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The twisted (h, q)-Genocchi numbers and polynomials with weight α and q-bernstein polynomials with weight α

Journal of Inequalities and Applications20122012:67

https://doi.org/10.1186/1029-242X-2012-67

Received: 16 December 2011

Accepted: 23 March 2012

Published: 23 March 2012

Abstract

In this article, we give some identities on the twisted (h, q)-Genocchi numbers and polynomials and q-Bernstein polynomials with weighted α.

Keywords

Genocchi numbers and polynomialstwisted (h, q)-Genocchi numbers and polynomials with weight αq-Bernstein polynomials

1 Introduction

Let p be a fixed odd prime number. The symbol, p , p , and p denote the ring of p-adic integers, the field of p-adic rational numbers and the completion of algebraic closure of p , respectively. Let be the set of natural numbers and + = { 0 } . As well known definition, the p-adic absolute value is given by |x| p = p -r , where x = p r t s with (t, p) = (s, p) = (t, s) = 1. When one talks of q-extension, q is variously considered as an indeterminate, a complex number q , or a p-adic number q p . In this article, we assume that q p with |1 - q| p < 1.

For f U D ( p ) = { f / f : p p is uniformly differentiable function}, Kim defined the fermionic p-adic q-integral on p as follows:
I - q ( f ) = p f ( x ) d μ - q ( x ) = lim N 1 [ p N ] - q x = 0 p N - 1 f ( x ) ( - q ) x .
(1.1)
For n , let f n (x) = f(x + n) be translation. As well known equation, by (1.1), we have
q n I - q ( f n ) = ( - 1 ) n I - q ( f ) + [ 2 ] q l - 0 n - 1 ( - 1 ) n - 1 - l q l f ( l ) , .
(1.2)
Throughout this article we use the notation:
[ x ] q = 1 - q x 1 - q .
limq→1|x| q = x for any x with |x| p 1 in the present p-adic case. To investigate relation of the twisted (h, q)-Genocchi numbers and polynomials with weight α and the q-Bernstein polynomials with weight α, we will use useful property for [ x ] q α as following;
[ x ] q α = 1 - [ 1 - x ] q - α [ 1 - x ] q - α = 1 - [ x ] q α
(1.3)
The twisted (h, q)-Genocchi numbers and polynomials with weight α are defined by the generating function, respectively:
G n , q , w ( h , α ) = n p q x ( h - 1 ) ϕ w ( x ) [ x ] q α n - 1 d μ - q ( x ) .
(1.4)
G n , q , w ( h , α ) ( x ) = n p q y ( h - 1 ) ϕ w ( y ) [ y + x ] q α n - 1 d μ - q ( y ) .
(1.5)

In the special case, x = 0 , G n , q , w ( h , α ) ( 0 ) = G n , q , w ( h , α ) are called the n th twisted (h, q)-Genocchi numbers with weight α (see [1]).

Let C p n = { w | w p n = 1 } be the cyclic group of order p n and let
T p = lim n C p n = n 1 C p n

see [15].

Kim defined the q-Bernstein polynomials with weight α of degree n as follows:
B k , n ( α ) ( x , q ) = ( k n ) [ x ] q α k [ 1 x ] q α n k , where  x [ 0 , 1 ] , n , k +
(1.6)

cf [612].

In this article, we investigate some properties for the twisted (h, q)-Genocchi numbers and polynomials with weight α. By using these properties, we give some interesting identities on the twisted (h, q)-Genocchi polynomials with weight α and q-Bernstein polynomials with weight α.

2 Twisted (h, q)-genocchi numbrs and polynomials with weight α and q-bernstein polynomials with weight α

From (1.2), we can get the following form for the twisted (h, q)-Genocchi numbers with weight α:
G 0 , q , w ( h , α ) = 0 , and q h w G n , q , w ( h , α ) ( 1 ) + G n , q , w ( h , α ) = [ 2 ] q , if n = 1 , 0 , if n > 1 ,
(2.1)
G 0 , q , w ( h , α ) = 0 , and q h w ( 1 + q α G q , w ( h , α ) ) n + q α G n , q , w ( h , α ) = q α [ 2 ] q , if n = 1 , 0 , if n > 1 ,
(2.2)
q α x G n + 1 , q , w ( h , α ) ( x ) = [ x ] q α + q α x G q , w ( h , α ) n + 1
(2.3)

with usual convention about replacing ( G q , w ( h , α ) ) n by G n , q , w ( h , α ) .

By (1.4), we can obtain
G n , q , w ( h , α ) ( x ) = n [ 2 ] q 1 1 - q α n - 1 l = 0 n - 1 n - 1 l ( - 1 ) l 1 1 + w q α l + h
(2.4)
By (2.4), we can get
G n , q - 1 , w - 1 ( h , α ) ( 1 - x ) = n [ 2 ] q - 1 1 1 - q - α n - 1 l = 0 n - 1 n - 1 l ( - 1 ) l ( q - 1 ) α l ( 1 - x ) 1 1 + w - 1 ( q - 1 ) α l + h = n 1 q [ 2 ] q 1 1 - q α n - 1 ( - 1 ) n - 1 q α n + α l = 0 n - 1 n - 1 l ( - 1 ) l q α l x w q h 1 + w q α l + h = n [ 2 ] q 1 1 - q α n - 1 l = 0 n - 1 n - 1 l ( - 1 ) l q α l x 1 1 + w q α l + h 1 q q α n - α ( - 1 ) n - 1 w q h = ( - 1 ) n - 1 w q α ( n - 1 ) + ( h - 1 ) G n , q , w ( h , α ) ( x ) .

So, we get the following theorem.

Theorem 1. Let n + . For w T p , we have
G n , q , w ( h , α ) ( x ) = ( - 1 ) n - 1 w - 1 q α ( 1 - n ) + ( 1 - h ) G n , q - 1 , w - 1 ( h , α ) ( 1 - x ) .
By (2.1), (2.2), and (2.3), we note that
G n , q , w ( h , α ) = - w q h G n , q , w ( h , α ) ( 1 ) = - w q h ( q - α ( 1 + g α G q , w ( h , α ) ) n ) = - w q h - α l = 0 n n l ( q α ) l G l , q , w ( h , α ) = - w q h n 1 G 1 , q , w ( h , α ) - w q h - α l = 2 n n l q α l - w q h G l , q , w ( h , α ) ( 1 ) = - w q h n 1 G 1 , q , w ( h , α ) - w q h - α l = 2 n n l q α l ( - w q h q - α ( 1 + q α G q , w ( h , α ) ) l ) = - n w q h G 1 , q , w ( h , α ) + w 2 q 2 h - 2 α l = 2 n n l q α l ( 1 + q α G q , w ( h , α ) ) l = - n w q h G 1 , q , w ( h , α ) + w 2 q 2 h - 2 α ( 1 + q α ( 1 + g α G q , w ( h , α ) ) ) n - n w 2 q 2 h - 2 α q α ( 1 + q α G q , w ( h , α ) ) 1 = - n w q h G 1 , q , w ( h , α ) + w 2 q 2 h - 2 α ( [ 2 ] q α + q 2 α G q , w ( h , α ) ) n - n w 2 q 2 h - α ( q α G 1 , q , w ( h , α ) ) = - n w q h G 1 , q , w ( h , α ) + w 2 q 2 h - 2 α ( [ 2 ] q α + q 2 α G q , w ( h , α ) ) n - n w 2 q 2 h G 1 , q , w ( h , α ) = - n w q h G 1 , q , w ( h , α ) + w 2 q 2 h G n , q , w ( h , α ) ( 2 ) - n w 2 q 2 h G 1 , q , w ( h , α )
(2.5)

Therefore, by (2.5), we obtain the theorem below.

Theorem 2. For n with n > 1, we have
G n , q , w ( h , α ) ( 2 ) = w - 2 q - 2 h G n , q , w ( h , α ) + w - 1 q - h n [ 2 ] q 1 + w q h + n [ 2 ] q 1 + w q h .
From Theorem 2,
G n + 1 , q , w ( h , α ) ( 2 ) n + 1 = 1 n + 1 ( n + 1 ) [ 2 ] q 1 + w q h + ( n + 1 ) w - 1 q - h [ 2 ] q 1 + w q h + w - 2 q - 2 h G n + 1 , q , w ( h , α ) n + 1 = [ 2 ] q 1 + w q h + w - 1 q - h [ 2 ] q 1 + w q h + w - 2 q - 2 h G n + 1 , q , w ( h , α ) n + 1

Therefore, we obtain the Corollary 3 by (1.5) and Theorem 2.

Corollary 3. For n , we have
p q y ( h - 1 ) ϕ w ( y ) [ y + 2 ] q α n q μ - q ( y ) = [ 2 ] q 1 + w q h + w - 1 q - h [ 2 ] q 1 + w q h + w - 2 q - 2 h G n + 1 , q , w ( h , α ) n + 1
By Theorems 1, 2 and fermionic integral on p , we note that
p q x ( h - 1 ) ϕ w ( x ) [ 1 - x ] q - α n q μ - q ( x ) = ( - 1 ) n q α n p q x ( h - 1 ) ϕ w ( x ) [ x - 1 ] q α n d μ - q ( x ) = ( - 1 ) n q α n G n + 1 , q , w ( h , α ) ( - 1 ) n + 1 = w - 1 q 1 - h G n + 1 , q - 1 , w - 1 ( h , α ) ( 2 ) n + 1 = w - 1 q 1 - h [ 2 ] q - 1 1 + q - h w - 1 + w q h [ 2 ] q - 1 1 + q - 1 w - 1 + w 2 q 2 h G n + 1 , q - 1 , w - 1 ( h , α ) n + 1 = [ 2 ] q 1 + w q h + w q h [ 2 ] q 1 + w q h + w q h + 1 G n + 1 , q - 1 , w - 1 ( h , α ) n + 1 = [ 2 ] q + w q h + 1 G n + 1 , q - 1 , w - 1 ( h , α ) n + 1 .
(2.6)

Hence, we get the following theorem.

Theorem 4. for n with n > 1, we have
p q x ( h - 1 ) ϕ w ( x ) [ 1 - x ] q - α n d μ - q ( x ) = [ 2 ] q + w q h + 1 G n + 1 , q - 1 , w - 1 ( h , α ) n + 1 .
(2.7)

Corollary 5.

From (1.3) and Theorem 4, we take the fermionic p-adic invariant integral on p for q-Bernstein polynomials as follows:
p q x ( h - 1 ) ϕ w ( x ) B k , n ( x , q ) d μ - q ( x ) = p q x ( h - 1 ) ϕ w ( x ) n k [ x ] q α k [ 1 - x ] q - α n - k d μ - q ( x ) = n k p q x ( h - 1 ) ϕ w ( x ) [ x ] q α k ( 1 - [ x ] q α ) n - k d μ - q ( x ) = n k p q x ( h - 1 ) ϕ w ( x ) [ x ] q α k ( 1 - [ x ] q α ) n - k d μ - q ( x ) = n k l = 0 n - k n n - k ( - 1 ) l p q x ( h - 1 ) ϕ w ( x ) [ x ] q α k + l d μ - q ( x ) = n k l = 0 n - k n - k l ( - 1 ) l G k + l + 1 , q , w ( h , α ) k + l + 1
(2.8)
And we get the following formula;
p q x ( h - 1 ) ϕ w ( x ) B k , n ( x , q ) d μ - q ( x ) = p q x ( h - 1 ) ϕ w ( x ) n k [ x ] q α n - k [ 1 - x ] q - α k d μ - q ( x ) = p q x ( h - 1 ) ϕ w ( x ) n k [ 1 - x ] q - α k ( 1 - [ 1 - x ] q - α ) n - k d μ - q ( x ) = n k l = 0 n - k n - k l ( - 1 ) n - k - l [ 1 - x ] q - α n - k - l p q x ( h - 1 ) ϕ w ( x ) [ 1 - x ] q - α k d μ - q ( x ) = n k l = 0 n - k n - k l ( - 1 ) n - k - l p q x ( h - 1 ) ϕ w ( x ) [ 1 - x ] q - α n - l d μ - q ( x ) = n k l = 0 n - k n - k l ( - 1 ) n - k - l [ 2 ] q + w q 1 + h G n - l + 1 , q - 1 , w - 1 ( h , α ) n - l + 1
(2.9)

Hence, we can get the following theorem by (2.8) and (2.9).

Theorem 5. for n with n > 1, we have
l = 0 n - k n - k l ( - 1 ) l G k + l + 1 , q , w ( h , α ) k + l + 1 = l = 0 n - k n - k l ( - 1 ) n - k - l [ 2 ] + w q 1 + h G n - l + 1 , q - 1 , w - 1 ( h , α ) n - l + 1 = l = 0 n - k n - k l ( - 1 ) n - k - l w - 1 q 1 - h G n - l + 1 , q - 1 , w - 1 ( h , α ) ( 2 ) n - l + 1
(2.10)
Also, we can see that
p q x ( h - 1 ) ϕ w ( x ) B k , n ( x , q ) d μ q ( x ) = n k l = 0 n - k n - k l ( - 1 ) l G k + l + 1 , q , w ( h , α ) k + l + 1 = n k l = 0 q x ( h - 1 ) ϕ w ( x ) [ 1 - x ] q - α n - k [ x ] q α k d μ - q ( x ) = n k l = 0 q x ( h - 1 ) ϕ w ( x ) [ 1 - x ] q - α n - k 1 - [ 1 - x ] q - α d μ - q ( x ) = n k l = 0 k k l ( - 1 ) k - 1 p q x ( h - 1 ) ϕ w ( x ) [ 1 - x ] q - α n - k d μ - q ( x ) = n k l = 0 k k l ( - 1 ) k - 1 [ 2 ] q + w q 1 + h G n - l + 1 , q - 1 , w - 1 ( h , α ) n - l + 1 .
(2.11)

Therefore, we have the theorem below.

Theorem 6. For n , k + with n > k + 1, we have
p q x ( h - 1 ) ϕ w ( x ) B k , n ( x , q ) d μ - q ( x ) = n k l = 0 k k l ( - 1 ) k - l [ 2 ] q + w q 1 + h G n - l + 1 , q - 1 , w - 1 ( h , α ) n - l + 1 .
(2.12)

By (2.7) and Theorem 6, we can get the theorem below.

Theorem 7. Let n , k + with n > k + 1. Then we have
l = 0 n - k n - k l ( - 1 ) l G k + l + 1 , q , w ( h , α ) k + l + 1 = l = 0 k k l ( - 1 ) k - 1 [ 2 ] q + w q 1 + h G n - l + 1 , q - 1 , w - 1 ( h , α ) n - l + 1 .
Let n 1 , n 2 , k + with n1 + n2> 2k + 1. Then we get
p q x ( h - 1 ) ϕ w ( x ) B k , n 1 ( α ) ( x , q ) B k , n 2 ( α ) ( x , q ) d μ - q ( x ) = p q x ( h - 1 ) ϕ w ( x ) n 1 k [ x ] q α k [ 1 - x ] q - α n 1 - k n 2 k [ x ] q α k [ 1 - x ] q - α n 2 - k d μ - q ( x ) = n 1 k n 2 k p q x ( h - 1 ) ϕ w ( x ) [ x ] q α 2 k [ 1 - x ] q - α n 1 + n 2 - k d μ - q ( x ) = n 1 k n 2 k l = 0 2 k 2 k l ( - 1 ) 2 k - l p q x ( h - 1 ) ϕ w ( x ) [ 1 - x ] q - α n 1 + n 2 - l d μ - q ( x ) = n 1 k n 2 k l = 0 2 k 2 k l ( - 1 ) 2 k - l [ 2 ] q + w q 1 + h G n 1 + n 2 - l + 1 , q - 1 , w - 1 ( h , α ) n 1 + n 2 - l + 1 .

Therefore, we obtain the theorem below.

Theorem 8. For n 1 , n 2 , k + , we have
p q x ( h - 1 ) ϕ w ( x ) B k , n 1 ( α ) ( x , q ) B k , n 2 ( α ) ( x , q ) d μ - q ( x ) = n 1 k n 2 k l = 0 2 k 2 k l ( - 1 ) 2 k - l [ 2 ] q + w q 1 + h G n 1 + n 2 - l + 1 , q - 1 , w - 1 ( h , α ) n 1 + n 2 - l + 1 = [ 2 ] q + w q 1 + h G n 1 + n 2 - l + 1 , q - 1 , w - 1 ( h , α ) n 1 + n 2 - l + 1 , if k = 0 , w q 1 + h n 1 k n 2 k l = 0 2 k 2 k l ( - 1 ) 2 k - l G n 1 + n 2 - l + 1 , q - 1 , w - 1 ( h , α ) n 1 + n 2 - l + 1 , if k > 0 ,
(2.13)
And we can easily have that
p q x ( h - 1 ) ϕ ω ( x ) B k , n 1 ( α ) ( x , q ) B k , n 2 ( α ) ( x , q ) d μ - q ( x ) = p q x ( h - 1 ) ϕ ω ( x ) n 1 k [ x ] q α k [ 1 - x ] q - α n 1 - k n 2 k [ x ] q α k [ 1 - x ] q - α n 2 - k d μ - q ( x ) = n 1 k n 2 k p q x ( h - 1 ) ϕ w ( x ) [ x ] q α 2 k [ 1 - x ] q - α n 1 + n 2 - k d μ - q ( x ) = n 1 k n 2 k p q x ( h - 1 ) ϕ w ( x ) [ x ] q α 2 k ( 1 - [ x ] q α ) n 1 + n 2 - 2 k d μ - q ( x ) = n 1 k n 2 k p q x ( h - 1 ) ϕ w ( x ) [ x ] q α 2 k l = 0 n 1 + n 2 - 2 k n 1 + n 2 - 2 k l ( - 1 ) l [ x ] q α l d μ - q ( x ) = n 1 k n 2 k l = 0 n 1 + n 2 - 2 k ( - 1 ) l n 1 + n 2 - 2 k l p q x ( h - 1 ) ϕ w ( x ) [ x ] q α 2 k + 1 d μ - q ( x ) = n 1 k n 2 k l = 0 n 1 + n 2 - 2 k ( - 1 ) l n 1 + n 2 - 2 k l G 2 k + l + 1 , q , w ( h , α ) 2 k + l + 1 , where n 1 , n 2 , k + .
(2.14)

Therefore, by (2.14) and Theorem 8, we obtain the theorem below.

Theorem 9. Let n 1 , n 2 , k + with n1 + n2> 2k + 1. Then we have
l = 0 2 k 2 k l ( - 1 ) 2 k - l [ 2 ] q + w q 1 + h G n 1 + n 2 - l + 1 , q - 1 , w - 1 ( h , α ) n 1 + n 2 - l + 1 = l = 0 n 1 + n 2 - 2 k ( - 1 ) l n 1 + n 2 - 2 k l G 2 k + l + 1 , q , w ( h , α ) 2 k + l + 1 .
For n 1 , n 2 , , n s , k + , n 1 + n 2 + + n s > s k + 1 , then by the symmetry of q-Bernstein polynomials with weight α, we see that
p q x ( h - 1 ) ϕ w ( x ) i = 1 s B k , n i ( α ) ( x , q ) d μ - q ( x ) = i = 1 s n i k p q x ( h - 1 ) ϕ w ( x ) [ x ] q α s k [ 1 - x ] q - α n 1 + n 2 + + n s - s k d μ - q ( x ) = i = 1 s n i k p q x ( h - 1 ) ϕ w ( x ) ( 1 - [ 1 - x ] q - α ) s k [ 1 - x ] q - α n 1 + n 2 + + n s - s k d μ - q ( x ) = i = 1 s n i k l = 0 s k s k l ( - 1 ) s k - l p q x ( h - 1 ) ϕ w ( x ) [ 1 - x ] q - α n 1 + n 2 + + n s - l d μ - q ( x ) = i = 1 s n i k l = 0 s k s k l ( - 1 ) s k - l [ 2 ] q + w q 1 + h G n 1 + n 2 + + n s - l + 1 , q - 1 , w - 1 ( h , α ) n 1 + n 2 + + n s - l + 1 .

Therefore, we have the theorem below.

Theorem 10. For n 1 , n 2 , n 3 , , n s , k + with n1 + n2 + . . . + n s > sk + 1, we have
p q x ( h - 1 ) ϕ w ( x ) i = 1 s B k , n i ( α ) ( x , q ) d μ - q ( x ) = i = 1 s n i k l = 0 s k s k l ( - 1 ) s k - l [ 2 ] q + w q 1 + h G n 1 + n 2 + + n s - l + 1 , q - 1 , w - 1 ( h , α ) n 1 + n 2 + + n s - l + 1 .
In the same manner as in (2.11), we can get the following relation:
p q x ( h - 1 ) ϕ w ( x ) i = 1 s B k , n i ( α ) ( x , q ) d μ - q ( x ) = i = 1 s n i k p q x ( h - 1 ) ϕ w ( x ) [ x ] q α s k ( 1 - [ x ] q α ) n 1 + n 2 + + n s - s k d μ - q ( x ) = i = 1 s n i k p q x ( h - 1 ) ϕ w ( x ) [ x ] q α s k l = 0 n 1 + n 2 + + n s - s k ( - 1 ) l n 1 + n 2 + + n s - s k l ( - 1 ) l [ x ] q α l d μ - q ( x ) = i = 1 s n i k l = 0 n 1 + n 2 + + n s - s k ( - 1 ) l n 1 + n 2 + + n s - s k l p q x ( h - 1 ) ϕ w ( x ) [ x ] q α s k + l d μ - q ( x ) = i = 1 s n i k l = 0 n 1 + n 2 + + n s - s k ( - 1 ) l n 1 + n 2 + + n s - s k l G s k + l + 1 , q , w ( h , α ) s k + l + 1 ,

where n 1 , n 2 , , n s , k + with n1 + n2 + . . . + n s > sk + 1.

By Theorem 11 and (2.9), we have the following corollary.

Corollary 11. Let m . For n 1 , n 2 , , n s , k + with n1 + . . . + n s > sk + 1, we have
l = 0 s k s k l ( - 1 ) s k - l [ 2 ] q + w q 1 + h G n 1 + n 2 + + n s - l + 1 , q - 1 , w - 1 ( h , α ) n 1 + n 2 + + n s - l + 1 = l = 0 n 1 + n 2 + + n s - s k ( - 1 ) l n 1 + n 2 + + n s - s k l G s k + l + 1 , q , w ( h , α ) s k + l + 1 ,

Declarations

Authors’ Affiliations

(1)
Department of Mathematics, Hannam University

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