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 {y_{1}(z),y_{2}(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 {y_{1}(z),y_{2}(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 |
K-Hypergeometric Equations, Frobenious Method, K-Hypergeometric Series Solutions, Regular Singular Point
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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
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
@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} }
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 -