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Some identities on the weighted q-Euler numbers and q-Bernstein polynomials

Abstract

Recently, Ryoo introduced the weighted q-Euler numbers and polynomials which are a slightly different Kim's weighted q-Euler numbers and polynomials(see C. S. Ryoo, A note on the weighted q-Euler numbers and polynomials, 2011]). In this paper, we give some interesting new identities on the weighted q-Euler numbers related to the q-Bernstein polynomials

2000 Mathematics Subject Classification - 11B68, 11S40, 11S80

1. Introduction

Let p be a fixed odd prime number. Throughout this paper p , p , and p will denote the ring of p-adic integers, the field of p-adic rational numbers, the complex number fields and the completion of algebraic closure of p , respectively. Let be the set of natural numbers and + = {0}. Let ν p be the normalized exponential valuation of p with |p | p = p - ν p ( p ) = 1 p . When one talks of q-extension, q is variously considered as an indeterminate, a complex number q , or a p-adic number q p . If q , then one normally assumes |q| < 1, and if q p , then one normally assumes |q - 1| p < 1. In this paper, the q-number is defined by

[ x ] q = 1 - q x 1 - q , .

(see [119])

Note that limq→1[x] q = x (see [119]). Let f be a continuous function on p . For α and k, n +, the weighted p-adic q-Bernstein operator of order n for f is defined by Kim as follows:

B n , q ( α ) ( f | x ) = k = 0 n n k f k n [ x ] q α k [ 1 - x ] q - α n - k (1) = k = 0 n f k n B k , n ( α ) ( x , q ) , . (2) (3)
(1)

see [4, 9, 19].

Here B k , n ( α ) ( x , q ) = n k [ x ] q α k [ 1 - x ] q - α n - k are called the q-Bernstein polynomials of degree n with weighted α.

Let C( p ) be the space of continuous functions on p . For f C( p ), the fermionic q-integral on p is defined by

I q ( f ) = p f ( x ) d μ q ( x ) = l i m N 1 + q 1 + q p N x = 0 p N 1 f ( x ) ( q ) x ,
(2)

see [519].

For n , by (2), we get

q n p f ( x + n ) d μ q ( x ) = ( 1 ) n p f ( x ) d μ q ( x ) + [ 2 ] q l = 0 n 1 ( 1 ) n 1 l q l f ( l ) ,
(3)

see [6, 7].

Recently, by (2) and (3), Ryoo considered the weighted q-Euler polynomials which are a slightly different Kim's weighted q-Euler polynomials as follows:

p [ x + y ] q α n d μ - q ( y ) = E n , q ( α ) ( x ) , for n + and α ,
(4)

see [17].

In the special case, x = 0, E n , q ( α ) ( 0 ) = E n , q ( α ) are called the n-th q-Euler numbers with weight α (see [14]).

From (4), we note that

E n , q ( α ) ( x ) = [ 2 ] q ( 1 - q α ) n l = 0 n n l ( - 1 ) l q α l x 1 + q α l + 1 ,
(5)

see [17].

and

E n , q ( α ) ( x ) = l = 0 n n l [ x ] q α n - l q α l x E l , q ( α ) ,
(6)

see [17].

That is, (6) can be written as

E n , q ( α ) ( x ) = ( q α x E q ( α ) + [ x ] q α ) n , n + .
(7)

with usual convention about replacing ( E q ( α ) ) n by E n , q ( α ) .

In this paper we study the weighted q-Bernstein polynomials to express the fermionic q-integral on p and investigate some new identities on the weighted q-Euler numbers related to the weighted q-Bernstein polynomials.

2. q-Euler numbers with weight α

In this section we assume that α and q with |q| < 1.

Let F q (t, x) be the generating function of q-Euler polynomials with weight α as followings:

F q ( t , x ) = n = 0 E n , q ( α ) ( x ) t n n ! .
(8)

By (5) and (8), we get

F q ( t , x ) = n = 0 [ 2 ] q ( 1 - q α ) n l = 0 n n l ( - 1 ) l q α l x 1 + q α l + 1 t n n ! (1) = [ 2 ] q m = 0 ( - 1 ) m q m e [ x + m ] q α t . (2) (3)
(9)

In the special case, x = 0, let F q (t, 0) = F q (t). Then we obtain the following difference equation.

q F q ( t , 1 ) + F q ( t ) = [ 2 ] q .
(10)

Therefore, by (8) and (10), we obtain the following proposition.

Proposition 1. For n +, we have

E 0 , q ( α ) = 1 , and q E n , q ( α ) ( 1 ) + E n , q ( α ) = 0 if n > 0 .

By (6), we easily get the following corollary.

Corollary 2. For n +, we have

E 0 , q ( α ) = 1 , and q ( q α E q ( α ) + 1 ) n + E n , q ( α ) = 0 if n > 0 ,

with usual convention about replacing ( E q ( α ) ) n by E n , q ( α ) .

From (9), we note that

F q - 1 ( t , 1 - x ) = F q ( - q α t , x ) .
(11)

Therefore, by (11), we obtain the following lemma.

Lemma 3. Let n +. Then we have

E n , q - 1 ( α ) ( 1 - x ) = ( - 1 ) n q α n E n , q ( α ) ( x ) .

By Corollary 2, we get

q 2 E n , q ( α ) ( 2 ) - q 2 - q = q 2 l = 0 n n l q α l ( q α E q ( α ) + 1 ) l - q 2 - q (1) = - q l = 1 n n l q α l E l , q ( α ) - q (2) = - q l = 0 n n l q α l E l , q ( α ) (3) = - q E n , q ( α ) ( 1 ) = E n , q ( α ) if n > 0 . (4) (5)
(12)

Therefore, by (12), we obtain the following theorem.

Theorem 4. For n , we have

E n , q ( α ) ( 2 ) = 1 q 2 E n , q ( α ) + 1 q + 1 .

Theorem 4 is important to study the relations between q-Bernstein polynomials and the weighted q-Euler number in the next section.

3. Weighted q-Euler numbers concerning q-Bernstein polynomials

In this section we assume that α p and q p with |1 - q| p < 1.

From (2), (3) and (4), we note that

q p [ 1 x ] q α n d μ q ( x ) = ( 1 ) n q α n + 1 p [ x 1 ] q α n d μ q ( x ) = q l = 0 n ( n l ) ( 1 ) l p [ x ] q α l d μ q ( x ) .
(13)

Therefore, by (13) and Lemma 3, we obtain the following theorem.

Theorem 5. For n +, we get

q p [ 1 x ] q α n d μ q ( x ) = ( 1 ) n q α n + 1 E n , q ( α ) ( 1 ) = q E n , q 1 ( α ) ( 2 ) = q l = 0 n ( n l ) ( 1 ) l E l , q ( α ) .

Let n . Then, by Theorem 4, we obtain the following corollary.

Corollary 6. For n , we have

p [ 1 - x ] q - α n d μ - q ( x ) = E n , q - 1 ( α ) ( 2 ) (1) = q 2 E n , q - 1 ( α ) + [ 2 ] q . (2) (3)

For x p , the p-adic q-Bernstein polynomials with weight α of degree n are given by

B k , n ( α ) ( x , q ) = n k [ x ] q α k [ 1 - x ] q - α n - k , where n , k + ,
(14)

see [9].

From (14), we can easily derive the following symmetric property for q-Bernstein polynomials:

B k , n ( α ) ( x , q ) = B n - k , n ( α ) ( 1 - x , q - 1 ) ,
(15)

see [11]

By (15), we get

p B k , n ( α ) ( x , q ) d μ - q ( x ) = p B n - k , n ( α ) ( 1 - x , q - 1 ) d μ - q ( x ) (1) = n k l = 0 k k l ( - 1 ) k + l p [ 1 - x ] q - α n - l d μ - q ( x ) . (2) (3)
(16)

Let n, k + with n > k. Then, by (16) and Corollary 6, we have

p B k , n ( α ) ( x , q ) d μ - q ( x ) = n k l = 0 k k l ( - 1 ) k + l q 2 E n - l , q - 1 ( α ) + [ 2 ] q = q 2 E n , q - 1 ( α ) + [ 2 ] q , if  k = 0 , q 2 n k l = 0 k k l ( - 1 ) k + l E n - l , q - 1 ( α ) , if  k > 0 .
(17)

Taking the fermionic q-integral on p for one weighted q-Bernstein polynomials in (14), we have

p B k , n ( α ) ( x , q ) d μ - q ( x ) = n k p [ x ] q α k [ 1 - x ] q - α n - k d μ - q ( x ) (1) = n k l = 0 n - k n - k l ( - 1 ) l p [ x ] q α k + l d μ - q ( x ) (2) = n k l = 0 n - k n - k l ( - 1 ) l E l + k , q ( α ) . (3) (4)
(18)

Therefore, by comparing the coefficients on the both sides of (17) and (18), we obtain the following theorem.

Theorem 7. For n, k + with n > k, we have

l = 0 n - k ( - 1 ) l n - k l E l + k , q ( α ) = q 2 E n , q - 1 ( α ) + [ 2 ] q , if k = 0 , q 2 l = 0 k k l ( - 1 ) k + l E n - l , q - 1 ( α ) , if  k > 0 .

Let n1, n2, k + with n1 + n2 > 2k. Then we see that

p 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 ) l + 2 k p [ 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 ) l + 2 k q 2 E n 1 + n 2 - l , q - 1 ( α ) + [ 2 ] q .
(19)

By the binomial theorem and definition of q-Bernstein polynomials, we get

p 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 n 1 + n 2 - 2 k ( - 1 ) l n 1 + n 2 - 2 k l p [ x ] q α 2 k + 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 E 2 k + l , q ( α ) .
(20)

By comparing the coefficients on the both sides of (19) and (20), we obtain the following theorem.

Theorem 8. Let n1, n2, k + with n1 + n2 > 2k. Then we have

l = 0 n 1 + n 2 - 2 k ( - 1 ) l n 1 + n 2 - 2 k l E 2 k + l , q ( α ) = q 2 E n 1 + n 2 , q - 1 ( α ) + [ 2 ] q , if  k = 0 , q 2 l = 0 2 k 2 k l ( - 1 ) 2 k + l E n 1 + n 2 - l , q - 1 ( α ) , if  k > 0 .

Let s with s ≥ 2. For n1, n2, ..., n s , k + with n1 + + n s > sk, we have

p B k , n 1 ( α ) ( x , q ) B k , n s ( α ) ( x , q ) s - t i m e s d μ - q ( x ) = n 1 k n s k p [ x ] q α s k [ 1 - x ] q - α n 1 + + n s - s k d μ - q ( x ) = n 1 k n s k l = 0 s k s k l ( - 1 ) l + s k p [ 1 - x ] q - α n 1 + + n s - l d μ - q ( x ) = n 1 k n s k l = 0 s k s k l ( - 1 ) l + s k q 2 E n 1 + + n s - l , q - 1 ( α ) + [ 2 ] q .
(21)

From the binomial theorem and the definition of q-Bernstein polynomials, we note that

p B k , n 1 ( α ) ( x , q ) B k , n s ( α ) ( x , q ) s - times d μ - q ( x ) = n 1 k n s k l = 0 n 1 + + n s - s k ( - 1 ) l n 1 + + n s - s k l p [ x ] q α s k + l d μ - q ( x ) = n 1 k n s k l = 0 n 1 + + n s - s k ( - 1 ) l n 1 + + n s - s k l E s k + l , q ( α ) .
(22)

Therefore, by (21) and (22), we obtain the following theorem.

Theorem 9. Let s with s ≥ 2. For n1, n2, ..., n s , k + with n1 + + n s > sk, we have

l = 0 n 1 + + n s - s k ( - 1 ) l n 1 + + n s - s k l E s k + l , q ( α ) = q 2 E n 1 + + n s , q - 1 ( α ) + [ 2 ] q , if k = 0 , q 2 l = 0 s k s k l ( - 1 ) l + s k E n 1 + + n s - l , q - 1 ( α ) , if  k > 0 .

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The authors would like to express their sincere gratitude to referee for his/her valuable comments.

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Kim, T., Kim, YH. & Ryoo, C.S. Some identities on the weighted q-Euler numbers and q-Bernstein polynomials. J Inequal Appl 2011, 64 (2011). https://doi.org/10.1186/1029-242X-2011-64

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Keywords

  • Euler numbers and polynomials
  • q-Euler numbers and polynomials
  • weighted
  • q-Euler numbers and polynomials
  • Bernstein polynomials
  • q-Bernstein polynomials