Some Identities of the Twisted -Genocchi Numbers and Polynomials with Weight and -Bernstein Polynomials with Weight
H. Y. Lee,1N. S. Jung,1and C. S. Ryoo1
Academic Editor: John Rassias
Received07 Jul 2011
Accepted22 Aug 2011
Published02 Nov 2011
Abstract
Recently mathematicians have studied some interesting relations between -Genocchi numbers, -Euler numbers, polynomials, Bernstein polynomials, and -Bernstein polynomials. In this paper, we give some interesting identities of the twisted -Genocchi numbers, polynomials, and -Bernstein polynomials with weighted .
1. Introduction
Throughout this paper, let be a fixed odd prime number. The symbols , , and denote the ring of -adic integers, the field of -adic rational numbers, and the completion of algebraic closure of . Let be the set of natural numbers and let . As a well-known definition, the -adic absolute value is given by , where with . When one talks of -extension, is variously considered as an indeterminate, a complex number , or a -adic number . In this paper we assume that with .
We assume that is the space of the uniformly differentiable function on . For , Kim defined the fermionic -adic -integral on as follows:
For , let be translation. As a well known equation, by (1.1), we have
compared [1–4]. Throughout this paper we use the notation: (cf. [1–16]). for any with in the present -adic case. To investigate relation of the twisted -Genocchi numbers and polynomials with weight and the Bernstein polynomials with weight , we will use useful property for as follows;
The twisted -Genocchi numbers and polynomials with weight are defined by the generating function as follows, respectively:
In the special case, , are called the th twisted -Genocchi numbers with weight (see [9]).
Let be the cyclic group of order and let
see [9, 12–15].
Kim defined the -Bernstein polynomials with weight of degree as follows:
compare [4, 7].
In this paper, we investigate some properties for the twisted -Genocchi numbers and polynomials with weight . By using these properties, we give some interesting identities on the twisted -Genocchi polynomials with weight and -Bernstein polynomials with weight .
2. Some Identities on the Twisted -Genocchi Polynomials with Weight and -Bernstein Polynomials with Weight
From (1.8), we can derive the following recurrence formula for the twisted -Genocchi numbers with weight :
with usual convention about replacing by .
By (1.5), we easily get
By (2.4), we obtain the theorem below.
By (1.6) and Theorem 2.2,
Hence, we obtain the corollary below.
Corollary 2.3. For , one has
By fermionic integral on ,Theorems 2.1 and 2.2, we note that
Therefore, we have the theorem below.
Theorem 2.4. For with , one has
By (1.4), Theorem 2.4, we take the fermionic -adic invariant integral on for one q-Bernstein polynomials as follows:
By symmetry of -Bernstein polynomials with weight of degree , we get the following formula;
Therefore, by (2.12) and (2.13), we have the theorem below.
Theorem 2.5. For with , one has
Also, we note that
Therefore, we have the theorem below.
Theorem 2.6. For with , one has
By (2.11) and Theorem 2.6, we have the theorem below.
Theorem 2.7. Let with . Then one has
Let with . Then we get
Therefore, we obtain the theorem below.
Theorem 2.8. For , one has
By simple calculation, we easily see that
Therefore, by (2.20) and Theorem 2.8, we obtain the theorem below.
Theorem 2.9. Let with . Then one has
For , , and let , then by the symmetry of -Bernstein polynomials with weight , we see that
Therefore, we have the theorem below.
Theorem 2.10. For with , one has
where .
In the same manner as in (2.15), we can get the following relation:
where with .
By Theorem 2.10 and (2.13), we have the following corollary.
Corollary 2.11. Let . For with , one has
where .
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