Skip to main content

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

Abstract

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

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 pn 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 >sk+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 ,

References

  1. Jung NS, Ryoo CS: On the twisted (h; q)-Genocchi numbers and polynomials associated with weight α. Proc. Jangjeon Math. Soc. 2012, 15: 1–9. 10.1155/2011/123483

    MathSciNet  Google Scholar 

  2. Rim SH, Jin JH, Moon EJ, Lee SJ: Some identities on the q -Genocchi polynomials of higher-order and q -Stirling numbers by the fermionic p -adic integral on p . International Journal of Mathematics and Mathematical Sciences 2010., 2010: Art. ID 860280

    Google Scholar 

  3. Ryoo CS: Some identities of the twisted q -Euler numbers and polynomials associated with q -Bernstein polynomials. Proc Jangjeon Math Soc 2011, 14: 239–248.

    MathSciNet  Google Scholar 

  4. Ryoo CS: Some relations between twisted q -Euler numbers and Bernstein polynomials. Adv Stud Contemp Math 2011, 21: 217–223.

    MathSciNet  Google Scholar 

  5. Simesk Y, Kurt V, Kim D: New approach to the complete sum of products of the twisted ( h, q )-Bernoulli numbers and polynomials. J Nonlinear Math Phys 2007, 14: 44–56. 10.2991/jnmp.2007.14.1.5

    Article  MathSciNet  Google Scholar 

  6. Kim T, Choi J, Kim Y-H: Some identities on the q -Bernstein polynomials, q -Stirling numbers and q -Bernoulli numbers. Adv Stud Contemp Math 2010, 20: 335–341.

    Google Scholar 

  7. Kim T, Jang LC, Yi H: A note on the modified q -Bernstein polynomials. Discrete Dynamics in Nature and Society 2010, 2010: 12. Article ID 706483

    Google Scholar 

  8. Kim T: Some identities on the q -Euler polynomials of higher order and q -Stirling numbers by the fermionic p -adic integral on p . Russian J Math Phys 2009, 16: 484–491. 10.1134/S1061920809040037

    Article  Google Scholar 

  9. Kim T: Note on the Euler numbers and polynomials. Adv Stud Contemp Math 2008, 17: 131–136.

    Google Scholar 

  10. Kim T: q -Volkenborn integration. Russ J Math Phys 2002, 9: 288–299.

    MathSciNet  Google Scholar 

  11. Kim T, Choi J, Kim YH, Ryoo CS: On the fermionic p -adic integral representation of Bernstein polynomials associated with Euler numbers and polynomials. J Inequal Appl 2010, 2010: 12. Art ID 864247

    Google Scholar 

  12. Simsek Y, Acikgoz M: A new generating function of q -Bernstein-type polynomials and their interpolation function. Abstract and Applied Analysis 2010, 2010: 12. Article ID 769095

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Nam-Soon Jung.

Additional information

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

All authors contributed equally to the manuscript and read and approved the final manuscript.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article

Cite this article

Jung, NS., Lee, HY., Kang, JY. et al. The twisted (h, q)-Genocchi numbers and polynomials with weight α and q-bernstein polynomials with weight α. J Inequal Appl 2012, 67 (2012). https://doi.org/10.1186/1029-242X-2012-67

Download citation

  • Received:

  • Accepted:

  • Published:

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

Keywords

  • Genocchi numbers and polynomials
  • twisted (h, q)-Genocchi numbers and polynomials with weight α
  • q-Bernstein polynomials