| Peer-Reviewed

Generalisation of Euler's Identity in the Form of K-Hypergeometric Functions

Received: 27 October 2022     Accepted: 28 November 2022     Published: 29 December 2022
Views:       Downloads:
Abstract

In this paper, we present a generalization of Euler’s identity associated to usual hypergeometric function in the form of an identity associated with the k-hypergeometric function. The second-order homogeneous k-hypergeometric differential equation , by Frobenious method yields a pair {y1(z),y2(z)} of linearly independent solutions in the form of k-hypergeometric function 2F1,k define as k-hypergeometric power series is convergent in the region ={z: |z|<1/k}. Here with suitable substitution to y(z), we deduce two other forms of solutions of this equation near the singularity z=0. Using the dependency of these forms on {y1(z),y2(z)}, we obtain the generalized Euler’s identity in the form of k-hypergeometric function and a new k-hypergeometric transformation formula. Our generalization pertains to the case when the generalized Euler’s identity reduced to the classical Euler’s identity. In the ultimate section of the paper, we obtain another reduction formula for a particular product difference of k-hypergeometric functions.

Published in American Journal of Applied Mathematics (Volume 10, Issue 6)
DOI 10.11648/j.ajam.20221006.13
Page(s) 240-243
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

K-Hypergeometric Equations, Frobenious Method, K-Hypergeometric Series Solutions, Regular Singular Point

References
[1] Coddington, E. A.; Levinson, N. Theory of Ordinary Differential Equations; McGraw-Hill: New York, NY, USA, 1955.
[2] Campos, L. On some solutions of extended confluent hypergeometric differential equation, Journal of computational and applied mathematics 2001, 137 (1) 177-200.
[3] Diaz, R.; and Pariguan, E. On hypergeometric function and Pochhammer k-symbol, Divulg. Mat. 2007, 15, 179-192.
[4] Gasper, G.; Rahman, M. Basic Hypergeometric Series, 2nd, ed.; Cambridge University Press: Cambridge, UK, 2004.
[5] Krasniqi, V. A limit for the k-gamma and k-beta function. Int. Math. Forum 2010, 5, 1613-1617.
[6] Mubeen, S.; Habibullah, G. M. An integral representation of some k-hypergeometric function. Int. Math. Forum 2012, 7, 203-207.
[7] Mubeen, S.; Rehman, A. A Note on k-Gamma function and Pochhammer k-symbol. J. Inf. Math. Sci. 2014, 6, 93-107.
[8] Mubeen, S.; Naz, M. A. Rehman, G. Rahman. Solution of k-hypergeometric differential equations. J. Appl. Math. 2014, 1-13. [Cross Ref].
[9] Rainville, E. D., Special Functions, The Macmillan Company, New York, 1960.
[10] Slater, L. J. Confluent Hypergeometric Functions, Cambridge University Press, Cambridge New York, 1960.
[11] Shengfeng, L.; and Dong, Y. k-Hypergeometric series solutions to one type of non-homogeneous k-Hypergeometric equations, Symmetry 2019, 11, 262.
[12] Whittaker, E. T.; and Watson, G. N. A Course of Modern Analysis, Cambridge University Press, 1950.
[13] Abdalla, M; Boulaaras, S.; Akel, M; Idris, S. A; Jain, S. Certain fractional formulas of the extended k-hypergeometric functions, Adv. Differ. Equ. 2021, 450 (2021).
[14] Ali, A.; Iqbal, M. Z.; Iqbal, T.; Hadir, M. Study of Generalised k-hypergeometric Functions. Int. J. Math. and Comp. Sci 2021, 16, 379-388.
[15] Yilmazer, R; Ali, K. Fractional Solutions of a k-hypergeometric Differential Equation. Conference Proceedings of Science and Technology 2019, 2 (3), 212-214.
Cite This Article
  • APA Style

    Nafis Ahmad, Mohd Sadiq Khan, Mohammad Imran Aziz. (2022). Generalisation of Euler's Identity in the Form of K-Hypergeometric Functions. American Journal of Applied Mathematics, 10(6), 240-243. https://doi.org/10.11648/j.ajam.20221006.13

    Copy | Download

    ACS Style

    Nafis Ahmad; Mohd Sadiq Khan; Mohammad Imran Aziz. Generalisation of Euler's Identity in the Form of K-Hypergeometric Functions. Am. J. Appl. Math. 2022, 10(6), 240-243. doi: 10.11648/j.ajam.20221006.13

    Copy | Download

    AMA Style

    Nafis Ahmad, Mohd Sadiq Khan, Mohammad Imran Aziz. Generalisation of Euler's Identity in the Form of K-Hypergeometric Functions. Am J Appl Math. 2022;10(6):240-243. doi: 10.11648/j.ajam.20221006.13

    Copy | Download

  • @article{10.11648/j.ajam.20221006.13,
      author = {Nafis Ahmad and Mohd Sadiq Khan and Mohammad Imran Aziz},
      title = {Generalisation of Euler's Identity in the Form of K-Hypergeometric Functions},
      journal = {American Journal of Applied Mathematics},
      volume = {10},
      number = {6},
      pages = {240-243},
      doi = {10.11648/j.ajam.20221006.13},
      url = {https://doi.org/10.11648/j.ajam.20221006.13},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20221006.13},
      abstract = {In this paper, we present a generalization of Euler’s identity associated to usual hypergeometric function in the form of an identity associated with the k-hypergeometric function. The second-order homogeneous k-hypergeometric differential equation , by Frobenious method yields a pair {y1(z),y2(z)} of linearly independent solutions in the form of k-hypergeometric function 2F1,k define as k-hypergeometric power series is convergent in the region ={z: |z|}. Here with suitable substitution to y(z), we deduce two other forms of solutions of this equation near the singularity z=0. Using the dependency of these forms on {y1(z),y2(z)}, we obtain the generalized Euler’s identity in the form of k-hypergeometric function and a new k-hypergeometric transformation formula. Our generalization pertains to the case when the generalized Euler’s identity reduced to the classical Euler’s identity. In the ultimate section of the paper, we obtain another reduction formula for a particular product difference of k-hypergeometric functions.},
     year = {2022}
    }
    

    Copy | Download

  • TY  - JOUR
    T1  - Generalisation of Euler's Identity in the Form of K-Hypergeometric Functions
    AU  - Nafis Ahmad
    AU  - Mohd Sadiq Khan
    AU  - Mohammad Imran Aziz
    Y1  - 2022/12/29
    PY  - 2022
    N1  - https://doi.org/10.11648/j.ajam.20221006.13
    DO  - 10.11648/j.ajam.20221006.13
    T2  - American Journal of Applied Mathematics
    JF  - American Journal of Applied Mathematics
    JO  - American Journal of Applied Mathematics
    SP  - 240
    EP  - 243
    PB  - Science Publishing Group
    SN  - 2330-006X
    UR  - https://doi.org/10.11648/j.ajam.20221006.13
    AB  - In this paper, we present a generalization of Euler’s identity associated to usual hypergeometric function in the form of an identity associated with the k-hypergeometric function. The second-order homogeneous k-hypergeometric differential equation , by Frobenious method yields a pair {y1(z),y2(z)} of linearly independent solutions in the form of k-hypergeometric function 2F1,k define as k-hypergeometric power series is convergent in the region ={z: |z|}. Here with suitable substitution to y(z), we deduce two other forms of solutions of this equation near the singularity z=0. Using the dependency of these forms on {y1(z),y2(z)}, we obtain the generalized Euler’s identity in the form of k-hypergeometric function and a new k-hypergeometric transformation formula. Our generalization pertains to the case when the generalized Euler’s identity reduced to the classical Euler’s identity. In the ultimate section of the paper, we obtain another reduction formula for a particular product difference of k-hypergeometric functions.
    VL  - 10
    IS  - 6
    ER  - 

    Copy | Download

Author Information
  • Department of Mathematics, Shibli National College, Azamgarh, India

  • Department of Mathematics, Shibli National College, Azamgarh, India

  • Department of Physics, Shibli National College, Azamgarh, India

  • Sections