International journal of

ADVANCED AND APPLIED SCIENCES

EISSN: 2313-3724, Print ISSN:2313-626X

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 Volume 5, Issue 2 (February 2018), Pages: 56-59

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 Original Research Paper

 Title: The volkenborn integral of the p-adic gamma function

 Author(s): Özge Çolakoğlu Havare *, Hamza Menken

 Affiliation(s):

 Department of Mathematics, Science and Arts Faculty, Mersin University, Mersin, Turkey

 https://doi.org/10.21833/ijaas.2018.02.009

 Full Text - PDF          XML

 Abstract:

In the present work the p-adic gamma function has been considered. The Volkenborn integral of the p-adic gamma function by using its Mahler expansion has been derived. Moreover, a new representation for the p-adic Euler constant has been given. 

 © 2017 The Authors. Published by IASE.

 This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

 Keywords: 𝑝-Adic gamma function, Volkenborn integral, Mahler coefficients

 Article History: Received 22 May 2017, Received in revised form 2 November 2017, Accepted 2 December 2017

 Digital Object Identifier: 

 https://doi.org/10.21833/ijaas.2018.02.009

 Citation:

 Havare ÖÇ and Menken H (2018). The volkenborn integral of the p-adic gamma function. International Journal of Advanced and Applied Sciences, 5(2): 56-59

 Permanent Link:

 http://www.science-gate.com/IJAAS/2018/V5I2/Havare.html

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 References (23)

  1. Araci S and Açikgöz M (2015). A note on the values of weighted q-Bernstein polynomials and weighted q-Genocchi numbers. Advances in Difference Equations, 2015(1): 30-39. https://doi.org/10.1186/s13662-015-0369-y 
  2. Baesky D (1981). On Morita's $ p $-adic $\Gamma $-function. Groupe de Travail D'analyse Ultramétrique, 5: 1977-1978. 
  3. Boyarsky M (1980). p-adic gamma functions and Dwork cohomology. Transactions of the American Mathematical Society, 257(2): 359-369. 
  4. Cohen H and Friedman E (2008). Raabe's formula for $ p $-adic gamma and zeta functions (Formules de Raabe pour les fonctions gamma et zêta $ p $-adiques). Annales de L'institut Fourier, 58(1): 363-376. https://doi.org/10.5802/aif.2353 
  5. Conrad K (1997). p-adic Gamma Functions. Ph.D. Dissertations, Harvard Mathematics Department, Harvard University, Cambridge, USA. 
  6. Diamond J (1977). The -adic log gamma function and -adic Euler constants. Transactions of the American Mathematical Society, 233: 321-337. 
  7. Gross BH and Koblitz N (1979). Gauss sums and the p-adic Γ-function. Annals of Mathematics, 109(3): 569-581. https://doi.org/10.2307/1971226 
  8. Hensel K (1897). Über eine neue Begründung der Theorie der algebraischen Zahlen. Jahresbericht der Deutschen Mathematiker-Vereinigung, 6: 83-88. 
  9. Kim DS, Kim T, and Seo JJ (2013a). A note on Changhee polynomials and numbers. Advanced Studies in Theoretical Physics, 7(20): 993-1003. https://doi.org/10.12988/astp.2013.39117 
  10. Kim T, Kim DS, Mansour T, Rim SH, and Schork M (2013b). Umbral calculus and Sheffer sequences of polynomials. Journal of Mathematical Physics, 54(8): 083504. https://doi.org/10.1063/1.4817853 
  11. Mahler K (1958). An interpolation series for continuous functions of a p-adic variable. Journal Reine Angew: Math, 199: 23-34. 
  12. Morita Y (1975). A p-adic analogue of the $\Gamma $-function. Journal Ournal of the Faculty of Science, the University of Tokyo (Sect. 1 A, Mathematics), 22(2): 255-266. 
  13. Robert AM (2000). A course in p-adic analysis. volume 198 of Graduate Texts in Mathematics. Springer-Verlag, New York, USA. https://doi.org/10.1007/978-1-4757-3254-2 
  14. Schikhof WH (1984). Ultrametric calculus: An introduction to p-adic analysis. Cambridge University Press, Cambridge, UK. 
  15. Shapiro I (2012). Frobenius map and the p-adic gamma function. Journal of Number Theory, 132(8): 1770-1779. https://doi.org/10.1016/j.jnt.2012.03.005 
  16. Simsek Y (2014). Special numbers on analytic functions. Applied Mathematics, 5(07): 1091-1098. https://doi.org/10.4236/am.2014.57102 
  17. Simsek Y and Yardimci A (2016). Applications on the Apostol-Daehee numbers and polynomials associated with special numbers, polynomials, and p-adic integrals. Advances in Difference Equations, 2016(1): 308-322. https://doi.org/10.1186/s13662-016-1041-x 
  18. Srivastava HM, Kurt B, and Simsek Y (2012). Some families of Genocchi type polynomials and their interpolation functions. Integral Transforms and Special Functions, 23(12): 919-938. https://doi.org/10.1080/10652469.2011.643627 
  19. Villegas FR (2007). Experimental number theory (No. 13). Oxford University Press, Oxford, UK. 
  20. Vladimirov VS and Volovich IV (1984). Superanalysis. I. Differential-Calculus. Theoretical and Mathematical Physics, 59(1): 317-335. https://doi.org/10.1007/BF01028510 
  21. Volkenborn A (1972). Ein p-adisches integral und seine anwendungen I. Manuscripta Mathematica, 7(4): 341-373. https://doi.org/10.1007/BF01644073 
  22. Volkenborn A (1974). Ein p-adisches integral und seine anwendungen II. Manuscripta Mathematica, 12(1): 17-46. https://doi.org/10.1007/BF01166232 
  23. Volovich IV (1987). Number theory as the ultimate physical theory. No. CERN-TH-4781-87, CERN Research Institute, Geneva, Switzerland.