Article ID: MTJPAM-D-24-00035

Title: Results concerning generalized gamma and beta functions and its application


Montes Taurus J. Pure Appl. Math. / ISSN: 2687-4814

Article ID: MTJPAM-D-24-00035; Volume 6 / Issue 3 / Year 2024, Pages 330-338

Document Type: Research Paper

Author(s): ‪Nabiullah Khan a , Rakibul Sk b , Talha Usman c

aDepartment of Applied Mathematics, Faculty of Engineering and Technology, Aligarh Muslim University, Aligarh 202002, India

bDepartment of Applied Mathematics, Faculty of Engineering and Technology, Aligarh Muslim University, Aligarh 202002, India

cPreparatory Studies Center, Mathematics and Computing Skills Unit, University of Technology and Applied Sciences, P.O. Box: 484, P.C: 411, Sur, Oman

Received: 5 March 2024, Accepted: 30 November 2024, Published: 27 February 2025

Corresponding Author: Talha Usman (Email address: talhausman.maths@gmail.com)

Full Text: PDF

Abstract

The main motive of this paper is to introduce the extension of generalized gamma and beta functions involving the seven-parameters Mittag-Leffler function and also we have discussed some particular cases of these functions. Later, we investigated crucial properties of the beta function including summation relations, integral representations, Mellin transform and differential formulas. Further, we have produced statistical applications of this beta function involving beta distribution, mean, variance, moment generating function, cumulative distribution and reliability function.

Keywords: Gamma function, beta function, integral representation, Mellin transform, beta distribution

References:
  1. A. A. Al-Gonah and W. K. Mohammed, A new extension of extended gamma and beta functions and their properties, J. Sci. Eng. Res. 5 (9), 257–270, 2018.
  2. G. E. Andrews, R. Askey and R. Roy, Special functions, Cambridge University Press, Cambridge, 1999.
  3. A. A. Atash, S. S. Barahmah and M. A. Kulib, On a new extension of extended gamma and beta functions, Int. J. Stat. Appl. Math. 3 (6), 14–18, 2018.
  4. D. Baleanu, K. Diethelm, E. Scalas and J. J. Trujillo, Fractional calculus: Models and numerical methods (Second Edition), World Scientific, USA, 2016.
  5. M. A. Chaudhry and S. M. Zubair, Generalized incomplete gamma function with application, J. Comput. Appl. Math. 55 (1), 99–124, 1994.
  6. M. A. Chaudhry, A. Qadir, M. Rafique and S. M. Zubair, Extension of Euler’s beta function, J. Comput. Appl. Math. 78 (1), 19–32, 1997.
  7. M. A. Chaudhry, A. Qadir, H. M. Srivastava and R. B. Paris, Extended hypergeometric and confluent hypergeometric functions, Appl. Math. Comput. 159 (2), 589–602, 2004.
  8. J. Choi, A. K. Rathie and R.K. Parmar, Extension of extended beta, hypergeometric and confluent hypergeometric functions, Honam Math. J. 36 (2), 357–385, 2014.
  9. A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher transcendental functions, McGraw-Hill, New York, NY, USA, 3, 1955.
  10. M. Ghayasuddin and N. U. Khan, Remarks on extended Gauss hypergeometric functions, Acta Univ. Apulensis Math. Inform. 49, 1–13, 2017.
  11. M. Ghayasuddin, N. U. Khan and M. Ali, A study of the extended beta, Gauss and confluent hypergeometric functions, Int. J. Appl. Math. 33 (10), 1–13, 2020.
  12. R. Hilfer, Applications of fractional calculus in physics, World Scientific, Singapore, 2000.
  13. A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier, the Netherlands, 2006.
  14. N. U. Khan, T. Usman, M. Kamarujjama, M. Aman and M. I. Khan, A new extension of beta function and its properties, Int. J. Math. Sci. 9, 51–63, 2018.
  15. N. U. Khan, T. Usman and M. Aman, Extended beta, hypergeometric and confluent hypergeometric functions, Trans. Natl. Acad. Sci. Azerb. Ser. Phys.-Tech. Math. Sci. 39 (1), 83–97, 2019.
  16. N. U. Khan, T. Usman and M. Aman, Extended Beta, Hypergeometric and confluent hypergeometric functions via multi-index Mittag-Leffler function, Proc. Jangjeon Math. Soc. 25 (1), 43–58, 2022.
  17. N. U. Khan and S. Husain, A note on extended beta function inolving generalized Mittag-Leffler function and its applications, TWMS J. Appl. Eng. Math. 12 (1), 71–81, 2022 .
  18. N. U. Khan, S. Husain and O. Khan, A novel kind of beta logarithmic function and their properties, Hacet. J. Math. Stat. 52 (4), 945–955, 2023.
  19. G. Mittag-Leffler, Sur la nouvelle function Eα(z), C. R. Acad. Sci. Paris 137, 554–558, 1903.
  20. E. Özergin, M. A. Özarslan and A. Altın, Extension of gamma, beta and hypergeometric functions, J. Comput. Appl. Math. 235 (16), 4601–4610, 2011.
  21. P. I. Pucheta, A new extended beta function, Int. J. Math. Appl. 5 (3-C), 255–260, 2017.
  22. E. D. Rainville, Special functions, Macmillan Company, New York, 1960.
  23. M. K. Rasheed and A. H. Majeed, Generalization of gamma and beta functions with certain properties and statistical application, Iraqi J. Sci. 64 (7), 4487–4495, 2023.
  24. M. K. Rasheed and A. H. Majeed, New seven-parameter Mittag-Leffler function with certain analytic properties, Nonlinear Funct. Anal. Appl. 29 (1), 99–111, 2024.
  25. H. M. Srivastava and H. L. Manocha, A treatise on generating functions, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, 1984.
  26. A. Wiman, Uber den fundamentals at zin der teorieder funktionen Eα(z), Acta Math. 29 (1), 191–207, 1905.

Cite this article

How to cite this article: N. Khan, R. Sk and T. Usman, Results concerning generalized gamma and beta functions and its application, Montes Taurus J. Pure Appl. Math. 6 (3), 330-338, 2024; Article ID: MTJPAM-D-24-00035.