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Research Article

Open Access Peer-reviewed

Özge Çolakoğlu Havare^{ }, Hamza Menken

Received December 23, 2017; Revised September 18, 2018; Accepted October 16, 2018

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.

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) |

a*nd*

(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*

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 de**fi**ned 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 de**fi**ned 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 de**fi**ned 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*

[1] | I. V. Volovich, Number theory as the ultimate physical theory, Preprint No. TH 4781/87, CERN, Geneva, (1987). | ||

In article | |||

[2] | V. S Vladimirov and I. V. Volovich, Superanalysis. I. Differential calculus, Theor. Math. Phys. 59, (1984) 317.335. | ||

In article | |||

[3] | S. Araci, E. Ağyüz, M. Acikgoz, On a q-analogue of some numbers and polynomials, J. Inequal. Appl. (2015) 2015: 19. | ||

In article | View Article | ||

[4] | S. Araci and M. Acikgöz, A note on the values of weighted q-Bernstein polynomials and weighted q-Genocchi numbers, Adv. Diffierence Equa., (2015) 2015: 30. | ||

In article | View Article | ||

[5] | I. N. Cangul, A. S. Cevik, Y. Simsek, Generalization of q-Apostol-type Eulerian numbers and polynomials, and their interpolation functions, Adv. Stud. Contemp. Math. 25 (2) (2015), 211-220. | ||

In article | |||

[6] | Y. Simsek, Special Numbers on Analytic Functions, Applied Mathematics, (2014), 5, 1091-1098. | ||

In article | View Article | ||

[7] | H. Srivastava, B. Kurt, Y. Simsek, Some Families of Genocchi Type Polynomials And Their Interpolation Functions, Integral Transforms and Special Functions, no.12, ( 2012), 919-938. | ||

In article | View Article | ||

[8] | H. M. Srivastava, T. Kim, Y. Simsek, q-Bernoulli numbers and polynomials associated with multiple q-zeta functions and basic L-series. Russ. J. Math. Phys., (2005), 12, 241-268. | ||

In article | |||

[9] | S. Araci, D. Erdal, J. J. Seo, A study on the fermionic p-adic q-integralrepresentation on associated with weighted q-Bernstein and q-Genocchi polynomials, Abstr. Appl. Anal. 2011 (2011) Article ID 649248, 10 pp. | ||

In article | |||

[10] | T. Kim, On the analogs of Euler numbers and polynomials associated with p-adic q-integral on at q = -1, J. Math. Anal. Appl., 331 (2007) pp 779-792. | ||

In article | View Article | ||

[11] | T. Kim. q-Volkenborn integration, Russian Journal of Mathematical Physics, vol. 9, no.3, (2002) pp. 288.299. | ||

In article | |||

[12] | T. Kim, Symmetry of power sum polynomials and multivariate fermionic p-adic invariant integral on , Russ. J. Math. Phys. 16 (1), (2009), 93-96. | ||

In article | View Article | ||

[13] | H. Ozden, I.N. Cangul and Y. Simsek, Generalized q-Stirling Numbers and Their Interpolation Functions. Axioms (2013), 2, 10-19. | ||

In article | View Article | ||

[14] | Y. Simsek, A. Yardimci, Applications on the Apostol-Daehe numbers and polynomials as-sociated with special numbers, polynomials, and p-adic integrals, Advances in Difference Equations (2016), 2016: 308. | ||

In article | View Article | ||

[15] | Y. Simsek, Analysis of the p-adic q-Volkenborn integrals: An approach to generalized Apostol-type special numbers and polynomials and their applications, Cogent Mathematics, (2016), 3: 1269393. | ||

In article | View Article | ||

[16] | Y. Morita, A p-adic analogue of the Γ-function, J. Fac. Science Univ., 22 (1975), 225-266. | ||

In article | |||

[17] | J. Diamond, The p-adic log gamma function and p-adic Euler constant, Trans. Amer. Math. Soc. 233 (1977), 321-337. | ||

In article | |||

[18] | D. Barsky, On Morita’s p-adic gamma Function, Groupe d’Etude d’Analyse Ultramétrique, 5 (1977/78), 3, 1-6. | ||

In article | |||

[19] | B. Dwork, A note on p-adic gamma function, Groupe de travail d’analyse ultramétrique, 9 (1981-1982), 3, J1-J10. | ||

In article | |||

[20] | W. H. Schikhof, Ultrametric Calculus: An Introduction to p-adic Analysis, Cambridge University Pres, 1984. | ||

In article | |||

[21] | K. Mahler, An Interpolation Series for Continuous Functions of a p-adic Variable, J. Reine Angew. Math., 199, (1958) 23-34. | ||

In article | |||

[22] | A. M. Robert, A Course in p-adic Analysis, Graduate Texts in Mathematics 198, Springer, 2000. | ||

In article | View Article | ||

[23] | T. Kim, D. S. Kim, Mansour, T., Rim, S. H., Schork, M., Umbral calculus and Sheffer sequences of polynomials, J. Math. Phys. 54, 083504 (2013). | ||

In article | View Article | ||

[24] | D. S. Kim, T. Kim, J. Seo, A note on Changhee Polynomials and Numbers, Adv. Studies Theor. Phys., vol. 7, no.20, (2013) 993-1003. | ||

In article | View Article | ||

Published with license by Science and Education Publishing, Copyright © 2018 Özge Çolakoğlu Havare and Hamza Menken

This work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

Özge Çolakoğlu Havare, Hamza Menken. On the *p*-adic Gamma Function and Changhee Polynomials. *Turkish Journal of Analysis and Number Theory*. Vol. 6, No. 4, 2018, pp 120-123. http://pubs.sciepub.com/tjant/6/4/3

Havare, Özge Çolakoğlu, and Hamza Menken. "On the *p*-adic Gamma Function and Changhee Polynomials." *Turkish Journal of Analysis and Number Theory* 6.4 (2018): 120-123.

Havare, Ö. Ç. , & Menken, H. (2018). On the *p*-adic Gamma Function and Changhee Polynomials. *Turkish Journal of Analysis and Number Theory*, *6*(4), 120-123.

Havare, Özge Çolakoğlu, and Hamza Menken. "On the *p*-adic Gamma Function and Changhee Polynomials." *Turkish Journal of Analysis and Number Theory* 6, no. 4 (2018): 120-123.

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[1] | I. V. Volovich, Number theory as the ultimate physical theory, Preprint No. TH 4781/87, CERN, Geneva, (1987). | ||

In article | |||

[2] | V. S Vladimirov and I. V. Volovich, Superanalysis. I. Differential calculus, Theor. Math. Phys. 59, (1984) 317.335. | ||

In article | |||

[3] | S. Araci, E. Ağyüz, M. Acikgoz, On a q-analogue of some numbers and polynomials, J. Inequal. Appl. (2015) 2015: 19. | ||

In article | View Article | ||

[4] | S. Araci and M. Acikgöz, A note on the values of weighted q-Bernstein polynomials and weighted q-Genocchi numbers, Adv. Diffierence Equa., (2015) 2015: 30. | ||

In article | View Article | ||

[5] | I. N. Cangul, A. S. Cevik, Y. Simsek, Generalization of q-Apostol-type Eulerian numbers and polynomials, and their interpolation functions, Adv. Stud. Contemp. Math. 25 (2) (2015), 211-220. | ||

In article | |||

[6] | Y. Simsek, Special Numbers on Analytic Functions, Applied Mathematics, (2014), 5, 1091-1098. | ||

In article | View Article | ||

[7] | H. Srivastava, B. Kurt, Y. Simsek, Some Families of Genocchi Type Polynomials And Their Interpolation Functions, Integral Transforms and Special Functions, no.12, ( 2012), 919-938. | ||

In article | View Article | ||

[8] | H. M. Srivastava, T. Kim, Y. Simsek, q-Bernoulli numbers and polynomials associated with multiple q-zeta functions and basic L-series. Russ. J. Math. Phys., (2005), 12, 241-268. | ||

In article | |||

[9] | S. Araci, D. Erdal, J. J. Seo, A study on the fermionic p-adic q-integralrepresentation on associated with weighted q-Bernstein and q-Genocchi polynomials, Abstr. Appl. Anal. 2011 (2011) Article ID 649248, 10 pp. | ||

In article | |||

[10] | T. Kim, On the analogs of Euler numbers and polynomials associated with p-adic q-integral on at q = -1, J. Math. Anal. Appl., 331 (2007) pp 779-792. | ||

In article | View Article | ||

[11] | T. Kim. q-Volkenborn integration, Russian Journal of Mathematical Physics, vol. 9, no.3, (2002) pp. 288.299. | ||

In article | |||

[12] | T. Kim, Symmetry of power sum polynomials and multivariate fermionic p-adic invariant integral on , Russ. J. Math. Phys. 16 (1), (2009), 93-96. | ||

In article | View Article | ||

[13] | H. Ozden, I.N. Cangul and Y. Simsek, Generalized q-Stirling Numbers and Their Interpolation Functions. Axioms (2013), 2, 10-19. | ||

In article | View Article | ||

[14] | Y. Simsek, A. Yardimci, Applications on the Apostol-Daehe numbers and polynomials as-sociated with special numbers, polynomials, and p-adic integrals, Advances in Difference Equations (2016), 2016: 308. | ||

In article | View Article | ||

[15] | Y. Simsek, Analysis of the p-adic q-Volkenborn integrals: An approach to generalized Apostol-type special numbers and polynomials and their applications, Cogent Mathematics, (2016), 3: 1269393. | ||

In article | View Article | ||

[16] | Y. Morita, A p-adic analogue of the Γ-function, J. Fac. Science Univ., 22 (1975), 225-266. | ||

In article | |||

[17] | J. Diamond, The p-adic log gamma function and p-adic Euler constant, Trans. Amer. Math. Soc. 233 (1977), 321-337. | ||

In article | |||

[18] | D. Barsky, On Morita’s p-adic gamma Function, Groupe d’Etude d’Analyse Ultramétrique, 5 (1977/78), 3, 1-6. | ||

In article | |||

[19] | B. Dwork, A note on p-adic gamma function, Groupe de travail d’analyse ultramétrique, 9 (1981-1982), 3, J1-J10. | ||

In article | |||

[20] | W. H. Schikhof, Ultrametric Calculus: An Introduction to p-adic Analysis, Cambridge University Pres, 1984. | ||

In article | |||

[21] | K. Mahler, An Interpolation Series for Continuous Functions of a p-adic Variable, J. Reine Angew. Math., 199, (1958) 23-34. | ||

In article | |||

[22] | A. M. Robert, A Course in p-adic Analysis, Graduate Texts in Mathematics 198, Springer, 2000. | ||

In article | View Article | ||

[23] | T. Kim, D. S. Kim, Mansour, T., Rim, S. H., Schork, M., Umbral calculus and Sheffer sequences of polynomials, J. Math. Phys. 54, 083504 (2013). | ||

In article | View Article | ||

[24] | D. S. Kim, T. Kim, J. Seo, A note on Changhee Polynomials and Numbers, Adv. Studies Theor. Phys., vol. 7, no.20, (2013) 993-1003. | ||

In article | View Article | ||