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

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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.

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,bR, tC. Here, xI 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 (mk,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 ,nN:={1,2,}

and by the Korovkin theorem, it is known that B n (f;x)f(x) for all fC[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 (mk,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,),nN.

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 nmk.

Theorem 1 If nmk, 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 (mk,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 (mk,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 xR. Note that when λ1, 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=2a=b=1 and 2β=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 λ1, 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 xR. 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 ) (1x;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 ) (1x).

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 (1x;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=2l (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;2l,m) t n n ! .

Letting tit, 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;2l,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=2l+1and m=2j (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;2l+1,2j) t n n ! .

Putting tit,

[ 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;2l+1,2j) ( 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=2l+1, m=2j+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;2l+1,2j+1) t n n ! .

Taking tit,

[ 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;2l+1,2j+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,bR, 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 tt+u in (1) and then using the fact that

n = 0 l = 0 A(l,n)= n = 0 l = 0 n A(l,nl),
(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 (mk,y)= p , r = 0 l , n ( n r ) ( l p ) B n + l r p (mk,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 (mk,y)= p , r = 0 l , n ( n r ) ( l p ) H n + l r p (mk,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 .

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Acknowledgements

Dedicated to Professor Hari M Srivastava.

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Correspondence to Mehmet Bozer.

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The authors declare that they have no competing interests.

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All authors completed the paper together. All authors read and approved the final manuscript.

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Keywords

  • generating function
  • Bernstein polynomials
  • Bernoulli polynomials
  • Euler polynomials
  • Genocchi polynomials
  • Stirling numbers of the second kind