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A Generalized Gamma-Type Functions Involving Confluent Hypergeometric Mittage-Leffler Function and Associated Probability Distributions

Received: 23 May 2022     Accepted: 20 June 2022     Published: 21 October 2022
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Abstract

Gamma function is one of the important special functions occurring in many branches of mathematical physics and is investigated in detail in a number of literatures in recent years. The main object of this paper is to present a systematic study of a unification and generalization of the gamma-type functions, which is defined here by means of the confluent hypergeometric Mittage- Leffler function. Motivated essentially by the sources of the applications of the Mittage-Leffler functions in many areas of science and engineering, the author present in a unified manner. During the last two decades Mittage – Leffler function has come into prominence after about nine decades of its discovery by a Swedish Mathematician Mittage-Leffler, due to the vast potential of this applications in solving the problems of physical, biological, engineering, and earth sciences. In this article we investigated a probability density function associated with the generalized gamma-type function, together with several other related results in the theory of probability and statistics, are also considered.

Published in International Journal of Theoretical and Applied Mathematics (Volume 8, Issue 4)
DOI 10.11648/j.ijtam.20220804.11
Page(s) 78-84
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2022. Published by Science Publishing Group

Keywords

Special Function, Gamma and Incomplete Gamma Functions, Confluent Hypergeometric Mittage-Leffler Function, Probability Density Function, Moment Generating Function

References
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[2] Kobayashi, K., On generalized Gamma functions occurring in diffraction theory. Journal of the Physical Society of Japan, 60 (1991), 1501–1512.
[3] Kobayashi, K., Plane wave diffraction by a strip: Exact and asymptotic solutions. Journal of the Physical Society of Japan, 60 (1991), 1891–1905.
[4] Al-Musallam, F., Kalla, S. L., Asymptotic expansions for generalized gamma and incomplete gamma function, Appl. Anal., 66 (1997), 173-187.
[5] Al-Musallam, F., Kalla, S. L., Further results on a generalized gamma function occurring in diffraction theory. Integral Transforms and Special function, 7 (1998), 175-190.
[6] Virchenko, N., Kalla, S. L., On some generalization of the functions of hypergeometric type, Fract. Calc and Appl. Anal. 2 (3) (1999), 233-244.
[7] Virchenko, N., Kalla, S. L. and Al-Zamel, A., Some results on a generalized hypergeometric function. Integral Transforms and Special Functions, 12, 89–100 (2001).
[8] Ghanim, F.; Al- Janaby, H. F. Inclusion and Convolution Features of Univalent Meromorphic functions Correlating with Mittage-Leffler function. Filomat 2020, 34, 2141-2150.
[9] Ghanim, F.; Al- Janaby, H. F. Some analytical merites of Kummer-Type function associated with Mittage-Leffler parameters. Arab. J. Basic Appl. Sci. 2021, 28, 255-263.
[10] Ghanim, F.; Bendak, S.; Hawarneh, A. A. Fractional Calculus Operator involving Mittage-Leffler confluent hypergeometric function. 2021, preprint.
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[19] Bazighifan, O.; Dassios, I. Riccati technique and asymptotic behaviour of fourth order advanced differential equations. Mathematics 2020, 8, 590.
[20] Mittage-Leffler, G. M. Sur la nouvelle function Eα(x). C. R. Acad. Sci. Paris 1903, 137, 554-558.
[21] Mittage-Leffler, G. M. Sur la repr’estation analytique d; une branche uniforme d’une function monogene: Cinquieme note. Acta Math. 1905, 29, 101-181.
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[25] Srivastava, H. M.; Tomovski, Ž. Fractional calculus with an integral operator containing a generalized Mittage-Leffler function in the kernel. J. Appl. Math. Comput. 2009, 211, 198-210.
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  • APA Style

    Naresh Dudi. (2022). A Generalized Gamma-Type Functions Involving Confluent Hypergeometric Mittage-Leffler Function and Associated Probability Distributions. International Journal of Theoretical and Applied Mathematics, 8(4), 78-84. https://doi.org/10.11648/j.ijtam.20220804.11

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    ACS Style

    Naresh Dudi. A Generalized Gamma-Type Functions Involving Confluent Hypergeometric Mittage-Leffler Function and Associated Probability Distributions. Int. J. Theor. Appl. Math. 2022, 8(4), 78-84. doi: 10.11648/j.ijtam.20220804.11

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    AMA Style

    Naresh Dudi. A Generalized Gamma-Type Functions Involving Confluent Hypergeometric Mittage-Leffler Function and Associated Probability Distributions. Int J Theor Appl Math. 2022;8(4):78-84. doi: 10.11648/j.ijtam.20220804.11

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  • @article{10.11648/j.ijtam.20220804.11,
      author = {Naresh Dudi},
      title = {A Generalized Gamma-Type Functions Involving Confluent Hypergeometric Mittage-Leffler Function and Associated Probability Distributions},
      journal = {International Journal of Theoretical and Applied Mathematics},
      volume = {8},
      number = {4},
      pages = {78-84},
      doi = {10.11648/j.ijtam.20220804.11},
      url = {https://doi.org/10.11648/j.ijtam.20220804.11},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijtam.20220804.11},
      abstract = {Gamma function is one of the important special functions occurring in many branches of mathematical physics and is investigated in detail in a number of literatures in recent years. The main object of this paper is to present a systematic study of a unification and generalization of the gamma-type functions, which is defined here by means of the confluent hypergeometric Mittage- Leffler function. Motivated essentially by the sources of the applications of the Mittage-Leffler functions in many areas of science and engineering, the author present in a unified manner. During the last two decades Mittage – Leffler function has come into prominence after about nine decades of its discovery by a Swedish Mathematician Mittage-Leffler, due to the vast potential of this applications in solving the problems of physical, biological, engineering, and earth sciences. In this article we investigated a probability density function associated with the generalized gamma-type function, together with several other related results in the theory of probability and statistics, are also considered.},
     year = {2022}
    }
    

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    T1  - A Generalized Gamma-Type Functions Involving Confluent Hypergeometric Mittage-Leffler Function and Associated Probability Distributions
    AU  - Naresh Dudi
    Y1  - 2022/10/21
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    DO  - 10.11648/j.ijtam.20220804.11
    T2  - International Journal of Theoretical and Applied Mathematics
    JF  - International Journal of Theoretical and Applied Mathematics
    JO  - International Journal of Theoretical and Applied Mathematics
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    PB  - Science Publishing Group
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    AB  - Gamma function is one of the important special functions occurring in many branches of mathematical physics and is investigated in detail in a number of literatures in recent years. The main object of this paper is to present a systematic study of a unification and generalization of the gamma-type functions, which is defined here by means of the confluent hypergeometric Mittage- Leffler function. Motivated essentially by the sources of the applications of the Mittage-Leffler functions in many areas of science and engineering, the author present in a unified manner. During the last two decades Mittage – Leffler function has come into prominence after about nine decades of its discovery by a Swedish Mathematician Mittage-Leffler, due to the vast potential of this applications in solving the problems of physical, biological, engineering, and earth sciences. In this article we investigated a probability density function associated with the generalized gamma-type function, together with several other related results in the theory of probability and statistics, are also considered.
    VL  - 8
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Author Information
  • Department of Mathematics and Statistics, Jai Narain Vyas University, Jodhur, India

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