Three General Double-Series Identities and Associated Reduction Formulas and Fractional Calculus
In this article, we introduce three general double-series identities using Whipple transformations for terminating generalized hypergeometric <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-09-01
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| Series: | Fractal and Fractional |
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| Online Access: | https://www.mdpi.com/2504-3110/7/10/700 |
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| author | Mohd Idris Qureshi Tafaz Ul Rahman Shah Junesang Choi Aarif Hussain Bhat |
| author_facet | Mohd Idris Qureshi Tafaz Ul Rahman Shah Junesang Choi Aarif Hussain Bhat |
| author_sort | Mohd Idris Qureshi |
| collection | DOAJ |
| description | In this article, we introduce three general double-series identities using Whipple transformations for terminating generalized hypergeometric <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow></mrow><mn>4</mn></msub><msub><mi>F</mi><mn>3</mn></msub></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow></mrow><mn>5</mn></msub><msub><mi>F</mi><mn>4</mn></msub></mrow></semantics></math></inline-formula> functions. Then, by employing the left-sided Riemann–Liouville fractional integral on these identities, we show the ability to derive additional identities of the same nature successively. These identities are used to derive transformation formulas between the Srivastava–Daoust double hypergeometric function (S–D function) and Kampé de Fériet’s double hypergeometric function (KDF function) with equal arguments. We also demonstrate reduction formulas from the S–D function or KDF function to the generalized hypergeometric function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow></mrow><mi>p</mi></msub><msub><mi>F</mi><mi>q</mi></msub></mrow></semantics></math></inline-formula>. Additionally, we provide general summation formulas for the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow></mrow><mi>p</mi></msub><msub><mi>F</mi><mi>q</mi></msub></mrow></semantics></math></inline-formula> and S–D function (or KDF function) with specific arguments. We further highlight the connections between the results presented here and existing identities. |
| first_indexed | 2024-03-10T21:14:34Z |
| format | Article |
| id | doaj.art-4a936d7972454a5d8e092819899009bd |
| institution | Directory Open Access Journal |
| issn | 2504-3110 |
| language | English |
| last_indexed | 2024-03-10T21:14:34Z |
| publishDate | 2023-09-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj.art-4a936d7972454a5d8e092819899009bd2023-11-19T16:34:00ZengMDPI AGFractal and Fractional2504-31102023-09-0171070010.3390/fractalfract7100700Three General Double-Series Identities and Associated Reduction Formulas and Fractional CalculusMohd Idris Qureshi0Tafaz Ul Rahman Shah1Junesang Choi2Aarif Hussain Bhat3Department of Applied Sciences and Humanities, Faculty of Engineering and Technology, Jamia Millia Islamia, A Central University, New Delhi 110025, IndiaDepartment of Applied Sciences and Humanities, Faculty of Engineering and Technology, Jamia Millia Islamia, A Central University, New Delhi 110025, IndiaDepartment of Mathematics, Dongguk University, Gyeongju 38066, Republic of KoreaDepartment of Applied Sciences and Humanities, Faculty of Engineering and Technology, Jamia Millia Islamia, A Central University, New Delhi 110025, IndiaIn this article, we introduce three general double-series identities using Whipple transformations for terminating generalized hypergeometric <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow></mrow><mn>4</mn></msub><msub><mi>F</mi><mn>3</mn></msub></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow></mrow><mn>5</mn></msub><msub><mi>F</mi><mn>4</mn></msub></mrow></semantics></math></inline-formula> functions. Then, by employing the left-sided Riemann–Liouville fractional integral on these identities, we show the ability to derive additional identities of the same nature successively. These identities are used to derive transformation formulas between the Srivastava–Daoust double hypergeometric function (S–D function) and Kampé de Fériet’s double hypergeometric function (KDF function) with equal arguments. We also demonstrate reduction formulas from the S–D function or KDF function to the generalized hypergeometric function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow></mrow><mi>p</mi></msub><msub><mi>F</mi><mi>q</mi></msub></mrow></semantics></math></inline-formula>. Additionally, we provide general summation formulas for the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow></mrow><mi>p</mi></msub><msub><mi>F</mi><mi>q</mi></msub></mrow></semantics></math></inline-formula> and S–D function (or KDF function) with specific arguments. We further highlight the connections between the results presented here and existing identities.https://www.mdpi.com/2504-3110/7/10/700Bailey quadratic transformationgeneralized hypergeometric functionKampé de Fériet’s double hypergeometric functionseries rearrangement techniqueSrivastava–Daoust double hypergeometric functionWhipple transformations |
| spellingShingle | Mohd Idris Qureshi Tafaz Ul Rahman Shah Junesang Choi Aarif Hussain Bhat Three General Double-Series Identities and Associated Reduction Formulas and Fractional Calculus Fractal and Fractional Bailey quadratic transformation generalized hypergeometric function Kampé de Fériet’s double hypergeometric function series rearrangement technique Srivastava–Daoust double hypergeometric function Whipple transformations |
| title | Three General Double-Series Identities and Associated Reduction Formulas and Fractional Calculus |
| title_full | Three General Double-Series Identities and Associated Reduction Formulas and Fractional Calculus |
| title_fullStr | Three General Double-Series Identities and Associated Reduction Formulas and Fractional Calculus |
| title_full_unstemmed | Three General Double-Series Identities and Associated Reduction Formulas and Fractional Calculus |
| title_short | Three General Double-Series Identities and Associated Reduction Formulas and Fractional Calculus |
| title_sort | three general double series identities and associated reduction formulas and fractional calculus |
| topic | Bailey quadratic transformation generalized hypergeometric function Kampé de Fériet’s double hypergeometric function series rearrangement technique Srivastava–Daoust double hypergeometric function Whipple transformations |
| url | https://www.mdpi.com/2504-3110/7/10/700 |
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