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 $k,m∈ Z + :={1,2,…}$, $a,b∈R$, $t∈C$. 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 $P n ( 0 , − 1 ) (x;k,m):= B n (mk,x)$ which were introduced and investigated in . 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 )

$B n (m,x):= B m n (x)= ( n m ) x k ( 1 − x ) n − m .$

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 .

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  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 .

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 $n≤mk$.

Theorem 1 If $n≥mk$, 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 :

$P n ( 0 , − 1 ) (x;k,m):= B n (mk,x)= 2 ( 1 − k ) m x m k ( n m k ) ( 1 − x ) n − m k .$

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 , 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 [, 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:

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  and investigated in [16, 17] and . 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:

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  and given by the generating relation

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 [, 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 :

$B m n (x)= x m m ! ∑ i = 0 m ( m i ) ( λ − 1 ) m − i λ i i! ∑ j = i n ( n j ) S(j,i) B n − j ( m ) (1−x;λ).$

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 $k=2l$ ($l∈ N 0$) 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 $t→it$, 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 $k=2l+1$and $m=2j$ ($l,j∈ N 0$) 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 $t→it$,

$[ 2 − 2 l x 2 l + 1 ( i t ) 2 l + 1 ( 1 + a x ) 2 l + 1 ] 2 j 1 ( 2 j ( 2 l + 1 ) ) ! e i t [ 1 + b x 1 + a x ] = ∑ n = 0 ∞ P n ( a , b ) (x;2l+1,2j) ( i t ) n n ! .$

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 $t→it$,

$[ 2 − 2 l x 2 l + 1 ( 1 + a x ) 2 l + 1 ] 2 j + 1 ( i t ) ( 2 l + 1 ) ( 2 j + 1 ) e i t [ 1 + b x 1 + a x ] [ ( 2 j + 1 ) ( 2 l + 1 ) ] ! = ∑ n = 0 ∞ P n ( a , b ) (x;2l+1,2j+1) ( i t ) n n ! .$

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 $n,l∈ N 0$; $a,b∈R$, 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:

$B n + l (mk,y)= ∑ p , r = 0 l , n ( n r ) ( l p ) B n + l − r − p (mk,x) [ x − y ] r + p$
(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.

Authors’ contributions

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

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