Abstract

The p-adic gamma function is considered to obtain its derivative and to evaluate its the fermionic p-adic integral. Furthermore the relationship between the p-adic gamma function and Changhee polynomials and also between the Changhee polynomials and p-adic Euler constants is obtained. In addition, the p-adic Euler constants are expressed in term of Mahler coefficients of the p-adic gamma function.

1. Introduction

The p-adic numbers introduced by the German mathematician Kurt Hensel (1861-1941), are widely used in mathematics: in number theory, algebraic geometry, representation theory, algebraic and arithmetical dynamics, and cryptography. The p-adic numbers have been used to applying fields with successfully applying in super.eld theory of p-adic numbers by Vladimirov and Volovich. In addition, the p-adic model of the universe, the p-adic quantum theory, the p-adic string theory such as areas occurred in physics (for detail see ^{1, 2}).

Special numbers and polynomials plays an important role in almost all areas of mathematics, in mathematical physics, computer science, engineering problems and other areas of science. The q-calculus (or quantum calculus) appeared in the 18th century and it continues to develop rapidly and has been studied by many scientists (cf. ^{3, 4, 5, 6, 7, 8}). Many generalizations of special functions with a q-parameter recently were obtained using p-adic q-integral on (cf. ^{9, 10, 11, 12, 13, 14, 15}).

Let p be chosen as a fixed odd prime number. Throughout this paper, and denote the ring of p-adic integers, the field of p-adic numbers and the completion of the algebraic closure of respectively.

In the year 1975, Morita ¹⁶ defined the gamma function over p-adic fields, denoted by by the following formula:

where approaches through positive integers. The p-adic gamma function is analytic on and satisfies the functional relation:

(1.1)

The p-adic Euler constant is defined by the formula:

(1.2)

The p-adic gamma function has a great interest and has a great interest and has been studied by Diamond (1977) ¹⁷, Barsky (1977) ¹⁸, Dwork (1983) ¹⁹ and cited references therein.

For the symbol is defined by and The functions form an orthonormal base of the space with respect the norm This orthonormal base have the following property:

(1.3)

[ ²⁰, p162]

In 1958, Mahler introduced an expansion for continuous functions of a p-adic variable using special polynomials as binomial coefficient polynomial ²¹. Means that for any there exist unique elements of such that

The base is called Mahler base of the space and the elements in are called Mahler coefficients of

The Mahler expansion of the p-adic gamma function and its Mahler coefficients are determined by the following proposition:

Proposition 1. ( ^{20, 22}) Let

(1.4)

and

(1.5)

where is the region of convergence of the power series

For the fermionic p-adic integral on is defined by Kim to be

(1.6)

(see ^{10, 11}). For any by (1.6), the following relation holds:

(1.7)

where .

The Changhee numbers and polynomials which are derived umbral calculus are defined by Kim et al. as the generating function to be

In the case when , stands for Changhee numbers, see ²³ for details. In ²⁴, Kim et al. obtained following theorems which will be useful in deriving the main results of this paper:

Theorem 1. For one has

Theorem 2. For one has

Theorem 3. For one has

2. Main Results

In the present work, the fermionic p-adic integral of p-adic gamma function and of derivative of p-adic gamma function are evaluated. The p-adic Euler constants are expressed in term of Mahler coefficients of the p-adic gamma function. The relationship between the Changhee polynomials and the p-adic Euler constants are obtained.

Theorem 4. Then the equality holds:

for where is defined by Proposition 1.

Proof. Let From Proposition 1, we have

(2.1)

From Theorem 1, we get

Using Theorem 3 we can rewrite (2.1) and we have the following corollory:

Corollary 1. For and

where is defined by Proposition 1.

Lemma 1. For and the following equality holds:

Proof. When in (1.7), we have

Form Theorem 3 we prove the theorem.

Theorem 5. The following relation is holds

where is defined by Proposition 1.

Proof. Let By Proposition 1, we have

By using Theorem 2 we can write

Theorem 6. If then

Proof. By using Proposition 1, we have

From Lemma 1, fermionic p-adic integral of is evaluated.

From Theorem 5 and Theorem 6, the following corollary is obtained.

Corollary 2. For

From Proposition 1 and (1.3), derivative of p-adic Gamma functions, is obtained as

(2.2)

where is defined by Proposition 1.

Theorem 7. The p-adic Euler constants have the expansion

Proof. When in (1.7), we get

From (2.2) and (1.2), we can write

Using Theorem 3 and Lemma 1 we can rewrite (2.3) as

By some computing steps, the proof is completed.

Theorem 8. Relationship between the Changhee polynomials and the p-adic Euler constants is as

and

Proof. we can rewrite (2.3) by

From Theorem 1 and Theorem 2, it is obtained

In addition, by using Corollary 2, we get

Theorem 9. If then

Proof. Let We have

By using Theorem 3 we can write

In the case s = 1 in Theorem 9 we obtain the following conclusion

Corollary 3. For

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