n i ) ( i + r l 1 + 1 , , l r + 1 ) a 1 l 1 + 1 a r l r + 1 x n i ,
where
( i + r l 1 + 1 , , l r + 1 ) = ( i + r ) ! ( l 1 + 1 ) ! ( l r + 1 ) ! .
Remember that the generalized Barnes-type Euler polynomials E n ( x | λ 1 , , λ s ; μ 1 , , μ s ) are defined by the generating function
j = 1 s ( 2 1 + e λ j t ) μ j e x t = n = 0 E n ( x | λ 1 , , λ s ; μ 1 , , μ s ) t n n ! .
If μ 1 = = μ s = 1 , then E n ( x | λ 1 , , λ s ) = E n ( x | λ 1 , , λ s ; 1 , , 1 ) are called the Barnes-type Euler polynomials. If further λ 1 = = λ s = 1 , then E n ( r ) ( x ) = E n ( x | 1 , , 1 ; 1 , , 1 ) are called the Euler polynomials of order s. If x = 0 , we write E n ( λ 1 , , λ s ; μ 1 , , μ s ) = E n ( 0 | λ 1 , , λ s ; μ 1 , , μ s ) . By [[1], Theorem 2.5.8] with y = 0 , we have
E n ( x | λ 1 , , λ s ; μ 1 , , μ s ) = m = 0 n ( n m ) E n m ( λ 1 , , λ s ; μ 1 , , μ s ) x m .
We note that
t F n ( x | a 1 , , a r ) = d d x F n ( x | a 1 , , a r ) = n F n 1 ( x | a 1 , , a r ) , t E n ( x | λ 1 , , λ s ; μ 1 , , μ s ) = d d x E n ( x | λ 1 , , λ s ; μ 1 , , μ s ) t E n ( x | λ 1 , , λ s ; μ 1 , , μ s ) = n E n 1 ( x | λ 1 , , λ s ; μ 1 , , μ s ) .
From the definition (1), NS ˆ n ( x | a 1 , , a r ; λ 1 , , λ s ; μ 1 , , μ s ) is the Sheffer sequence for the pair
g ( t ) = i = 1 r ( t e a i t e a i t 1 ) j = 1 s ( 1 + e λ j t e λ j t ) μ j and f ( t ) = e t 1 .
So,
NS ˆ n ( x | a 1 , , a r ; λ 1 , , λ s ; μ 1 , , μ s ) ( i = 1 r ( t e a i t e a i t 1 ) j = 1 s ( 1 + e λ j t e λ j t ) μ j , e t 1 ) .
(12)

3.1 Explicit expressions

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

NS ˆ n ( x | a 1 , , a r ; λ 1 , , λ s ; μ 1 , , μ s ) = 2 j = 1 s μ j m = 0 n l = 0 m ( 1 ) m ( m l ) S 1 ( n , m ) E m l ( λ 1 , , λ s ; μ 1 , , μ s ) × F l ( x | a 1 , , a r )
(13)
= 2 j = 1 s μ j m = 0 n l = 0 m ( m l ) S 1 ( n , m ) E m l ( λ 1 , , λ s ; μ 1 , , μ s ) × F l ( x + j = 1 s λ j μ j i = 1 r a i | a 1 , , a r )
(14)
= 2 j = 1 s μ j m = 0 n l = 0 m ( 1 ) m ( m l ) S 1 ( n , m ) F m l ( a 1 , , a r ) × E l ( x | λ 1 , , λ s ; μ 1 , , μ s )
(15)
= j = 0 n l = j n ( n l ) S 1 ( l , j ) NS ˆ n l x j
(16)
= l = 0 n ( n l ) s ˆ n l ( λ 1 , , λ r ; μ 1 , , μ r ) N ˆ l ( x | a 1 , , a r ) ,
(17)
= l = 0 n ( n l ) N ˆ n l ( a 1 , , a r ) s ˆ l ( x | λ 1 , , λ s ; μ 1 , , μ s ) .
(18)
Proof Since
i = 1 r ( t e a i t e a i t 1 ) j = 1 s ( 1 + e λ j t e λ j t ) μ j NS ˆ n ( x | a 1 , , a r ; λ 1 , , λ s ; μ 1 , , μ s ) ( 1 , e t 1 )
(19)
and
( x ) n ( 1 , e t 1 ) ,
(20)
we have
NS ˆ n ( x | a 1 , , a r ; λ 1 , , λ s ; μ 1 , , μ s ) = i = 1 r ( e a i t 1 t e a i t ) j = 1 s ( e λ j t 1 + e λ j t ) μ j ( x ) n = m = 0 n S 1 ( n , m ) i = 1 r ( e a i t 1 t e a i t ) j = 1 s ( e λ j t 1 + e λ j t ) μ j x m = 2 j = 1 s μ j m = 0 n S 1 ( n , m ) i = 1 r ( e a i t 1 t ) j = 1 s ( 2 1 + e λ j t ) μ j x m = 2 j = 1 s μ j m = 0 n S 1 ( n , m ) i = 1 r ( e a i t 1 t ) ( 1 ) m E m ( x | λ 1 , , λ s ; μ 1 , , μ s ) = 2 j = 1 s μ j m = 0 n S 1 ( n , m ) i = 1 r ( e a i t 1 t ) ( 1 ) m × l = 0 m ( m l ) E m l ( λ 1 , , λ s ; μ 1 , , μ s ) ( x ) l = 2 j = 1 s μ j m = 0 n ( 1 ) m S 1 ( n , m ) × l = 0 m ( 1 ) l ( m l ) E m l ( λ 1 , , λ s ; μ 1 , , μ s ) i = 1 r ( e a i t 1 t ) x l = 2 j = 1 s μ j m = 0 n ( 1 ) m S 1 ( n , m ) × l = 0 m ( 1 ) l ( m l ) E m l ( λ 1 , , λ s ; μ 1 , , μ s ) ( 1 ) l F l ( x | a 1 , , a r ) .

So, we get (13).

We can obtain an alternative expression (14) for NS ˆ n ( x ) as follows:
NS ˆ n ( x ) = 2 j = 1 s μ j m = 0 n S 1 ( n , m ) e ( j = 1 s λ j μ j i = 1 r a i ) t × i = 1 r ( e a i t 1 t ) j = 1 s ( 2 1 + e λ j t ) μ j x m = 2 j = 1 s μ j m = 0 n S 1 ( n , m ) e ( j = 1 s λ j μ j i = 1 r a i ) t × i = 1 r ( e a i t 1 t ) E m ( x | λ 1 , , λ s ; μ 1 , , μ s ) = 2 j = 1 s μ j m = 0 n S 1 ( n , m ) e ( j = 1 s λ j μ j i = 1 r a i ) t × i = 1 r ( e a i t 1 t ) l = 0 m ( m l ) E m l ( λ 1 , , λ s ; μ 1 , , μ s ) x l = 2 j = 1 s μ j m = 0 n l = 0 m ( m l ) S 1 ( n , m ) E m l ( λ 1 , , λ s ; μ 1 , , μ s ) × e ( j = 1 s λ j μ j i = 1 r a i ) t i = 1 r ( e a i t 1 t ) x l = 2 j = 1 s μ j m = 0 n l = 0 m ( m l ) S 1 ( n , m ) E m l ( λ 1 , , λ s ; μ 1 , , μ s ) × F l ( x + j = 1 s λ j μ j i = 1 r a i | a 1 , , a r ) .
From the proof of (13),
NS ˆ n ( x | a 1 , , a r ; λ 1 , , λ s ; μ 1 , , μ s ) = 2 j = 1 s μ j m = 0 n S 1 ( n , m ) j = 1 s ( 2 1 + e λ j t ) μ j i = 1 r ( e a i t 1 t ) x m = 2 j = 1 s μ j m = 0 n S 1 ( n , m ) j = 1 s ( 2 1 + e λ j t ) μ j ( 1 ) m F m ( x | a 1 , , a r ) = 2 j = 1 s μ j m = 0 n ( 1 ) m S 1 ( n , m ) j = 1 s ( 2 1 + e λ j t ) μ j l = 0 m ( m l ) F m l ( a 1 , , a r ) ( x ) l = 2 j = 1 s μ j m = 0 n ( 1 ) m S 1 ( n , m ) l = 0 m ( m l ) F m l ( a 1 , , a r ) ( 1 ) l j = 1 s ( 2 1 + e λ j t ) μ j x l = 2 j = 1 s μ j m = 0 n ( 1 ) m S 1 ( n , m ) l = 0 m ( m l ) F m l ( a 1 , , a r ) E l ( x | λ 1 , , λ s ; μ 1 , , μ s ) = 2 j = 1 s μ j m = 0 n l = 0 m ( 1 ) m ( m l ) S 1 ( n , m ) F m l ( a 1 , , a r ) E l ( x | λ 1 , , λ s ; μ 1 , , μ s ) ,

which is the identity (15).

By (12), we have
g ( f ¯ ( t ) ) 1 f ¯ ( t ) j | x n = i = 1 r ( ( 1 + t ) a i 1 ( 1 + t ) a i ln ( 1 + t ) ) j = 1 s ( ( 1 + t ) λ j 1 + ( 1 + t ) λ j ) μ j ( ln ( 1 + t ) ) j | x n = i = 1 r ( ( 1 + t ) a i 1 ( 1 + t ) a i ln ( 1 + t ) ) j = 1 s ( ( 1 + t ) λ j 1 + ( 1 + t ) λ j ) μ j | j ! l = j S 1 ( l , j ) t l l ! x n = j ! l = j n ( n l ) S 1 ( l , j ) i = 1 r ( ( 1 + t ) a i 1 ( 1 + t ) a i ln ( 1 + t ) ) j = 1 s ( ( 1 + t ) λ j 1 + ( 1 + t ) λ j ) μ j | x n l = j ! l = j n ( n l ) S 1 ( l , j ) i = 0 NS ˆ i t i i ! | x n l = j ! l = j n ( n l ) S 1 ( l , j ) NS ˆ n l .

By (9), we get the identity (16).

Next,
NS ˆ n ( y | a 1 , , a r ; λ 1 , , λ s ; μ 1 , , μ s ) = i = 0 NS ˆ i ( y | a 1 , , a r ; λ 1 , , λ s ; μ 1 , , μ s ) t i i ! | x n = i = 1 r ( ( 1 + t ) a i 1 ( 1 + t ) a i ln ( 1 + t ) ) j = 1 s ( ( 1 + t ) λ j 1 + ( 1 + t ) λ j ) μ j ( 1 + t ) y | x n = j = 1 s ( ( 1 + t ) λ j 1 + ( 1 + t ) λ j ) μ j | i = 1 r ( ( 1 + t ) a i 1 ( 1 + t ) a i ln ( 1 + t ) ) ( 1 + t ) y x n = j = 1 s ( ( 1 + t ) λ j 1 + ( 1 + t ) λ j ) μ j | l = 0 N ˆ l ( y | a 1 , , a r ) t l l ! x n = l = 0 n ( n l ) N ˆ l ( y | a 1 , , a r ) j = 1 s ( ( 1 + t ) λ j 1 + ( 1 + t ) λ j ) μ j | x n l = l = 0 n ( n l ) N ˆ l ( y | a 1 , , a r ) i = 0 s ˆ i ( λ 1 , , λ r ; μ 1 , , μ r ) t i i ! | x n l = l = 0 n ( n l ) N ˆ l ( y | a 1 , , a r ) s ˆ n l ( λ 1 , , λ r ; μ 1 , , μ r ) .

Thus, we obtain (17).

Finally, we obtain
NS ˆ n ( y | a 1 , , a r ; λ 1 , , λ s ; μ 1 , , μ s ) = i = 0 NS ˆ i ( y | a 1 , , a r ; λ 1 , , λ s ; μ 1 , , μ s ) t i i ! | x n = i = 1 r ( ( 1 + t ) a i 1 ( 1 + t ) a i ln ( 1 + t ) ) j = 1 s ( ( 1 + t ) λ j 1 + ( 1 + t ) λ j ) μ j ( 1 + t ) y | x n = i = 1 r ( ( 1 + t ) a i 1 ( 1 + t ) a i ln ( 1 + t ) ) | j = 1 s ( ( 1 + t ) λ j 1 + ( 1 + t ) λ j ) μ j ( 1 + t ) y x n = i = 1 r ( ( 1 + t ) a i 1 ( 1 + t ) a i ln ( 1 + t ) ) | l = 0 s ˆ l ( y | λ 1 , , λ s ; μ 1 , , μ s ) t l l ! x n = l = 0 n s ˆ l ( y | λ 1 , , λ s ; μ 1 , , μ s ) ( n l ) i = 1 r ( ( 1 + t ) a i 1 ( 1 + t ) a i ln ( 1 + t ) ) | x n l = l = 0 n s ˆ l ( y | λ 1 , , λ s ; μ 1 , , μ s ) ( n l ) i = 0 N ˆ i ( a 1 , , a r ) t i i ! | x n l = l = 0 n ( n l ) s ˆ l ( y | λ 1 , , λ s ; μ 1 , , μ s ) N ˆ n l ( a 1 , , a r ) .

Thus, we get the identity (18). □

3.2 Sheffer identity

Theorem 2
NS ˆ n ( x + y | a 1 , , a r ; λ 1 , , λ s ; μ 1 , , μ s ) = l = 0 n ( n l ) NS ˆ l ( x | a 1 , , a r ; λ 1 , , λ s ; μ 1 , , μ s ) ( y ) n l .
(21)
Proof By (12) with
p n ( x ) = i = 1 r ( t e a i t e a i t 1 ) j = 1 s ( 1 + e λ j t e λ j t ) μ j NS ˆ n ( x | a 1 , , a r ; λ 1 , , λ s ; μ 1 , , μ s ) = ( x ) n ( 1 , e t 1 ) ,

using (10), we have (21). □

3.3 Difference relations

Theorem 3
NS ˆ n ( x + 1 | a 1 , , a r ; λ 1 , , λ s ; μ 1 , , μ s ) NS ˆ n ( x | a 1 , , a r ; λ 1 , , λ s ; μ 1 , , μ s ) = n NS ˆ n 1 ( x | a 1 , , a r ; λ 1 , , λ s ; μ 1 , , μ s ) .
(22)
Proof By (8) with (12), we get
( e t 1 ) NS ˆ n ( x | a 1 , , a r ; λ 1 , , λ s ; μ 1 , , μ s ) = n NS ˆ n 1 ( x | a 1 , , a r ; λ 1 , , λ s ; μ 1 , , μ s ) .

By (7), we have (22). □

3.4 Recurrence

Theorem 4
NS ˆ n + 1 ( x | a 1 , , a r ; λ 1 , , λ s ; μ 1 , , μ s ) = ( x + j = 1 s μ j λ j i = 1 r a i ) NS ˆ n ( x 1 | a 1 , , a r ; λ 1 , , λ s ; μ 1 , , μ s ) + 2 j = 1 s μ j m = 0 n l = 0 m ( 1 ) m + 1 ( m l ) l + 1 S 1 ( n , m ) E m l ( λ 1 , , λ s ; μ 1 , , μ s ) × ( i = 1 r a i F l + 1 ( 1 x | a 1 , , a i 1 , a i + 1 , , a r ) r F l + 1 ( 1 x | a 1 , , a r ) ) + 2 1 j = 1 s μ j m = 0 n l = 0 m ( 1 ) m + 1 ( m l ) S 1 ( n , m ) F m l ( a 1 , , a r ) × i = 1 s μ i λ i E l ( 1 x | λ ; μ + e i ) .
(23)
Proof By applying
s n + 1 ( x ) = ( x g ( t ) g ( t ) ) 1 f ( t ) s n ( x )
(24)
[[1], Corollary 3.7.2] with (12), we get
NS ˆ n + 1 ( x ) = x NS ˆ n ( x 1 ) e t g ( t ) g ( t ) NS ˆ n ( x ) .
Observe that
g ( t ) g ( t ) = ( ln g ( t ) ) = ( r ln t + ( i = 1 r a i ) t i = 1 r ln ( e a i t 1 ) + j = 1 s μ j ln ( 1 + e λ j t ) ( j = 1 s μ j λ j ) t ) = r t + i = 1 r a i i = 1 r a i e a i t e a i t 1 + j = 1 s μ j λ j e λ j t 1 + e λ j t j = 1 s μ j λ j = r i = 1 r a i t e a i t e a i t 1 t + i = 1 r a i + j = 1 s μ j λ j e λ j t 1 + e λ j t j = 1 s μ j λ j ,
where
r i = 1 r a i t e a i t e a i t 1 = 1 2 ( i = 1 r a i ) t +
has the order at least one. Since from the proofs of (13) and (15)
NS ˆ n ( x ) = 2 j = 1 s μ j m = 0 n ( 1 ) m S 1 ( n , m ) × l = 0 m ( 1 ) l ( m l ) E m l ( λ 1 , , λ s ; μ 1 , , μ s ) i = 1 r ( e a i t 1 t ) x l = 2 j = 1 s μ j m = 0 n ( 1 ) m S 1 ( n , m ) × l = 0 m ( m l ) F m l ( a 1 , , a r ) ( 1 ) l j = 1 s ( 2 1 + e λ j t ) μ j x l ,
we have
g ( t ) g ( t ) NS ˆ n ( x ) = ( i = 1 r a i j = 1 s μ j λ j ) NS ˆ n ( x ) + 2 j = 1 s μ j m = 0 n ( 1 ) m S 1 ( n , m ) l = 0 m ( 1 ) l ( m l ) E m l ( λ 1 , , λ s ; μ 1 , , μ s ) × i = 1 r ( e a i t 1 t ) r i = 1 r a i t 1 e a i t t x l + 2 j = 1 s μ j m = 0 n ( 1 ) m S 1 ( n , m ) l = 0 m ( m l ) F m l ( a 1 , , a r ) ( 1 ) l × i = 1 s μ i λ i 1 + e λ i t j = 1 s ( 2 1 + e λ j t ) μ j x l .
The third term is equal to
2 j = 1 s μ j m = 0 n l = 0 m ( 1 ) m ( m l ) S 1 ( n , m ) F m l ( a 1 , , a r ) ( 1 ) l × i = 1 s μ i λ i 2 ( 1 ) l E l ( x | λ ; μ + e i ) = 2 1 j = 1 s μ j m = 0 n l = 0 m ( 1 ) m ( m l ) S 1 ( n , m ) F m l ( a 1 , , a r ) × i = 1 s μ i λ i E l ( x | λ ; μ + e i ) ,
where λ = ( λ 1 , , λ s ) , μ = ( μ 1 , , μ s ) , and . The second term is
2 j = 1 s μ j m = 0 n ( 1 ) m S 1 ( n , m ) l = 0 m ( 1 ) l l + 1 ( m l ) E m l ( λ 1 , , λ s ; μ 1 , , μ s ) × i = 1 r ( e a i t 1 t ) ( r i = 1 r a i t 1 e a i t ) x l + 1 = r 2 j = 1 s μ j m = 0 n ( 1 ) m S 1 ( n , m ) l = 0 m ( 1 ) l l + 1 ( m l ) E m l ( λ 1 , , λ s ; μ 1 , , μ s ) × i = 1 r ( e a i t 1 t ) x l + 1 2 j = 1 s μ j m = 0 n ( 1 ) m S 1 ( n , m ) l = 0 m ( 1 ) l l + 1 ( m l ) E m l ( λ 1 , , λ s ; μ 1 , , μ s ) × i = 1 r a i t e a i t 1 i = 1 r ( e a i t 1 t ) x l + 1 = r 2 j = 1 s μ j m = 0 n ( 1 ) m S 1 ( n , m ) l = 0 m ( 1 ) l l + 1 ( m l ) E m l ( λ 1 , , λ s ; μ 1 , , μ s ) × ( 1 ) l + 1 F l + 1 ( x | a 1 , , a r ) 2 j = 1 s μ j m = 0 n ( 1 ) m S 1 ( n , m ) l = 0 m ( 1 ) l l + 1 ( m l ) E m l ( λ 1 , , λ s ; μ 1 , , μ s ) × i = 1 r a i ( 1 ) l + 1 F l + 1 ( x | a 1 , , a i 1 , a i + 1 , , a r ) = 2 j = 1 s μ j m = 0 n l = 0 m ( 1 ) m + 1 ( m l ) l + 1 S 1 ( n , m ) E m l ( λ 1 , , λ s ; μ 1 , , μ s ) × ( r F l + 1 ( x | a 1 , , a r ) i = 1 r a i F l + 1 ( x | a 1 , , a i 1 , a i + 1 , , a r ) ) .
Therefore, we obtain
NS ˆ n + 1 ( x ) = ( x + j = 1 s μ j λ j i = 1 r a i ) NS ˆ n ( x 1 ) + 2 j = 1 s μ j m = 0 n l = 0 m ( 1 ) m + 1 ( m l ) l + 1 S 1 ( n , m ) E m l ( λ 1 , , λ s ; μ 1 , , μ s ) × ( i = 1 r a i F l + 1 ( 1 x | a 1 , , a i 1 , a i + 1 , , a r ) r F l + 1 ( 1 x | a 1 , , a r ) ) + 2 1 j = 1 s μ j m = 0 n l = 0 m ( 1 ) m + 1 ( m l ) S 1 ( n , m ) F m l ( a 1 , , a r ) × i = 1 s μ i λ i E l ( 1 x | λ ; μ + e i ) ,

which is the identity (23). □

3.5 Differentiation

Theorem 5
d d x NS ˆ n ( x | a 1 , , a r ; λ 1 , , λ s ; μ 1 , , μ s ) = n ! l = 0 n 1 ( 1 ) n l 1 l ! ( n l ) NS ˆ l ( x | a 1 , , a r ; λ 1 , , λ s ; μ 1 , , μ s ) .
(25)
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. [[1], 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 NS ˆ n ( x | a 1 , , a r ; λ 1 , , λ s ; μ 1 , , μ s ) = l = 0 n 1 ( n l ) ( 1 ) n l 1 ( n l 1 ) ! NS ˆ l ( x | a 1 , , a r ; λ 1 , , λ s ; μ 1 , , μ s ) = n ! l = 0 n 1 ( 1 ) n l 1 l ! ( n l ) NS ˆ l ( x | a 1 , , a r ; λ 1 , , λ s ; μ 1 , , μ s ) ,

which is the identity (25). □

3.6 One more relation

The classical Cauchy numbers of the first kind c n are defined by
t ln ( 1 + t ) = n = 0 c n t n n !

(see e.g. [3, 4]).

Theorem 6
NS ˆ n ( x | a ; λ ; μ ) = ( x i = 1 r a i ) NS ˆ n 1 ( x 1 | a ; λ ; μ ) + j = 1 s λ j μ j NS ˆ n 1 ( x λ j 1 | a ; λ ; μ + e j ) + 1 n l = 0 n ( n l ) c l ( i = 1 r a i NS ˆ n l ( x 1 | a 1 , , a i 1 , a i + 1 , , a r ; λ ; μ ) r NS ˆ n l ( x 1 | a ; λ ; μ ) ) .
(26)
Proof For n 1 , we have
NS ˆ n ( y | a 1 , , a r ; λ 1 , , λ s ; μ 1 , , μ s ) = l = 0 NS ˆ l ( y | a 1 , , a r ; λ 1 , , λ s ; μ 1 , , μ s ) t l l ! | x n = i = 1 r ( ( 1 + t ) a i 1 ( 1 + t ) a i ln ( 1 + t ) ) j = 1 s ( ( 1 + t ) λ j 1 + ( 1 + t ) λ j ) μ j ( 1 + t ) y | x n = t ( i = 1 r ( ( 1 + t ) a i 1 ( 1 + t ) a i ln ( 1 + t ) ) j = 1 s ( ( 1 + t ) λ j 1 + ( 1 + t ) λ j ) μ j ( 1 + t ) y ) | x n 1 = ( t i = 1 r ( ( 1 + t ) a i 1 ( 1 + t ) a i ln ( 1 + t ) ) ) j = 1 s ( ( 1 + t ) λ j 1 + ( 1 + t ) λ j ) μ j ( 1 + t ) y | x n 1 + i = 1 r ( ( 1 + t ) a i 1 ( 1 + t ) a i ln ( 1 + t ) ) ( t j = 1 s ( ( 1 + t ) λ j 1 + ( 1 + t ) λ j ) μ j ) ( 1 + t ) y | x n 1 + i = 1 r ( ( 1 + t ) a i 1 ( 1 + t ) a i ln ( 1 + t ) ) j = 1 s ( ( 1 + t ) λ j 1 + ( 1 + t ) λ j ) μ j ( t ( 1 + t ) y ) | x n 1 .
The third term is
y i = 1 r ( ( 1 + t ) a i 1 ( 1 + t ) a i ln ( 1 + t ) ) j = 1 s ( ( 1 + t ) λ j 1 + ( 1 + t ) λ j ) μ j ( 1 + t ) y 1 | x n 1 = y NS ˆ n 1 ( y 1 | a 1 , , a r ; λ 1 , , λ s ; μ 1 , , μ s ) .
Since
t i = 1 r ( ( 1 + t ) a i 1 ( 1 + t ) a i ln ( 1 + t ) ) = 1 1 + t i = 1 r ( ( 1 + t ) a i 1 ( 1 + t ) a i ln ( 1 + t ) ) ν = 1 r ( a ν t ( 1 + t ) a ν ( 1 + t ) a ν 1 t ln ( 1 + t ) ) t 1 1 + t i = 1 r ( ( 1 + t ) a i 1 ( 1 + t ) a i ln ( 1 + t ) ) ( l = 1 r a l ) ,
with
ν = 1 r ( a ν t ( 1 + t ) a ν ( 1 + t ) a ν 1 t ln ( 1 + t ) )
having order ≥1, the first term is
i = 1 r ( ( 1 + t ) a i 1 ( 1 + t ) a i ln ( 1 + t ) ) j = 1 s ( ( 1 + t ) λ j 1 + ( 1 + t ) λ j ) μ j ( 1 + t ) y 1 | ν = 1 r ( a ν t ( 1 + t ) a ν ( 1 + t ) a ν 1 t ln ( 1 + t ) ) t x n 1 ( l = 1 r a l ) i = 1 r ( ( 1 + t ) a i 1 ( 1 + t ) a i ln ( 1 + t ) ) j = 1 s ( ( 1 + t ) λ j 1 + ( 1 + t ) λ j ) μ j ( 1 + t ) y 1 | x n 1 = ( l = 1 r a l ) NS ˆ n 1 ( y 1 ) + 1 n i = 1 r ( ( 1 + t ) a i 1 ( 1 + t ) a i ln ( 1 + t ) ) j = 1 s ( ( 1 + t ) λ j 1 + ( 1 + t ) λ j ) μ j × ( 1 + t ) y 1 | ν = 1 r ( a ν t ( 1 + t ) a ν ( 1 + t ) a ν 1 t ln ( 1 + t ) ) x n = ( l = 1 r a l ) NS ˆ n 1 ( y 1 ) + 1 n ( ν = 1 r a ν ( 1 + t ) a ν ln ( 1 + t ) ( 1 + t ) a ν 1 i = 1 r ( ( 1 + t ) a i 1 ( 1 + t ) a i ln ( 1 + t ) ) j = 1 s ( ( 1 + t ) λ j 1 + ( 1 + t ) λ j ) μ j × ( 1 + t ) y 1 | t ln ( 1 + t ) x n r i = 1 r ( ( 1 + t ) a i 1 ( 1 + t ) a i ln ( 1 + t ) ) j = 1 s ( ( 1 + t ) λ j 1 + ( 1 + t ) λ j ) μ j ( 1 + t ) y 1 | t ln ( 1 + t ) x n ) = ( l = 1 r a l ) NS ˆ n 1 ( y 1 ) + 1 n ( ν = 1 r a ν ( 1 + t ) a ν ln ( 1 + t ) ( 1 + t ) a ν 1 i = 1 r ( ( 1 + t ) a i 1 ( 1 + t ) a i ln ( 1 + t ) ) j = 1 s ( ( 1 + t ) λ j 1 + ( 1 + t ) λ j ) μ j × ( 1 + t ) y 1 | l = 0 c l t l l ! x n r i = 1 r ( ( 1 + t ) a i 1 ( 1 + t ) a i ln ( 1 + t ) ) j = 1 s ( ( 1 + t ) λ j 1 + ( 1 + t ) λ j ) μ j ( 1 + t ) y 1 | l = 0 c l t l l ! x n ) = ( l = 1 r a l ) NS ˆ n 1 ( y 1 ) + 1 n ( ν = 1 r l = 0 n ( n l ) a ν c l NS ˆ n l ( y 1 | a 1 , , a ν 1 , a ν + 1 , , a r ; λ ; μ ) r l = 0 n ( n l ) c l NS ˆ n l ( y 1 | a ; λ ; μ ) ) = ( ν = 1 r a ν ) NS ˆ n 1 ( y 1 ) + 1 n l = 0 n ( n l ) c l ( ν = 1 r a ν NS ˆ n l ( y 1 | a 1 , , a ν 1 , a ν + 1 , , a r ; λ ; μ ) r NS ˆ n l ( y 1 | a ; λ ; μ ) ) .
Since
t j = 1 s ( ( 1 + t ) λ j 1 + ( 1 + t ) λ j ) μ j = l = 1 s λ l μ l ( 1 + t ) λ l 1 ( ( 1 + t ) λ l 1 + ( 1 + t ) λ l ) j = 1 s ( ( 1 + t ) λ j 1 + ( 1 + t ) λ j ) μ j ,
the second term is
l = 1 s λ l μ l i = 1 r ( ( 1 + t ) a i 1 ( 1 + t ) a i ln ( 1 + t ) ) ( ( 1 + t ) λ l 1 + ( 1 + t ) λ l ) j = 1 s ( ( 1 + t ) λ j 1 + ( 1 + t ) λ j ) μ j ( 1 + t ) y λ l 1 | x n 1 = l = 1 s λ l μ l NS ˆ n 1 ( y λ l 1 | a ; λ ; μ + e l ) .
Therefore, we obtain
NS ˆ n ( x | a ; λ ; μ ) = ( x i = 1 r a i ) NS ˆ n 1 ( x 1 | a ; λ ; μ ) + j = 1 s λ j μ j NS ˆ n 1 ( x λ j 1 | a ; λ ; μ + e j ) + 1 n l = 0 n ( n l ) c l ( i = 1 r a i NS ˆ n l ( x 1 | a 1 , , a i 1 , a i + 1 , , a r ; λ ; μ ) r NS ˆ n l ( x 1 | a ; λ ; μ ) ) ,

which is the identity (26). □

3.7 A relation involving the Stirling numbers of the first kind

Theorem 7 For n 1 m 1 , we have
l = 0 n m ( n l ) S 1 ( n l , m ) NS ˆ l ( a ; λ ; μ ) = ( i = 1 r a i ) l = 0 n m 1 ( n 1 l ) S 1 ( n l 1 , m ) NS ˆ l ( 1 | a ; λ ; μ ) + 1 n k = 0 n m l = k n m ( n l ) ( l k ) S 1 ( n l , m ) c l k × ( i = 1 r a i NS ˆ k ( 1 | a 1 , , a i 1 , a i + 1 , , a r ; λ ; μ ) r NS ˆ k ( 1 | a ; λ ; μ ) ) + j = 1 s λ j μ j l = 0 n m 1 ( n 1 l ) S 1 ( n l 1 , m ) NS ˆ l ( λ j 1 | a ; λ ; μ + e j ) + l = 0 n m ( n 1 l ) S 1 ( n l 1 , m 1 ) NS ˆ l ( 1 | a ; λ ; μ ) .
(27)
Proof We shall compute
i = 1 r ( ( 1 + t ) a i 1 ( 1 + t ) a i ln ( 1 + t ) ) j = 1 s ( ( 1 + t ) λ j 1 + ( 1 + t ) λ j ) μ j ( ln ( 1 + t ) ) m | x n
in two different ways. On the one hand, it is equal to
i = 1 r ( ( 1 + t ) a i 1 ( 1 + t ) a i ln ( 1 + t ) ) j = 1 s ( ( 1 + t ) λ j 1 + ( 1 + t ) λ j ) μ j | ( ln ( 1 + t ) ) m x n = i = 1 r ( ( 1 + t ) a i 1 ( 1 + t ) a i ln ( 1 + t ) ) j = 1 s ( ( 1 + t ) λ j 1 + ( 1 + t ) λ j ) μ j | m ! l = m S 1 ( l , m ) t l l ! x n = m ! l = m n ( n l ) S 1 ( l , m ) i = 1 r ( ( 1 + t ) a i 1 ( 1 + t ) a i ln ( 1 + t ) ) j = 1 s ( ( 1 + t ) λ j 1 + ( 1 + t ) λ j ) μ j | x n l = m ! l = m n ( n l ) S 1 ( l , m ) i = 0 NS ˆ i ( a 1 , , a r ; λ 1 , , λ s ; μ 1 , , μ s ) t i i ! | x n l = m ! l = m n ( n l ) S 1 ( l , m ) NS ˆ n l ( a 1 , , a r ; λ 1 , , λ s ; μ 1 , , μ s ) = m ! l = 0 n m ( n l ) S 1 ( n l , m ) NS ˆ l ( a 1 , , a r ; λ 1 , , λ s ; μ 1 , , μ s ) .
On the other hand, it is equal to
t ( i = 1 r ( ( 1 + t ) a i 1 ( 1 + t ) a i ln ( 1 + t ) ) j = 1 s ( ( 1 + t ) λ j 1 + ( 1 + t ) λ j ) μ j ( ln ( 1 + t ) ) m ) | x n 1 = ( t i = 1 r ( ( 1 + t ) a i 1 ( 1 + t ) a i ln ( 1 + t ) ) ) j = 1 s ( ( 1 + t ) λ j 1 + ( 1 + t ) λ j ) μ j ( ln ( 1 + t ) ) m | x n 1 + i = 1 r ( ( 1 + t ) a i 1 ( 1 + t ) a i ln ( 1 + t ) ) ( t j = 1 s ( ( 1 + t ) λ j 1 + ( 1 + t ) λ j ) μ j ) ( ln ( 1 + t ) ) m | x n 1 + i = 1 r ( ( 1 + t ) a i 1 ( 1 + t ) a i ln ( 1 + t ) ) j = 1 s ( ( 1 + t ) λ j 1 + ( 1 + t ) λ j ) μ j ( t ( ln ( 1 + t ) ) m ) | x n 1 .
(28)
The third term of (28) is equal to
m i = 1 r ( ( 1 + t ) a i 1 ( 1 + t ) a i ln ( 1 + t ) ) j = 1 s ( ( 1 + t ) λ j 1 + ( 1 + t ) λ j ) μ j ( 1 + t ) 1 | ( ln ( 1 + t ) ) m 1 x n 1 = m i = 1 r ( ( 1 + t ) a i 1 ( 1 + t ) a i ln ( 1 + t ) ) j = 1 s ( ( 1 + t ) λ j 1 + ( 1 + t ) λ j ) μ j × ( 1 + t ) 1 | ( m 1 ) ! l = m 1 S 1 ( l , m 1 ) t l l ! x n 1 = m ! l = m 1 n 1 ( n 1 l ) S 1 ( l , m 1 ) × i = 1 r ( ( 1 + t ) a i 1 ( 1 + t ) a i ln ( 1 + t ) ) j = 1 s ( ( 1 + t ) λ j 1 + ( 1 + t ) λ j ) μ j ( 1 + t ) 1 | x n 1 l = m ! l = m 1 n 1 ( n 1 l ) S 1 ( l , m 1 ) × NS ˆ n 1 l ( 1 | a 1 , , a r ; λ 1 , , λ s ; μ 1 , , μ s ) = m ! l = 0 n m ( n 1 l ) S 1 ( n l 1 , m 1 ) × NS ˆ l ( 1 | a 1 , , a r ; λ 1 , , λ s ; μ 1 , , μ s ) .
Since
t i = 1 r ( ( 1 + t ) a i 1 ( 1 + t ) a i ln ( 1 + t ) ) = 1 1 + t i = 1 r ( ( 1 + t ) a i 1 ( 1 + t ) a i ln ( 1 + t ) ) ν = 1 r ( a ν t ( 1 + t ) a ν ( 1 + t ) a ν 1 t ln ( 1 + t ) ) t 1 1 + t i = 1 r ( ( 1 + t ) a i 1 ( 1 + t ) a i ln ( 1 + t ) ) ( ν = 1 r a ν ) ,
the first term of (28) is equal to
1 n i = 1 r ( ( 1 + t ) a i 1 ( 1 + t ) a i ln ( 1 + t ) ) j = 1 s ( ( 1 + t ) λ j 1 + ( 1 + t ) λ j ) μ j × ( 1 + t ) 1 ν = 1 r ( a ν t ( 1 + t ) a ν ( 1 + t ) a ν 1 t ln ( 1 + t ) ) | ( ln ( 1 + t ) ) m x n ( ν = 1 r a ν ) i = 1 r ( ( 1 + t ) a i 1 ( 1 + t ) a i ln ( 1 + t ) ) j = 1 s ( ( 1 + t ) λ j 1 + ( 1 + t ) λ j ) μ j × ( 1 + t ) 1 | ( ln ( 1 + t ) ) m x n 1 = 1 n i = 1 r ( ( 1 + t ) a i 1 ( 1 + t ) a i ln ( 1 + t ) ) j = 1 s ( ( 1 + t ) λ j 1 + ( 1 + t ) λ j ) μ j ( 1 + t ) 1 × ν = 1 r ( a ν t ( 1 + t ) a ν ( 1 + t ) a ν 1 t ln ( 1 + t ) ) | m ! l = m S 1 ( l , m ) t l l ! x n ( ν = 1 r a ν ) i = 1 r ( ( 1 + t ) a i 1 ( 1 + t ) a i