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The twisted (h, q)-Genocchi numbers and polynomials with weight α and q-bernstein polynomials with weight α
Journal of Inequalities and Applications volume 2012, Article number: 67 (2012)
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, , and denote the ring of p-adic integers, the field of p-adic rational numbers and the completion of algebraic closure of , respectively. Let be the set of natural numbers and . As well known definition, the p-adic absolute value is given by |x| p = p-r , where with (t, p) = (s, p) = (t, s) = 1. When one talks of q-extension, q is variously considered as an indeterminate, a complex number , or a p-adic number . In this article, we assume that with |1 - q| p < 1.
For is uniformly differentiable function}, Kim defined the fermionic p-adic q-integral on as follows:
For , let f n (x) = f(x + n) be translation. As well known equation, by (1.1), we have
Throughout this article we use the notation:
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 as following;
The twisted (h, q)-Genocchi numbers and polynomials with weight α are defined by the generating function, respectively:
In the special case, are called the n th twisted (h, q)-Genocchi numbers with weight α (see [1]).
Let be the cyclic group of order pn and let
Kim defined the q-Bernstein polynomials with weight α of degree n as follows:
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:
with usual convention about replacing by .
By (1.4), we can obtain
By (2.4), we can get
So, we get the following theorem.
Theorem 1. Let . For w ∈ T p , we have
By (2.1), (2.2), and (2.3), we note that
Therefore, by (2.5), we obtain the theorem below.
Theorem 2. For with n > 1, we have
From Theorem 2,
Therefore, we obtain the Corollary 3 by (1.5) and Theorem 2.
Corollary 3. For , we have
By Theorems 1, 2 and fermionic integral on , we note that
Hence, we get the following theorem.
Theorem 4. for with n > 1, we have
Corollary 5.
From (1.3) and Theorem 4, we take the fermionic p-adic invariant integral on for q-Bernstein polynomials as follows:
And we get the following formula;
Hence, we can get the following theorem by (2.8) and (2.9).
Theorem 5. for with n > 1, we have
Also, we can see that
Therefore, we have the theorem below.
Theorem 6. For with n > k + 1, we have
By (2.7) and Theorem 6, we can get the theorem below.
Theorem 7. Let with n > k + 1. Then we have
Let with n1 + n2> 2k + 1. Then we get
Therefore, we obtain the theorem below.
Theorem 8. For , we have
And we can easily have that
Therefore, by (2.14) and Theorem 8, we obtain the theorem below.
Theorem 9. Let with n1 + n2> 2k + 1. Then we have
For , then by the symmetry of q-Bernstein polynomials with weight α, we see that
Therefore, we have the theorem below.
Theorem 10. For with n1 + n2 + . . . + n s > sk + 1, we have
In the same manner as in (2.11), we can get the following relation:
where with n1 + n2 + . . . + n s > sk + 1.
By Theorem 11 and (2.9), we have the following corollary.
Corollary 11. Let . For with n1 + . . . + n s > sk + 1, we have
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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
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DOI: https://doi.org/10.1186/1029-242X-2012-67