Keywords: modified q-Genocchi numbers with weight alpha and beta, modified q-Euler numbers with weight alpha and beta, p-adic log gamma functions
Turkish Journal of Analysis and Number Theory, 2013 1 (1),
pp 9-12.
DOI: 10.12691/tjant-1-1-3
Received August 14, 2013; Revised September 17, 2013; Accepted September 25, 2013
Copyright © 2013 Science and Education Publishing. All Rights Reserved.
1. Introduction
Assume that p is a fixed odd prime number. Throughout this paper 

and
will denote by the ring of integers, the field of p-adic rational numbers and the completion of the algebraic closure of
respectively. Also we denote
and
Let
denote the p-adic valuation of
normalized so that
. The absolute value on
will be denoted as
, and
for
When one talks of q-extensions,
is considered in many ways, e.g. as an indeterminate, a complex number
or a p-adic number
If
we assume that
If
we assume
so that
for
We use the following notation.
 | (1.1) |
where
cf. [1-23][1].
For a fixed positive integer
with
we set
and
where
satisfies the condition
.
It is known that
is a distribution on
for
with 
Let
be the set of uniformly differentiable function on
We say that
is a uniformly differentiable function at a point
if the difference quotient
has a limit
as
and denote this by
. The p-adic q-integral of the function
is defined by
 | (1.2) |
The bosonic integral is considered by Kim as the bosonic limit
. Similarly, the p-adic fermionic integration on
defined by Kim as follows:
Let
then we have p-adic fermionic integral on
as follows:
Stirling asymptotic series are known as
 | (1.3) |
where
are familiar n-th Bernoulli numbers cf. [6, 8, 9, 23].
Recently, Araci et al. defined modified q-Genocchi numbers and polynomials with weight
and
in [4, 5] by the means of generating function:
 | (1.4) |
So from the above, we easily get Witt's formula of modified q-Genocchi numbers and polynomials with weight
and
as follows:
 | (1.5) |
where
are modified q-extension of Genocchi numbers with weight
and
cf. [4, 5].
In [21], Rim and Jeong are defined modified q-Euler numbers with weight
as follows:
 | (1.6) |
From expressions of (1.5) and (1.6), we get the following Proposition 1.
Proposition 1. The following
is true.
In previous paper [6], Araci, Acikgoz and Park introduced weighted q-analogue of p-adic lod gamma type functions and derived some interesting identities. They were motivated from paper of T. Kim by "On a q-analogue of the p-adic log gamma functions and related integrals, J. Number Theory, 76 (1999), no. 2, 320-329." By the same motivation, we introduce q-analogue of p-adic log gamma type function with weight
and
We derive in this paper some interesting identities including this type of functions.
2. On P-Adic log Γ Function with Weight α and β
In this part, from (1.2), we start at the following nice identity:
 | (2.1) |
where
and
(see [4]).
In particular for
into (2.1), we easily see that
 | (2.2) |
By simple an application, it is easy to indicate as follows:
 | (2.3) |
where
.
By expression of (2.3), we can derive
 | (2.4) |
where
is constant.
If we take
so we get
By expression of (2.3) and (2.4), we easily see that,
 | (2.5) |
It is considered by T. Kim for q-analogue of p-adic locally analytic function on
as follows:
 | (2.6) |
(for detail, see [5, 6]).
By the same motivation of (2.6), in previous paper [6], q-analogue of p-adic locally analytic function on
with weight
is considered
 | (2.7) |
In particular
in (2.7), we easily see that, 
With the same manner, we introduce q-analogue of p-adic locally analytic function on p with weight
and
as follows:
 | (2.8) |
From expressions of (2.2) and (2.8), we state the following Theorem:
Theorem 1. The following identity holds:
It is easy to show that,
 | (2.9) |
Substituting
into (2.5) and by using (2.9), we get interesting formula:
 | (2.10) |
If we substitute
into (2.10), we get Kim's q-analogue of p-adic log gamma function (for detail, see [8]). From expression of (1.2) and (2.10), we obtain the following worthwhile and interesting theorems.
Theorem 2. For
the following
is true.
Corollary 1. Taking
in Theorem 2, we get nice identity:
where
are called famous Genocchi numbers.
Theorem 3. The following nice identity
is true.
Corollary 2. Putting
into Theorem 3, we have the following identity:
where
are familiar Euler numbers.
References
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| In article | |
|
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