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

Nafis Ahmad; Mohd Sadiq Khan; Mohammad Imran Aziz

doi:doi:10.11648/j.ajam.20221006.13

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

Nafis Ahmad, Mohd Sadiq Khan, Mohammad Imran Aziz

Published in American Journal of Applied Mathematics (Volume 10, Issue 6)

Received: 27 October 2022 Accepted: 28 November 2022 Published: 29 December 2022

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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), 2024. 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.

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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

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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

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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

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@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}
}

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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
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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  -

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Author Information

Nafis Ahmad

Department of Mathematics, Shibli National College, Azamgarh, India
Mohd Sadiq Khan

Department of Mathematics, Shibli National College, Azamgarh, India
Mohammad Imran Aziz

Department of Physics, Shibli National College, Azamgarh, India

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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

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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

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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