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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