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Relations Among Certain Generalized Hyper-Geometric Functions Suggested by N-fractional Calculus

Received: 27 August 2016     Accepted: 19 November 2016     Published: 20 December 2016
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Abstract

The subject of fractional calculus has gained importance and popularity during the past three decades. Based upon the N-fractional calculus we introduce a new N-fractional operators involving hyper-geometric function. By means of these N-fractional operators a number of operational relations among the hyper-geometric functions of two, three, four and several variables are then found. Other closely-related results are also considered.

Published in Mathematics Letters (Volume 2, Issue 6)
DOI 10.11648/j.ml.20160206.12
Page(s) 47-57
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), 2016. Published by Science Publishing Group

Keywords

N-fractional Calculus Operators, Horn’s Functions, Appell Functions, Saran Functions, Quadruple Functions, Hyper-Geometric of Several Variables

References
[1] Appell, P., Sur les séries hypérgeométriques de deux variables, et sur des équations différentielles linéaires aux déivées partielles. C. R. Acad. Sci. Paris 90, (1880), 296-298.
[2] Maged G. Bin-Saad, Maisoon A. Hussein, Operational images and relations of two and three variable hypergeometric series, Volume 2, Issue 1., J. Prog. Res. Math. (JPRM), 2395-0218.
[3] Bin-Saad, Maged G., A new multiple hyper-geometric function related to Lauricella’s F(n)A and F(n)D (communicated for publication).
[4] Chhaya Sharma and C. L. Parihar, Hypergeometric functions of four variables (I), J. Indian Acad. Math. Vol. 11, No.2, (1989), 121-133.
[5] Chen, M. P., Srivastava, H. M. (1997). Fractional calculus operators and their applications involving power functions and summation of series. Appl. Math. Comput., 81, 283-304.
[6] Goyal, S. P., Jain, R. M., Gaur, N. (1992). Fractional integral operators involving a product of generalized hypergeometric functions and a general class of polynomials II. Indian J. Pure Appl. Math., 23, 121-128.
[7] Goyal, S. P., Jain, R. M., Gaur, N. (1991). Fractional integral operators involving a product of generalized hypergeometric functions and a general class of polynomials. Indian J. Pure Appl. Math., 22, 403-411.
[8] Exton, H., Multiple hypergeometric functions and applications, Ellis Horwood Ltd., Chichester, New York, 1976.
[9] Exton, H., Hypergeometric functions of three variables, J. Indian Acad. Math. Vol. 4, (1982), 111-119.
[10] Horn, J., Hypergeometrische Funktionen zweier veranderlichen, Math. Ann. 105, (1931), 381-407.
[11] Kalla, S. L. (1970). Integral operators of fractional integration. Mat. Notae 22, 89-93.
[12] Kalla, S. L., Saxena, R. K. (1969). Integral operators involving hypergeometric functions. Math. Zeitschr.108 231-234.
[13] Kalla, S. L., Saxena, R. K. (1974). Integral operators involving hypergeometric functions II. Univ. Nac. Tucuman Rev. Ser. A 24, 31-36.
[14] Kant, S., Koul, C. L. (1991). On fractional integral operators. J. Indian Math. Soc. (N. S) 56, 97-107.
[15] Lauricella, G., Sulle funzioni ipergeometriche a piu variabili. Rend. Circ. Math. Palermo 7, (1898), 111-158.
[16] Nishimoto, K., Fractional Calculus Vol. I, Descartes Press Co., Koriyuma, Japan, 963(1984).
[17] Nishimoto, K., Fractional Calculus Vol. II, Descartes Press Co., Koriyuma, Japan, 963 (1984).
[18] Saran, S., Hypergeometric functions of three variables, Ganita 5, (1954), 77-91.
[19] Samko, S. G. and A. A. Kilbas, and O. I. Marichev. Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach, Amsterdam, 1993.
[20] Sharma, C., Parihar C. L., Hypergeometric functions of four variables (I), Indian Acad. Math. 11 (1989) 121–133.
[21] Srivastava, H. M. and Karlsson P. W., Multiple Gaussian hypergeometric series, Halsted Press, Bristone, London, New York, 1985.
[22] Srivastava, H. M. and Manocha, H. L., A treatise on generating functions, Halsted Press, New York, Brisbane and Toronto, 1984.
Cite This Article
  • APA Style

    Maged Gumaan Bin-Saad. (2016). Relations Among Certain Generalized Hyper-Geometric Functions Suggested by N-fractional Calculus. Mathematics Letters, 2(6), 47-57. https://doi.org/10.11648/j.ml.20160206.12

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    ACS Style

    Maged Gumaan Bin-Saad. Relations Among Certain Generalized Hyper-Geometric Functions Suggested by N-fractional Calculus. Math. Lett. 2016, 2(6), 47-57. doi: 10.11648/j.ml.20160206.12

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    AMA Style

    Maged Gumaan Bin-Saad. Relations Among Certain Generalized Hyper-Geometric Functions Suggested by N-fractional Calculus. Math Lett. 2016;2(6):47-57. doi: 10.11648/j.ml.20160206.12

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  • @article{10.11648/j.ml.20160206.12,
      author = {Maged Gumaan Bin-Saad},
      title = {Relations Among Certain Generalized Hyper-Geometric Functions Suggested by N-fractional Calculus},
      journal = {Mathematics Letters},
      volume = {2},
      number = {6},
      pages = {47-57},
      doi = {10.11648/j.ml.20160206.12},
      url = {https://doi.org/10.11648/j.ml.20160206.12},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ml.20160206.12},
      abstract = {The subject of fractional calculus has gained importance and popularity during the past three decades. Based upon the N-fractional calculus we introduce a new N-fractional operators involving hyper-geometric function. By means of these N-fractional operators a number of operational relations among the hyper-geometric functions of two, three, four and several variables are then found. Other closely-related results are also considered.},
     year = {2016}
    }
    

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    AU  - Maged Gumaan Bin-Saad
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    AB  - The subject of fractional calculus has gained importance and popularity during the past three decades. Based upon the N-fractional calculus we introduce a new N-fractional operators involving hyper-geometric function. By means of these N-fractional operators a number of operational relations among the hyper-geometric functions of two, three, four and several variables are then found. Other closely-related results are also considered.
    VL  - 2
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    ER  - 

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Author Information
  • Department of Mathematics, Aden University, Aden, Kohrmaksar, Yemen

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