AM  Vol.8 No.1 , January 2017
The Generalized r-Whitney Numbers
Abstract
In this paper, we define the generalized r-Whitney numbers of the first and second kind. Moreover, we drive the generalized Whitney numbers of the first and second kind. The recurrence relations and the generating functions of these numbers are derived. The relations between these numbers and generalized Stirling numbers of the first and second kind are deduced. Furthermore, some special cases are given. Finally, matrix representation of the relations between Whitney and Stirling numbers is given.

1. Introduction

The r-Whitney numbers of the first and second kind were introduced, respec- tively, by Mezö [1] as

m n ( x ) n = k = 0 n w m , r ( n , k ) ( m x + r ) k , (1)

( m x + r ) n = k = 0 n W m , r ( n , k ) m k ( x ) k . (2)

Many properties of these numbers and their combinatorial interpretations can be seen in Mezö [2] and Cheon [3] . At r = 1 the r-Whitney numbers are reduced to the Whitney numbers of Dowling lattice introduced by Dowling [4] and Benoumhani [5] .

In this paper we use the following notations ( see [6] [7] [8] ):

Let α ¯ = ( α 0 , α 1 , , α n 1 ) where α i , i = 0 , 1 , , n 1 are real numbers.

( α i ) n = j = 0 , j i n 1 ( α i α j ) , ( α i ) 0 = 1 and ( x ; α ¯ ) n = i = 0 n 1 ( x α i ) , ( x ; α ¯ ) 0 = 1. (3)

[ x ; α ¯ ] n _ , q = i = 0 n 1 [ x α i ] q = q i = 0 n 1 α i ( [ x ] q ; [ α ¯ ] q ) n , (4)

where ( [ x n .

Corollary 2. The generalized Whitney numbers of the second kind W ˜ m ; α ¯ ( n , k ) satisfy the recurrence relation

W ˜ m ; α ¯ ( n , k ) = W ˜ m ; α ¯ ( n 1, k 1 ) + ( 1 + m α k ) W ˜ m ; α ¯ ( n 1, k ) , (50)

for k 1 , and W ˜ m ; α ¯ ( n ,0 ) = ( 1 + m α 0 ) n .

Proof. The proof follows directly by setting r = 1 in Equation (22).

Corollary 3. The generalized Whitney numbers of the second kind have the exponential generating function

φ k ( t ; α ¯ ) = i = 0 k e ( 1 + m α i ) t j = 0 , j i k ( m α i m α j ) . (51)

Proof. The proof follows directly by setting r = 1 in Equation (23).

Corollary 4. The generalized Whitney numbers of the second kind have the explicit formula

W ˜ m ; α ¯ ( n , k ) = i = 0 k 1 m k ( α i ) k ( 1 + m α i ) n . (52)

Proof. The proof follows directly by setting r = 1 in Equation (32).

Special cases:

1. Setting α i = 0 , for i = 0 , 1 , , n 1 , in Equation (49), then we get

( k n ) = W ˜ m ; 0 ¯ ( n , k ) , (53)

where W ˜ m ; 0 ¯ ( n , k ) are the Pascal numbers.

2. Setting α i = α , for i = 0 , 1 , , n 1 , in Equation (49), then we get

( i n ) m i = k = i n ( i k ) W ˜ m ; α ( n , k ) m k . (54)

3. Setting α i = i , for i = 0 , 1 , , n 1 , in Equation (49), then we get

( m x + 1 ) n = k = 0 n W ˜ m ; i ( n , k ) m k ( x ) k , (55)

where W ˜ m ; i ( n , k ) = W m ( n , k ) are the Whitney numbers of the second kind.

Remark 10. Setting α i = i and r = 1 in Equation (23) we obtain the exponential generating function of Whitney numbers of the second kind, see [4] .

4. Setting α i = ( p i ) , for i = p 1 , p , , n + p 2 , in Equation (49), we get

( i n ) m i = k = i 1 n W ˜ m ; p ¯ ( n , k ) m k s 2 ( k , i , p ) .

5. Setting α i = [ α i ] q and x = [ x ] q for i = 0 , 1 , , n 1 , in Equation (49), we get

W ˜ m ; [ α ¯ ] q ( n , k ) m k = i = k n ( i n ) m i S q , α ¯ ( i , k ) . (56)

5. Relations between Whitney Numbers and Some Types of Numbers

This section is devoted to drive many important relations between the gene- ralized r-Whitney numbers and different types of Stirling numbers of the first and second kind and the generalized harmonic numbers.

1. Comtet [7] , [16] defined the generalized Stirling numbers of the first and second kind, respectively by,

( x ; α ¯ ) n = i = 0 n s α ¯ ( n , i ) x i , (57)

x k = i = 0 k S α ¯ ( k , i ) ( x ; α ¯ ) i , (58)

substituting Equation (57) in Equation (5), we obtain

m n i = 0 n s α ¯ ( n , i ) x i = k = 0 n w m , r ; α ¯ ( n , k ) ( m x + r ) k = i = 0 n ( k = i n ( i k ) w m , r ; α ¯ ( n , k ) r k i ) ( m x ) i .

Equating the coefficients of x i on both sides, we have

s α ¯ ( n , i ) = 1 m n i k = i n ( i k ) r k i w m , r ; α ¯ ( n , k ) . (59)

This equation gives the generalized Stirling numbers of the first kind in terms of the generalized r-Whitney numbers of the first kind. Moreover, setting r = 1 , we get

s α ¯ ( n , i ) = 1 m n i k = i n ( i k ) w ˜ m ; α ¯ ( n , k ) . (60)

2. From Equation (21) and Equation (58), we have

i = 0 n W m , r ; α ¯ ( n , i ) m i ( x ; α ¯ ) i = k = 0 n ( k n ) m k x k r n k = i = 0 n ( k = i n ( k n ) m k r n k S α ¯ ( k , i ) ) ( x ; α ¯ ) i .

Equating the coefficients of ( x ; α ¯ ) i on both sides, we have

W m , r ; α ¯ ( n , i ) m i = k = i n ( k n ) m k r n k S α ¯ ( k , i ) , (61)

which gives the generalized r-Whitney numbers of the second kind in terms of the generalized Stirling numbers of the second kind. Moreover setting r = 1 , we get

W ˜ m ; α ¯ ( n , i ) m i = k = i n ( k n ) m k S α ¯ ( k , i ) . (62)

3. El-Desouky [17] defined the multiparameter noncentral Stirling numbers of the first and second kind, respectively by,

( x ) n = k = 0 n s ( n , k ; α ¯ ) ( x ; α ¯ ) k , (63)

( x ; α ¯ ) n = k = 0 n S ( n , k ; α ¯ ) ( x ) k , (64)

using Equation (21) and Equation (2), we have

k = 0 n W m , r ( n , k ) m k ( x ) k = i = 0 n W m , r ; α ¯ ( n , i ) m i ( x ; α ¯ ) i , (65)

from Equation (63) we get

i = 0 n m i W m , r ; α ¯ ( n , i ) ( x ; α ¯ ) i = k = 0 n W m , r ( n , k ) m k i = 0 k s ( k , i ; α ¯ ) ( x ; α ¯ ) i = i = 0 n ( k = i n m k W m , r ( n , k ) s ( k , i ; α ¯ ) ) ( x ; α ¯ ) i .

Equating the coefficients of ( x ; α ¯ ) i on both sides, we have

W m , r ; α ¯ ( n , i ) = k = i n m k i W m , r ( n , k ) s ( k , i ; α ¯ ) . (66)

This equation gives the generalized r-Whitney numbers of the second kind in terms of r-Whitney numbers of the second kind and the multiparameter noncentral Stirling numbers of the first kind. Moreover setting r = 1 , we get

k = i n m k i W m ( n , k ) s ( k , i ; α ¯ ) = W ˜ m ; α ¯ ( n , i ) . (67)

4. From Equation (64) and Equation (5), we have

m n i = 0 n S ( n , i ; α ¯ ) ( x ) i = k = 0 n w m , r ; α ¯ ( n , k ) ( m x + r ) k = i = 0 n ( k = i n w m , r ; α ¯ ( n , k ) W m , r ( k , i ) ) m i ( x ) i .

Equating the coefficients of ( x ) i on both sides, we get

S ( n , i ; α ¯ ) = 1 m n i k = i n w m , r ; α ¯ ( n , k ) W m , r ( k , i ) , (68)

which gives the multiparameter noncentral Stirling numbers of the second kind in terms of the generalized r-Whitney numbers of the first kind and r-Whitney numbers of the second kind. Also, setting r = 1 , we get

S ( n , i ; α ¯ ) = 1 m n i k = i n w ˜ m ; α ¯ ( n , k ) W m ( k , i ) . (69)

5. Similarly, from Equation (65) and Equation (64), we get

W m , r ( n , k ) = i = k n W m , r ; α ¯ ( n , i ) S ( i , k ; α ¯ ) m i k . (70)

Equation (70) gives r-Whitney numbers of the second kind in terms of the multiparameter noncentral Stirling numbers and the generalized r-Whitney numbers of the second kind. Setting r = 1 , we have

W m ( n , k ) = i = k n W ˜ m ; α ¯ ( n , i ) S ( i , k ; α ¯ ) m i k . (71)

6. Cakić [18] defined the generalized harmonic numbers as

H n ( k ; α ¯ ) = i = 0 n 1 1 ( α i ) k .

From Eq (5), we have

m n ( x ; α ¯ ) n = k = 0 n w m , r ; α ¯ ( n , k ) ( m x + r ) k = j = 0 ( k = j n ( i k ) r k j w m , r ; α ¯ ( n , k ) ) ( m x ) j . (72)

Also,

( x ; α ¯ ) n = i = 0 n ( x α i ) = i = 0 n ( α i ) ( 1 x α i ) = i = 0 n ( α i ) . exp ( i = 0 n 1 log ( 1 x α i ) ) = i = 0 n ( α i ) . exp ( k = 1 x k k i = 0 n 1 ( 1 α i ) k ) = i = 0 n ( α i ) . exp ( k = 1 x k k H n ( k ; α ¯ ) ) = i = 0 n ( α i ) l = 0 ( 1 ) l l ! ( k = 1 x k k H n ( k ; α ¯ ) ) l ,

using Cauchy rule product, this lead to

( k = 1 x k k H = n ( k ; α ¯ ) ) l = j = 1 l ( k j = 1 x k j k j H n ( k j ; α ¯ ) ) = j = l ( k 1 + k 2 + + k l = j 1 k 1 k 2 k l ι = 1 l H n ( k ι ; α ¯ ) ) x j ,

therefore, we get

( x ; α ¯ ) n = i = 0 n ( α i ) l = 0 ( 1 ) l l ! j = l ( k 1 + k 2 + + k l = j 1 k 1 k 2 k l ι = 1 l H n ( k ι ; α ¯ ) ) x j = i = 0 n ( α i ) j = 0 ( l = 0 ( 1 ) l l ! ( k 1 + k 2 + + k l = j 1 k 1 k 2 k l ι = 1 l H n ( k ι ; α ¯ ) ) ) x j . (73)

From Equation (72) and Equation (73) we have the following identity

k = j n ( i k ) r k j w m , r ; α ¯ ( n , k ) = m n j i = 0 n ( α i ) l = 0 ( 1 ) l l ! ( k 1 + k 2 + + k l = j 1 k 1 k 2 k l ι = 1 l H n ( k ι ; α ¯ ) ) (74)

From Equation (59) and Equation (74) we have

s α ¯ ( n , j ) = i = 0 n ( α i ) l = 0 ( 1 ) l l ! ( k 1 + k 2 + + k l = j 1 k 1 k 2 k l ι = 1 l H n ( k ι ; α ¯ ) ) , (75)

this equation gives the generalized Stirling numbers of the first kind in terms of the generalized Harmonic numbers.

6. Matrix Representation

In this section we drive a matrix representation for some given relations.

1. Equation (66) can be represented in matrix form as

W ^ m , r s ( α ¯ ) = W ^ m , r ; α ¯ , (76)

where W ^ m , r ( n , k ) = m k W m , r ( n , k ) , W ^ m , r ; α ¯ ( n , i ) = m i W m , r ; α ¯ ( n , i ) and W m , r , s ( α ¯ ) and W m , r ; α ¯ are n × n lower triangle matrices whose entries are, respectively, the r-Whitney numbers of the second kind, the multiparameter noncentral Stirling numbers of the first kind and the generalized r-Whitney numbers of the second kind.

For example if 0 n , k , i 3 , and using matrix representation given in [19] , hence Equation (76) can be written as

( 1 0 0 0 r m 0 0 r 2 m ( 2 r + m ) m 2 0 r 3 m ( 3 r 2 + 3 m r + m 2 ) m 2 ( 3 r + 3 m ) m 3 ) ( 1 0 0 0 α 0 1 0 0 α 0 ( α 0 1 ) α 0 + α 1 1 1 0 s ( 3,0 ; α ¯ ) s ( 3,1 ; α ¯ ) s ( 3,2 ; α ¯ ) 1 ) = ( 1 0 0 0 ( r + m α 0 ) m 0 0 ( r + m α 0 ) 2 ( 2 r + m α 0 + m α 1 ) m m 2 0 ( r + m α 0 ) 3 W ^ m , r ; α ¯ ( 3,1 ) W ^ m , r ; α ¯ ( 3,2 ) m 3 )

where

s ( 3,0 ; α ¯ ) = α 0 ( α 0 1 ) ( α 0 2 ) ,

s ( 3,1 ; α ¯ ) = α 0 ( α 0 1 ) + ( α 1 2 ) ( α 0 + α 1 1 ) ,

s ( 3,2 ; α ¯ ) = α 0 + α 1 + α 2 3 ,

W ^ m , r ; α ¯ ( 3,1 ) = ( ( r + m α 0 ) 2 + ( r + m α 1 ) ( 2 r + m α 0 + m α 1 ) ) m ,

W ^ m , r ; α ¯ ( 3,2 ) = ( 3 r + m α 0 + m α 1 + m α 2 ) m 2 .

2. Equation (68) can be represented in a matrix form as

w m , r ; α ¯ W ^ m , r = S ^ ( α ¯ ) , (77)

where S ^ ( n , i ; α ¯ ) = m n S ( n , i ; α ¯ ) , and w m , r ; α ¯ and S ( α ¯ ) are n × n lower triangle matrices whose entries are, respectively, the generalized r-Whitney numbers of the first kind and the multiparameter noncentral Stirling numbers of the second kind.

For example if 0 n , k , i 3, hence Equation (77) can be written as

( 1 0 0 0 r m α 0 1 0 0 ( r + m α 0 ) ( r + m α 1 ) 2 r m α 0 m α 1 1 0 w m , r ; α ¯ ( 3 , 0 ) w m , r ; α ¯ ( 3 , 1 ) w m , r ; α ¯ ( 3 , 2 ) 1 ) ( 1 0 0 0 r m 0 0 r 2 m ( 2 r + m ) m 2 0 r 3 m ( 3 r 2 + 3 m r + m 2 ) m 2 ( 3 r + 3 m ) m 3 ) = ( 1 0 0 0 m α 0 m 0 0 m 2 α 0 α 1 m 2 ( α 0 α 1 + 1 ) m 2 0 m 3 α 0 α 1 α 2 S ^ ( 3,1 ; α ¯ ) S ^ ( 3,2 ; α ¯ ) m 3 )

where

w m , r ; α ¯ ( 3,0 ) = ( r + m α 0 ) ( r + m α 1 ) ( r + m α 2 ) ,

w m , r ; α ¯ ( 3 , 1 ) = ( r + m α 0 ) ( r + m α 1 ) + ( 2 r + m α 0 + m α 1 ) ( r + m α 2 ) ,

w m , r ; α ¯ ( 3,2 ) = 3 r m α 0 m α 1 m α 2 ,

S ^ ( 3,1 ; α ¯ ) = m 3 ( α 0 α 1 + α 0 α 2 + α 1 α 2 α 0 α 1 α 2 + 1 ) ,

S ^ ( 3,2 ; α ¯ ) = m 3 ( α 0 α 1 α 2 + 3 ) .

3. Equation (70) can be represented in a matrix form as

W ^ m , r ; α ¯ S ( α ¯ ) = W ^ m , r , (78)

For example if 0 n , k , i 3 , hence Equation (77) can be written as

( 1 0 0 0 ( r + m α 0 ) m 0 0 ( r + m α 0 ) 2 ( 2 r + m α 0 + m α 1 ) m m 2 0 ( r + m α 0 ) 3 W ^ m , r ; α ¯ ( 3,1 ) W ^ m , r ; α ¯ ( 3,2 ) m 3 ) ( 1 0 0 0 α 0 1 0 0 α 0 α 1 α 0 α 1 + 1 1 0 α 0 α 1 α 2 S ( 3,1 ; α ¯ ) S ( 3,2 ; α ¯ ) 1 ) = ( 1 0 0 0 r m 0 0 r 2 m ( 2 r + m ) m 2 0 r 3 m ( 3 r 2 + 3 m r + m 2 ) m 2 ( 3 r + 3 m ) m 3 )

where

W ^ m , r ; α ¯ ( 3 , 1 ) = ( ( r + m α 0 ) 2 + ( r + m α 1 ) ( 2 r + m α 0 + m α 1 ) ) m ,

W ^ m , r ; α ¯ ( 3 , 2 ) = ( 3 r + m α 0 + m α 1 + m α 2 ) m 2 ,

S ( 3,1 ; α ¯ ) = α 0 α 1 + α 0 α 2 + α 1 α 2 α 0 α 1 α 2 + 1 ,

S ( 3,2 ; α ¯ ) = α 0 α 1 α 2 + 3.

Cite this paper
El-Desouky, B. , Shiha, F. and Shokr, E. (2017) The Generalized r-Whitney Numbers. Applied Mathematics, 8, 117-132. doi: 10.4236/am.2017.81010.
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