Some identities on the weighted q-Euler numbers and q-Bernstein polynomials
© Kim et al; licensee Springer. 2011
Received: 18 February 2011
Accepted: 20 September 2011
Published: 20 September 2011
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
KeywordsEuler numbers and polynomials q-Euler numbers and polynomials weighted q-Euler numbers and polynomials Bernstein polynomials q-Bernstein polynomials
Here are called the q-Bernstein polynomials of degree n with weighted α.
In the special case, x = 0, are called the n-th q-Euler numbers with weight α (see ).
with usual convention about replacing by .
In this paper we study the weighted q-Bernstein polynomials to express the fermionic q-integral on ℤ 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 ∈ ℕ and q ∈ ℂ with |q| < 1.
Therefore, by (8) and (10), we obtain the following proposition.
By (6), we easily get the following corollary.
with usual convention about replacing by .
Therefore, by (11), we obtain the following lemma.
Therefore, by (12), we obtain the following theorem.
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 ∈ ℤ p and q ∈ ℂ p with |1 - q| p < 1.
Therefore, by (13) and Lemma 3, we obtain the following theorem.
Let n ∈ ℕ. Then, by Theorem 4, we obtain the following corollary.
Therefore, by comparing the coefficients on the both sides of (17) and (18), we obtain the following theorem.
By comparing the coefficients on the both sides of (19) and (20), we obtain the following theorem.
Therefore, by (21) and (22), we obtain the following theorem.
The authors would like to express their sincere gratitude to referee for his/her valuable comments.
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