Abstract
By using fermionic p-adic q-integral on , we give some interesting relationship between the twisted (h, q)-Euler numbers with weight α and the q-Bernstein polynomials.
1. Introduction
Let be a fixed odd prime number. Throughout this paper, we always make use of the following notations: denotes the ring of rational integers, denotes the ring of -adic rational integers, denotes the field of -adic rational numbers, and denotes the completion of algebraic closure of , respectively. Let be the set of natural numbers and . Let be the cyclic group of order , and let (see [1–22]), be the locally constant space. For , we denote by the locally constant function . The -adic absolute value is defined by , where . In this paper, we assume that with as an indeterminate. The -number is defined by (see [1–22]). Note that . For the fermionic -adic -integral on is defined by Kim as follows: (see [1–7]). From (1.4), we note that where for .
For and , Kim defined -Bernstein polynomials, which are different -Bernstein polynomials of Phillips, as follows: (see [5]). In [9], the -adic extension of (1.6) is given by For , , , and with , twisted -Euler numbers with weight are defined by In the special case, , are called the -th twisted -Euler numbers with weight .
In this paper, we investigate some relations between the -Bernstein polynomials and the twisted -Euler numbers with weight . From these relations, we derive some interesting identities on the twisted -Euler numbers and polynomials with weight .
2. Twisted -Euler Numbers and Polynomials with Weight
By using -adic -integral and (1.8), we obtain We set By (2.1) and (2.2), we have Since , we obtain
Therefore, we obtain the following theorem.
Theorem 2.1. For and , we have
Furthermore,
with usual convention about replacing with .
Let . Then we see that
In the special case, , let .
By (2.1), we get
From (2.3) and (2.7), we note that
By (2.9), we get the following recurrence formula:
By (2.10) and Theorem 2.1, we obtain the following theorem.
Theorem 2.2. For and , we have
with usual convention about replacing with .
By (2.4), Theorem 2.1, and Theorem 2.2, we have
Therefore, we obtain the following theorem.
Theorem 2.3. For , we have
By (2.8), we see that
Therefore, we obtain the following theorem.
Theorem 2.4. For , we have
Let . By Theorems 2.3 and 2.4, we get
From (2.16), we have
Therefore, we obtain the following corollary.
Corollary 2.5. For , we have
Kim defined the -Bernstein polynomials with weight of degree as below.
For , the -adic -Bernstein polynomials with weight of degree are given by
compare [5, 10, 22] By (2.19), we get the symmetry of -Bernstein polynomials as follows:
see [8]. Thus, by Corollary 2.5, (2.19), and (2.20), we see that
For with , we have
Let us take the fermionic -integral on for the -Bernstein polynomials with weight of degree as follows:
Therefore, by (2.22) and (2.23), we obtain the following theorem.
Theorem 2.6. Let with . Then we have
Moreover,
Let with . Then we get
Therefore, by (2.26), we obtain the following theorem.
Theorem 2.7. For with , we have
From the binomial theorem, we can derive the following equation:
Thus, by (2.28) and Theorem 2.7, we obtain the following corollary.
Corollary 2.8. Let with . Then we have
For and with , let with . Then we take the fermionic -adic -integral on for the -Bernstein polynomials with weight of degree as follows:
Therefore, by (2.30), we obtain the following theorem.
Theorem 2.9. For with , let with . Then we get
By the definition of -Bernstein polynomials with weight and the binomial theorem, we easily get
Therefore, we have the following corollary.
Corollary 2.10. For with , let with . Then we have