Skip to content

Advertisement

Open Access

Unified Bernstein and Bleimann-Butzer-Hahn basis and its properties

Advances in Difference Equations20132013:55

https://doi.org/10.1186/1687-1847-2013-55

Received: 30 November 2012

Accepted: 31 January 2013

Published: 13 March 2013

Abstract

In this paper we introduce the unification of Bernstein and Bleimann-Butzer-Hahn basis via the generating function. We give the representation of this unified family in terms of Apostol-type polynomials and Stirling numbers of the second kind. More generating functions of trigonometric type are also obtained to this unification.

MSC:11B65, 11B68, 41A10, 30C15.

Keywords

generating functionBernstein polynomialsBernoulli polynomialsEuler polynomialsGenocchi polynomialsStirling numbers of the second kind

1 Introduction

In this paper, we introduce a two-parameter generating function, which generates not only the Bernstein basis polynomials, but also the Bleimann-Butzer-Hahn basis functions. The generating function that we propose is given by
G a , b ( t , x ; k , m ) : = [ 2 1 k x k t k ( 1 + a x ) k ] m 1 ( m k ) ! e t [ 1 + b x 1 + a x ] = n = 0 P n ( a , b ) ( x ; k , m ) t n n ! ,
(1)
where k , m Z + : = { 1 , 2 , } , a , b R , t C . Here, x I where I is a subinterval of such that the expansion in (1) is valid. The following two cases will be important for us.
  1. 1.
    The case a = 0 , b = 1 . In this case, we let x [ 0 , 1 ] and we see that
    G 0 , 1 ( t , x ; k , m ) = [ 2 1 k x k t k ] m 1 ( m k ) ! e t [ 1 x ] = n = 0 P n ( 0 , 1 ) ( x ; k , m ) t n n !
     
generates the unifying Bernstein basis polynomials P n ( 0 , 1 ) ( x ; k , m ) : = B n ( m k , x ) which were introduced and investigated in [1]. We should note further that G 0 , 1 ( t , x ; 1 , m ) gives
G 0 , 1 ( t , x ; 1 , m ) = [ x t ] m 1 m ! e t [ 1 x ] = n = 0 B n ( m , x ) t n n !
which generates the celebrated Bernstein basis polynomials (see [28])
B n ( m , x ) : = B m n ( x ) = ( n m ) x k ( 1 x ) n m .
Note that the Bernstein operators B n : C [ 0 , 1 ] C [ 0 , 1 ] are given by
B n ( f ; x ) = m = 0 n f ( m n ) ( n m ) x k ( 1 x ) n m , n N : = { 1 , 2 , }
and by the Korovkin theorem, it is known that B n ( f ; x ) f ( x ) for all f C [ 0 , 1 ] , where C [ 0 , 1 ] denotes the space of continuous functions defined on [ 0 , 1 ] , and the notation ‘’ denotes the uniform convergence with respect to the usual supremum norm on C [ 0 , 1 ] . Very recently, interesting properties of Bernstein polynomials were discussed in [7, 911] and [12].
  1. 2.
    The case a = 1 , b = 0 . In this case, we let x [ 0 , ) and define
    G 1 , 0 ( t , x ; k , m ) : = [ 2 1 k x k t k ( 1 + x ) k ] m 1 ( m k ) ! e t [ 1 1 + x ] = n = 0 P n ( 1 , 0 ) ( x ; k , m ) t n n ! .
     
We will see that this generating function produces the generalized Bleimann-Butzer-Hahn basis functions P n ( 1 , 0 ) ( x ; k , m ) : = H n ( m k , x ) . Furthermore, the special case
G 1 , 0 ( t , x ; 1 , m ) = [ x t ( 1 + x ) ] m 1 ( m k ) ! e t [ 1 1 + x ] = n = 0 H n ( m , x ) t n n !
generates the well-known Bleimann-Butzer-Hahn basis functions:
H n ( m , x ) : = H m n ( x ) = ( n m ) x m ( 1 + x ) n .
The Bleimann-Butzer-Hahn operators were introduced in [5] and defined by
L n ( f ; x ) = 1 ( 1 + x ) n m = 0 n f ( m n ) ( n m ) x m ; x [ 0 , ) , n N .

Denoting C B [ 0 , ) by the space of real-valued bounded continuous functions defined on [ 0 , ) , they proved that L n ( f ) f as n . On the other hand, the convergence is uniform on each compact subset of [ 0 , ) , where the norm is the usual supremum norm of C B [ 0 , ) . For the review of the results concerning the Bleimann-Butzer-Hahn operators obtained in the period 1980-2009, we refer to [13].

The following theorem gives the explicit representation of the basis family defined in (1). Note that throughout the paper, we let P n ( a , b ) ( x ; k , m ) : = 0 for n m k .

Theorem 1 If n m k , we have
P n ( a , b ) ( x ; k , m ) = 2 ( 1 k ) m x m k ( n m k ) ( 1 + b x ) n m k ( 1 + a x ) n .

Proof

Direct calculations give
G a , b ( t , x ; k , m ) = [ 2 1 k x k t k ( 1 + a x ) k ] m 1 ( m k ) ! e t [ 1 + b x 1 + a x ] = 2 ( 1 k ) m ( m k ) ! ( x t 1 + a x ) m k n = 0 ( 1 + b x 1 + a x ) n t n n ! = 2 ( 1 k ) m x m k n = m k ( n m k ) ( 1 + b x ) n m k ( 1 + a x ) n t n n ! .
(2)

Comparing (1) and (2), we get the result. □

Corollary 2 By taking a = 0 , b = 1 in Theorem  1, we obtain the explicit representation of the unifying Bernstein basis polynomials [1]:
P n ( 0 , 1 ) ( x ; k , m ) : = B n ( m k , x ) = 2 ( 1 k ) m x m k ( n m k ) ( 1 x ) n m k .

Furthermore, B n ( m , x ) = B m n ( x ) is the well-known Bernstein basis.

Corollary 3 Taking a = 1 , b = 0 in Theorem  1, we get the explicit representation of the generalized Bleimann-Butzer-Hahn basis:
P n ( 1 , 0 ) ( x ; k , m ) : = H n ( m k , x ) = 2 ( 1 k ) m x m k ( n m k ) 1 ( 1 + x ) n .

Moreover, H n ( m , x ) = H m n ( x ) is the Bleimann-Butzer-Hahn basis function.

We organize the paper as follows. In Section 2, we obtain the representation of this unified family in terms of Apostol-type polynomials and Stirling numbers of the second kind. In Section 3, we give more trigonometric generating functions for this unification and obtain a certain summation formula. All the special cases are listed at the end of each theorem.

2 Representation in terms of Apostol-type polynomials and Stirling numbers

Recently [14], the first author introduced the unification of the Apostol-Bernoulli, Euler and Genocchi polynomials by
(3)

For the convergence of the series in (3), we refer to [[14], p.2453].

Some of the well-known polynomials included by Q n , β ( α ) ( x ; k , a , b ) are listed below.

Remark 4 Having k = a = b = 1 and β = λ in (3), we get
Q n , λ ( α ) ( x ; 1 , 1 , 1 ) = B n ( α ) ( x ; λ ) .
Note that B n ( α ) ( x ; λ ) are the generalized Apostol-Bernoulli polynomials defined through the following generating relation:
( t λ e t 1 ) α e x t = n = 0 B n ( α ) ( x ; λ ) t n n ! ( | t | < 2 π  when  λ = 1 ; | t | < | log λ |  when  λ 1 ) ,
where α and λ are arbitrary real or complex parameters and x R . Note that when λ 1 , the order α should be restricted to nonnegative integer values. These polynomials were introduced by Luo and Srivastava [15] and investigated in [16, 17] and [18]. The Apostol-Bernoulli polynomials and numbers are obtained by the generalized Apostol-Bernoulli polynomials, respectively, as follows:
B n ( x ; λ ) = B n ( 1 ) ( x ; λ ) , B n ( λ ) = B n ( 0 ; λ ) ( n N 0 ) .

Taking λ = 1 in the above relations, we obtain the classical Bernoulli polynomials B n ( x ) and Bernoulli numbers B n .

Remark 5 Letting k = 2 a = b = 1 and 2 β = λ in (3), we get
Q n , λ 2 ( α ) ( x ; 1 , 1 2 , 1 ) = G n α ( x ; λ ) ,
the Apostol-Genocchi polynomial of order α (arbitrary real or complex) which was defined by [19, 20]. Here the parameter λ is arbitrary real or complex. These polynomials are given as follows:
( 2 t λ e t + 1 ) α e x t = n = 0 G n α ( x ; λ ) t n n ! ( | t | < π  when  λ = 1 ; | t | < | log ( λ ) |  when  λ 1 ) .
Note that when λ 1 , the order α should be restricted to nonnegative integer values. The Apostol-Genocchi polynomials and numbers are respectively given by
G n ( x ; λ ) = G n 1 ( x ; λ ) , G n ( λ ) = G n ( 0 ; λ ) .

When λ = 1 , the above relations give the classical Genocchi polynomials G n ( x ) and Genocchi numbers G n .

Although our results do not contain the Apostol-Euler polynomials, for the sake of completeness, we give their definitions as a special case of the polynomial family Q n , β ( α ) ( x ; k , a , b ) .

Remark 6 Setting k + 1 = a = b = 1 and β = λ in (3), we get
Q n , λ ( α ) ( x ; 0 , 1 , 1 ) = E n ( α ) ( x ; λ ) .
Recall that the Apostol-Euler polynomials E n ( α ) ( x ; λ ) are generalized by Luo [21] and given by the generating relation
( 2 λ e t + 1 ) α e x t = n = 0 E n α ( x ; λ ) t n n ! ( | t | < π  when  λ = 1 ; | t | < | log ( λ ) |  when  λ 1 ; 1 α : = 1 )
for arbitrary real or complex parameters α and λ and x R . The Apostol-Euler polynomials and numbers are given respectively by
E n ( x ; λ ) = E n 1 ( x ; λ ) , E n ( λ ) = E n ( 1 ; λ ) .

When λ = 1 , the above relations give the classical Euler polynomials E n ( x ) and Euler numbers E n .

Now, recall that the Stirling numbers of the second kind are denoted by S ( j , i ) and defined by (see [[22], p.58 (15)])
( e t 1 ) i = i ! j = i S ( j , i ) t j j ! .

The following theorem states an interesting explicit representation of the unified basis in terms of Apostol-type polynomials and relation between Stirling numbers of the second kind.

Theorem 7 The following representation:
P n ( a , b ) ( x ; k , m ) = 1 ( m k ) ! ( x 1 + a x ) m k i = 0 m ( m i ) ( β d c d ) m i β i d i ! × j = i n ( n j ) S ( j , i ) Q n j , β ( m ) ( 1 + b x 1 + a x ; k , c , d )

holds true between the unified Bernstein and Bleimann-Butzer-Hahn basis and Apostol-type polynomials.

Proof We get, using (1), that
(4)
On the other hand, since
( β d c d + β d [ e t 1 ] ) m = i = 0 m ( m i ) ( β d c d ) m i β i d [ e t 1 ] i = i = 0 m ( m i ) ( β d c d ) m i β i d i ! j = i S ( j , i ) t j j ! ,
we can write from (4) that
n = 0 P n ( a , b ) ( x ; k , m ) t n n ! = 1 ( m k ) ! ( x 1 + a x ) m k [ 2 1 k t k β b e t a b ] m e t [ 1 + b x 1 + a x ] × i = 0 m ( m i ) ( β b a b ) m i β i b i ! j = i S ( j , i ) t j j ! .
Now, using (3) in the above relation, we get
n = 0 P n ( a , b ) ( x ; k , m ) t n n ! = 1 ( m k ) ! ( x 1 + a x ) m k n = 0 Q n , β ( m ) ( 1 + b x 1 + a x ; k , c , d ) t n n ! × i = 0 m ( m i ) ( β d c d ) m i β i d i ! j = i S ( j , i ) t j j ! = 1 ( m k ) ! ( x 1 + a x ) m k n = 0 { i = 0 m ( m i ) ( β d c d ) m i β i d i ! × j = i n ( n j ) S ( j , i ) Q n j , β ( m ) ( 1 + b x 1 + a x ; k , c , d ) } t n n ! .

Whence the result. □

Now, we list some important corollaries of the above theorem.

Corollary 8 Since P n ( 0 , 1 ) ( x ; 1 , m ) = B m n ( x ) and Q n , λ ( α ) ( x ; 1 , 1 , 1 ) = B n ( α ) ( x ; λ ) , we obtain the following [1]:
B m n ( x ) = x m m ! i = 0 m ( m i ) ( λ 1 ) m i λ i i ! j = i n ( n j ) S ( j , i ) B n j ( m ) ( 1 x ; λ ) .
Furthermore, for λ = 1 , we have the following known relation:
B m n ( x ) = x m j = m n ( n j ) S ( j , m ) B n j ( m ) ( 1 x ) .
Corollary 9 Since P n ( 0 , 1 ) ( x ; 1 , m ) = B m n ( x ) and Q n , λ 2 ( α ) ( x ; 1 , 1 2 , 1 ) = G n α ( x ; λ ) , we get
B m n ( x ) = x m 2 m m ! i = 0 m ( m i ) ( λ + 1 ) m i λ i i ! j = i n ( n j ) S ( j , i ) G n j m ( 1 x ; λ ) .
Corollary 10 Since P n ( 1 , 0 ) ( x ; 1 , m ) = H m n ( x ) and Q n , λ ( α ) ( x ; 1 , 1 , 1 ) = B n ( α ) ( x ; λ ) , we obtain
H m n ( x ) = 1 m ! ( x 1 + x ) m i = 0 m ( m i ) ( λ 1 ) m i λ i i ! × j = i n ( n j ) S ( j , i ) B n j ( m ) ( 1 1 + x ; λ ) .
Furthermore, when λ = 1 , we have the following:
H m n ( x ) = ( x 1 + x ) m j = m n ( n j ) S ( j , m ) B n j ( m ) ( 1 1 + x ) .
Corollary 11 Since P n ( 1 , 0 ) ( x ; 1 , m ) = H m n ( x ) and Q n , λ 2 ( α ) ( x ; 1 , 1 2 , 1 ) = G n α ( x ; λ ) , we get
H m n ( x ) = 1 2 m m ! ( x 1 + x ) m i = 0 m ( m i ) ( λ 1 ) m i λ i i ! × j = i n ( n j ) S ( j , i ) G n j m ( 1 1 + x ; λ ) .

3 Generating functions of trigonometric type

In this section, we obtain a trigonometric generating relation for the unified Bernstein and Bleimann-Butzer-Hahn basis. Furthermore, we give a certain summation formula for this unification. We start with the following theorem.

Theorem 12 For the unified family, we have the following implicit summation formulae:
[ 2 1 2 l x 2 l ( 1 + a x ) 2 l ] m ( t 2 ) l m ( 2 l m ) ! cos t ( 1 + b x 1 + a x ) = n = 0 ( 1 ) n P 2 n ( a , b ) ( x ; 2 l , m ) t 2 n ( 2 n ) ! , [ 2 1 2 l x 2 l ( 1 + a x ) 2 l ] m ( t 2 ) l m ( 2 l m ) ! sin t ( 1 + b x 1 + a x ) = n = 0 ( 1 ) n P 2 n + 1 ( a , b ) ( x ; 2 l , m ) t 2 n + 1 ( 2 n + 1 ) !
(5)
and
[ 2 2 l x 2 l + 1 ( 1 + a x ) 2 l + 1 ] 2 j ( t 2 ) ( 2 l + 1 ) j ( 2 j ( 2 l + 1 ) ) ! cos t ( 1 + b x 1 + a x ) = n = 0 ( 1 ) n P 2 n ( a , b ) ( x ; 2 l + 1 , 2 j ) t 2 n ( 2 n ) ! , [ 2 2 l x 2 l + 1 ( 1 + a x ) 2 l + 1 ] 2 j ( t 2 ) ( 2 l + 1 ) j ( 2 j ( 2 l + 1 ) ) ! sin t ( 1 + b x 1 + a x ) = n = 0 ( 1 ) n P 2 n + 1 ( a , b ) ( x ; 2 l + 1 , 2 j ) t 2 n + 1 ( 2 n + 1 ) ! .
(6)
Finally,
[ 2 2 l x 2 l + 1 ( 1 + a x ) 2 l + 1 ] 2 j + 1 ( t 2 ) ( 2 l j + l + j ) [ ( 2 j + 1 ) ( 2 l + 1 ) ] ! t sin t ( 1 + b x 1 + a x ) = n = 0 ( 1 ) n P 2 n ( a , b ) ( x ; 2 l + 1 , 2 j + 1 ) t 2 n ( 2 n ) ! , [ 2 2 l x 2 l + 1 ( 1 + a x ) 2 l + 1 ] 2 j + 1 ( t 2 ) ( 2 l j + l + j ) [ ( 2 j + 1 ) ( 2 l + 1 ) ] ! t cos t ( 1 + b x 1 + a x ) = n = 0 ( 1 ) n P 2 n + 1 ( a , b ) ( x ; 2 l + 1 , 2 j + 1 ) t 2 n + 1 ( 2 n + 1 ) ! .
(7)
Proof Writing k = 2 l ( l N 0 ) in (1), we get
[ 2 1 2 l x 2 l t 2 l ( 1 + a x ) 2 l ] m 1 ( 2 l m ) ! e t [ 1 + b x 1 + a x ] = n = 0 P n ( a , b ) ( x ; 2 l , m ) t n n ! .
Letting t i t , we get
[ 2 1 2 l x 2 l ( 1 + a x ) 2 l ] m ( i t ) 2 l m ( 2 l m ) ! e i t [ 1 + b x 1 + a x ] = n = 0 P n ( a , b ) ( x ; 2 l , m ) ( i t ) n n !
and hence
[ 2 1 2 l x 2 l ( 1 + a x ) 2 l ] m ( t 2 ) l m ( 2 l m ) ! { cos t ( 1 + b x 1 + a x ) + i sin t ( 1 + b x 1 + a x ) } = n = 0 P 2 n ( a , b ) ( x ; 2 l , m ) ( i t ) 2 n ( 2 n ) ! + n = 0 P 2 n + 1 ( a , b ) ( x ; 2 l , m ) ( i t ) 2 n + 1 ( 2 n + 1 ) ! = n = 0 ( 1 ) n P 2 n ( a , b ) ( x ; 2 l , m ) t 2 n ( 2 n ) ! + i n = 0 ( 1 ) n P 2 n + 1 ( a , b ) ( x ; 2 l , m ) t 2 n + 1 ( 2 n + 1 ) ! .

Equating real and imaginary parts, we get (5).

Now, taking k = 2 l + 1 and m = 2 j ( l , j N 0 ) in (1), we obtain
[ 2 1 ( 2 l + 1 ) x 2 l + 1 t 2 l + 1 ( 1 + a x ) 2 l + 1 ] 2 j 1 ( 2 j ( 2 l + 1 ) ) ! e t [ 1 + b x 1 + a x ] = n = 0 P n ( a , b ) ( x ; 2 l + 1 , 2 j ) t n n ! .
Putting t i t ,
[ 2 2 l x 2 l + 1 ( i t ) 2 l + 1 ( 1 + a x ) 2 l + 1 ] 2 j 1 ( 2 j ( 2 l + 1 ) ) ! e i t [ 1 + b x 1 + a x ] = n = 0 P n ( a , b ) ( x ; 2 l + 1 , 2 j ) ( i t ) n n ! .
Therefore, we get
[ 2 2 l x 2 l + 1 ( 1 + a x ) 2 l + 1 ] 2 j ( t 2 ) ( 2 l + 1 ) j ( 2 j ( 2 l + 1 ) ) ! { cos t ( 1 + b x 1 + a x ) + i sin t ( 1 + b x 1 + a x ) } = n = 0 ( 1 ) n P 2 n ( a , b ) ( x ; 2 l + 1 , 2 j ) t 2 n ( 2 n ) ! + i n = 0 ( 1 ) n P 2 n + 1 ( a , b ) ( x ; 2 l + 1 , 2 j ) t 2 n + 1 ( 2 n + 1 ) ! ,

which is precisely (6).

Finally, for k = 2 l + 1 , m = 2 j + 1 ,
[ 2 2 l x 2 l + 1 t 2 l + 1 ( 1 + a x ) 2 l + 1 ] 2 j + 1 e t [ 1 + b x 1 + a x ] [ ( 2 j + 1 ) ( 2 l + 1 ) ] ! = n = 0 P n ( a , b ) ( x ; 2 l + 1 , 2 j + 1 ) t n n ! .
Taking t i t ,
[ 2 2 l x 2 l + 1 ( 1 + a x ) 2 l + 1 ] 2 j + 1 ( i t ) ( 2 l + 1 ) ( 2 j + 1 ) e i t [ 1 + b x 1 + a x ] [ ( 2 j + 1 ) ( 2 l + 1 ) ] ! = n = 0 P n ( a , b ) ( x ; 2 l + 1 , 2 j + 1 ) ( i t ) n n ! .
Thus,
[ 2 2 l x 2 l + 1 ( 1 + a x ) 2 l + 1 ] 2 j + 1 ( t 2 ) ( 2 l j + l + j ) [ ( 2 j + 1 ) ( 2 l + 1 ) ] ! [ t sin t ( 1 + b x 1 + a x ) + i t cos t ( 1 + b x 1 + a x ) ] = n = 0 ( 1 ) n P 2 n ( a , b ) ( x ; 2 l + 1 , 2 j + 1 ) t 2 n ( 2 n ) ! + i n = 0 ( 1 ) n P 2 n + 1 ( a , b ) ( x ; 2 l + 1 , 2 j + 1 ) t 2 n + 1 ( 2 n + 1 ) ! .

Equating real and imaginary parts we get (7). □

Since we obtain the unified Bernstein family in the case a = 0 , b = 1 , we have the following corollary at once.

Corollary 13 For the unified Bernstein family, we have the following implicit summation formulae:
( 2 1 2 l x 2 l ) m ( t 2 ) l m ( 2 l m ) ! cos t ( 1 x ) = n = 0 ( 1 ) n B 2 n ( 2 l m , x ) t 2 n ( 2 n ) ! , ( 2 1 2 l x 2 l ) m ( t 2 ) l m ( 2 l m ) ! sin t ( 1 x ) = n = 0 ( 1 ) n B 2 n + 1 ( 2 l m , x ) t 2 n + 1 ( 2 n + 1 ) !
and
( 2 2 l x 2 l + 1 ) 2 j ( t 2 ) ( 2 l + 1 ) j ( 2 j ( 2 l + 1 ) ) ! cos t ( 1 x ) = n = 0 ( 1 ) n B 2 n ( ( 2 l + 1 ) ( 2 j ) , x ) t 2 n ( 2 n ) ! , ( 2 2 l x 2 l + 1 ) 2 j ( t 2 ) ( 2 l + 1 ) j ( 2 j ( 2 l + 1 ) ) ! sin t ( 1 x ) = n = 0 ( 1 ) n B 2 n + 1 ( ( 2 l + 1 ) ( 2 j ) , x ) t 2 n + 1 ( 2 n + 1 ) ! .
(8)
Finally,
[ 2 2 l x 2 l + 1 ] 2 j + 1 ( t 2 ) ( 2 l j + l + j ) [ ( 2 j + 1 ) ( 2 l + 1 ) ] ! t sin t ( 1 x ) = n = 0 ( 1 ) n B 2 n ( ( 2 l + 1 ) ( 2 j + 1 ) , x ) t 2 n ( 2 n ) ! , [ 2 2 l x 2 l + 1 ] 2 j + 1 ( t 2 ) ( 2 l j + l + j ) [ ( 2 j + 1 ) ( 2 l + 1 ) ] ! t cos t ( 1 x ) = n = 0 ( 1 ) n B 2 n + 1 ( ( 2 l + 1 ) ( 2 j + 1 ) , x ) t 2 n + 1 ( 2 n + 1 ) ! .
(9)
On the other hand, taking l = 0 in (8) and (9), we get the following relations for the Bernstein basis:
x 2 j ( t 2 ) j ( 2 j ) ! cos t ( 1 x ) = n = 0 ( 1 ) n B 2 j 2 n ( x ) t 2 n ( 2 n ) ! , x 2 j ( t 2 ) j ( 2 j ) ! sin t ( 1 x ) = n = 0 ( 1 ) n B 2 j 2 n + 1 ( x ) t 2 n + 1 ( 2 n + 1 ) !
and
x 2 j + 1 ( t 2 ) j ( 2 j + 1 ) ! t sin t ( 1 x ) = n = 0 ( 1 ) n B 2 j + 1 2 n ( x ) t 2 n ( 2 n ) ! , x 2 j + 1 ( t 2 ) j ( 2 j + 1 ) ! t cos t ( 1 x ) = n = 0 ( 1 ) n B 2 j + 1 2 n + 1 ( x ) t 2 n + 1 ( 2 n + 1 ) ! .

Since the case a = 1 , b = 0 gives the unified Bleimann-Butzer-Hahn family, we immediately obtain the following corollary.

Corollary 14 For the unified Bleimann-Butzer-Hahn family, we have the following implicit summation formulae:
[ 2 1 2 l x 2 l ( 1 + x ) 2 l ] m ( t 2 ) l m ( 2 l m ) ! cos ( t 1 + x ) = n = 0 ( 1 ) n H 2 n ( 2 l m , x ) t 2 n ( 2 n ) ! , [ 2 1 2 l x 2 l ( 1 + x ) 2 l ] m ( t 2 ) l m ( 2 l m ) ! sin ( t 1 + x ) = n = 0 ( 1 ) n H 2 n + 1 ( 2 l m , x ) t 2 n + 1 ( 2 n + 1 ) !
and
[ 2 2 l x 2 l + 1 ( 1 + x ) 2 l + 1 ] 2 j ( t 2 ) ( 2 l + 1 ) j ( 2 j ( 2 l + 1 ) ) ! cos ( t 1 + x ) = n = 0 ( 1 ) n H 2 n ( ( 2 l + 1 ) ( 2 j ) , x ) t 2 n ( 2 n ) ! , [ 2 2 l x 2 l + 1 ( 1 + x ) 2 l + 1 ] 2 j ( t 2 ) ( 2 l + 1 ) j ( 2 j ( 2 l + 1 ) ) ! sin ( t 1 + x ) = n = 0 ( 1 ) n H 2 n + 1 ( ( 2 l + 1 ) ( 2 j ) , x ) t 2 n + 1 ( 2 n + 1 ) ! .
(10)
Finally,
(11)
Taking l = 0 in (10) and (11), we get the following relations for the Bleimann-Butzer-Hahn basis:
[ x 1 + x ] 2 j ( t 2 ) j ( 2 j ) ! cos ( t 1 + x ) = n = 0 ( 1 ) n H 2 j 2 n ( x ) t 2 n ( 2 n ) ! , [ x 1 + x ] 2 j ( t 2 ) j ( 2 j ) ! sin ( t 1 + x ) = n = 0 ( 1 ) n H 2 j 2 n + 1 t 2 n + 1 ( 2 n + 1 ) ! .
Finally,
[ x 1 + x ] 2 j + 1 ( t 2 ) j ( 2 j + 1 ) ! t sin ( t 1 + x ) = n = 0 ( 1 ) n H 2 j + 1 2 n ( x ) t 2 n ( 2 n ) ! , [ x 1 + x ] 2 j + 1 ( t 2 ) j ( 2 j + 1 ) ! t cos ( t 1 + x ) = n = 0 ( 1 ) n H 2 j + 1 2 n + 1 ( x ) t 2 n + 1 ( 2 n + 1 ) ! .

Finally, we obtain a summation formula for the unified Bernstein and Bleimann-Butzer-Hahn basis as follows.

Theorem 15 For all n , l N 0 ; a , b R , the following implicit summation formula holds true:
P n + l ( a , b ) ( y ; k , m ) = p , r = 0 l , n ( n r ) ( l p ) P n + l r p ( a , b ) ( x ; k , m ) [ 1 + b y 1 + a y 1 + b x 1 + a x ] r + p .
Proof Letting t t + u in (1) and then using the fact that
n = 0 l = 0 A ( l , n ) = n = 0 l = 0 n A ( l , n l ) ,
(12)
we get
[ 2 1 k x k ( t + u ) k ( 1 + a x ) k ] m 1 ( m k ) ! e ( t + u ) [ 1 + b x 1 + a x ] = n = 0 P n ( a , b ) ( x ; k , m ) ( t + u ) n n ! = n = 0 P n ( a , b ) ( x ; k , m ) l = 0 n t n l u l l ! ( n l ) ! = n , l = 0 P n + l ( a , b ) ( x ; k , m ) t n u l n ! l !
(13)
and hence
[ 2 1 k x k ( t + u ) k ( 1 + a x ) k ] m 1 ( m k ) ! = e ( t + u ) [ 1 + b x 1 + a x ] n , l = 0 P n + l ( a , b ) ( x ; k , m ) t n u l n ! l ! .
Multiplying both sides by e ( t + u ) [ 1 + b y 1 + a y ] and then expanding the function e ( t + u ) [ 1 + b y 1 + a y 1 + b x 1 + a x ] , we get, after using (12) twice, that
[ 2 1 k x k ( t + u ) k ( 1 + a x ) k ] m 1 ( m k ) ! e ( t + u ) [ 1 + b y 1 + a y ] = e ( t + u ) [ 1 + b y 1 + a y 1 + b x 1 + a x ] n , l = 0 P n + l ( a , b ) ( x ; k , m ) t n u l n ! l ! = n , l = 0 r = 0 P n + l ( a , b ) ( x ; k , m ) [ 1 + b y 1 + a y 1 + b x 1 + a x ] r r ! ( t + u ) r t n u l n ! l ! = n , l , p , r = 0 P n + l ( a , b ) ( x ; k , m ) [ 1 + b y 1 + a y 1 + b x 1 + a x ] r + p t n + r u p + l n ! l ! r ! p ! .
Now, using (12) with the index pairs ( n , r ) and ( l , p ) , we get
(14)
Since the left-hand side is equal by (13) to
[ 2 1 k x k ( t + u ) k ( 1 + a x ) k ] m 1 ( m k ) ! e ( t + u ) [ 1 + b y 1 + a y ] = n , l = 0 P n + l ( a , b ) ( y ; k , m ) t n u l n ! l ! ,
(15)

the proof is completed by comparing the coefficients of t n u l n ! l ! in (14) and (15). □

In the case a = 0 , b = 1 , we obtain the following result for the unified Bernstein family at once.

Corollary 16 For all n , l N 0 , the following implicit summation formula:
B n + l ( m k , y ) = p , r = 0 l , n ( n r ) ( l p ) B n + l r p ( m k , x ) [ x y ] r + p
(16)
holds true for the unified Bernstein family. Taking k = 1 in (16), we get the following relation for the Bernstein basis:
B m n + l ( y ) = p , r = 0 l , n ( n r ) ( l p ) B m n + l r p ( x ) [ x y ] r + p .

Since the case a = 1 , b = 0 gives the unified Bleimann-Butzer-Hahn family, we have the following result.

Corollary 17 For all n , l N 0 , the following implicit summation formula:
H n + l ( m k , y ) = p , r = 0 l , n ( n r ) ( l p ) H n + l r p ( m k , x ) [ x y ] r + p
(17)
holds true for the unified Bleimann-Butzer-Hahn family. Upon taking k = 1 in (17), we get the following relation for the Bleimann-Butzer-Hahn basis:
H m n + l ( y ) = p , r = 0 l , n ( n r ) ( l p ) H m n + l r p ( x ) [ x y ] r + p .

Declarations

Acknowledgements

Dedicated to Professor Hari M Srivastava.

Authors’ Affiliations

(1)
Eastern Mediterranean University, Gazimagusa, Turkey

References

  1. Simsek Y: Constructing a new generating function of Bernstein type polynomials. Appl. Math. Comput. 2011, 218: 1072-1076. 10.1016/j.amc.2011.01.074MATHMathSciNetView ArticleGoogle Scholar
  2. Acikgoz M, Aracı S: On generating function of the Bernstein polynomials. AIP Conf. Proc. 2010, 1281: 1141-1143. Proceedings of the International Conference on Numerical Analysis and Applied MathematicsView ArticleGoogle Scholar
  3. Bayad A, Kim T: Identities involving values of Bernstein, q -Bernoulli, and q -Euler polynomials. Russ. J. Math. Phys. 2011, 18(2):133-143. 10.1134/S1061920811020014MATHMathSciNetView ArticleGoogle Scholar
  4. Bernstein SN: Démonstration du théorème de Weierstrass fondée sur la calcul des probabilités. Commun. Soc. Math. Kharkov 1912-13, 13: 1-2.Google Scholar
  5. Bleimann G, Butzer PL, Hahn L: A Bernstein-type operator approximating continuous functions on the semi-axis. Indag. Math. 1980, 42: 255-262.MATHMathSciNetView ArticleGoogle Scholar
  6. Busé L, Goldman R: Division algorithms for Bernstein polynomials. Comput. Aided Geom. Des. 2008, 25: 850-865. 10.1016/j.cagd.2007.10.003MATHView ArticleGoogle Scholar
  7. Kim M-S, Kim T, Lee B, Ryoo C-S: Some identities of Bernoulli numbers and polynomials associated with Bernstein polynomials. Adv. Differ. Equ. 2010., 2010: Article ID 305018Google Scholar
  8. Kim T, Jang L-J, Yi H: A note on the modified q -Bernstein polynomials. Discrete Dyn. Nat. Soc. 2010. doi:10.1155/2010/706483Google Scholar
  9. Morin G, Goldman R: On the smooth convergence of subdivision and degree elevation for Bézier curves. Comput. Aided Geom. Des. 2001, 18: 657-666. 10.1016/S0167-8396(01)00059-0MATHMathSciNetView ArticleGoogle Scholar
  10. Phillips GM CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC 14. In Interpolation and Approximation by Polynomials. Springer, New York; 2003.View ArticleGoogle Scholar
  11. Simsek Y, Acikgoz M: A new generating function of ( q -) Bernstein-type polynomials and their interpolation function. Abstr. Appl. Anal. 2010., 2010: Article ID 769095Google Scholar
  12. Zorlu S, Aktuglu H, Ozarslan MA: An estimation to the solution of an initial value problem via q -Bernstein polynomials. J. Comput. Anal. Appl. 2010, 12: 637-645.MATHMathSciNetGoogle Scholar
  13. Ulrich A, Mircea I: The Bleimann-Butzer-Hahn operators old and new results. Appl. Anal. 2011, 90(3-4):483-491. 10.1080/00036810903517639MATHMathSciNetView ArticleGoogle Scholar
  14. Ozarslan MA: Unified Apostol-Bernoulli, Euler and Genocchi polynomials. Comput. Math. Appl. 2011, 62(6):2452-2462. 10.1016/j.camwa.2011.07.031MathSciNetView ArticleGoogle Scholar
  15. Luo Q-M, Srivastava HM: Some generalizations of the Apostol-Bernoulli and Apostol-Euler polynomials. J. Math. Anal. Appl. 2005, 308(1):290-302. 10.1016/j.jmaa.2005.01.020MATHMathSciNetView ArticleGoogle Scholar
  16. Luo Q-M: On the Apostol-Bernoulli polynomials. Cent. Eur. J. Math. 2004, 2(4):509-515. 10.2478/BF02475959MATHMathSciNetView ArticleGoogle Scholar
  17. Luo Q-M, Srivastava HM: Some relationships between the Apostol-Bernoulli and Apostol-Euler polynomials. Comput. Math. Appl. 2006, 51(3-4):631-642. 10.1016/j.camwa.2005.04.018MATHMathSciNetView ArticleGoogle Scholar
  18. Srivastava HM: Some formulas for the Bernoulli and Euler polynomials at rational arguments. Math. Proc. Camb. Philos. Soc. 2000, 129(1):77-84. 10.1017/S0305004100004412MATHView ArticleGoogle Scholar
  19. Luo Q-M: Fourier expansions and integral representations for the Genocchi polynomials. J. Integer Seq. 2009., 12: Article ID 09.1.4Google Scholar
  20. Luo Q-M: Extension for the Genocchi polynomials and its Fourier expansions and integral representations. Osaka J. Math. 2011, 48(2):291-309.MATHMathSciNetGoogle Scholar
  21. Luo Q-M: Apostol-Euler polynomials of higher order and Gaussian hypergeometric functions. Taiwan. J. Math. 2006, 10: 917-925.MATHGoogle Scholar
  22. Srivastava HM, Choi J: Series Associated with the Zeta and Related Functions. Kluwer Academic, Dordrecht; 2001.MATHView ArticleGoogle Scholar

Copyright

© Özarslan and Bozer; licensee Springer 2013

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/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Advertisement