Reduction Formulas for Generalized Hypergeometric Series Associated with New Sequences and Applications
In this paper, by introducing two sequences of <i>new</i> numbers and their derivatives, which are closely related to the Stirling numbers of the first kind, and choosing to employ six known generalized Kummer’s summation formulas for <inline-formula><math xmlns="http://www...
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| Format: | Article |
| Language: | English |
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MDPI AG
2021-10-01
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| Series: | Fractal and Fractional |
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| Online Access: | https://www.mdpi.com/2504-3110/5/4/150 |
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| author | Junesang Choi Mohd Idris Qureshi Aarif Hussain Bhat Javid Majid |
| author_facet | Junesang Choi Mohd Idris Qureshi Aarif Hussain Bhat Javid Majid |
| author_sort | Junesang Choi |
| collection | DOAJ |
| description | In this paper, by introducing two sequences of <i>new</i> numbers and their derivatives, which are closely related to the Stirling numbers of the first kind, and choosing to employ six known generalized Kummer’s summation formulas for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow></mrow><mn>2</mn></msub><msub><mi>F</mi><mn>1</mn></msub><mrow><mo>(</mo><mo>−</mo><mn>1</mn><mo>)</mo></mrow></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>2</mn></msub><msub><mi>F</mi><mn>1</mn></msub><mrow><mo>(</mo><mn>1</mn><mo>/</mo><mn>2</mn><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, we establish six classes of generalized summation formulas for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow></mrow><mrow><mi>p</mi><mo>+</mo><mn>2</mn></mrow></msub><msub><mi>F</mi><mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></semantics></math></inline-formula> with arguments <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>−</mo><mn>1</mn></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></semantics></math></inline-formula> for any positive integer <i>p</i>. Next, by differentiating both sides of six chosen formulas presented here with respect to a specific parameter, among numerous ones, we demonstrate six identities in connection with finite sums of <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><mo>(</mo><mo>−</mo><mn>1</mn><mo>)</mo></mrow></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>4</mn></msub><msub><mi>F</mi><mn>3</mn></msub><mrow><mo>(</mo><mn>1</mn><mo>/</mo><mn>2</mn><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. Further, we choose to give simple particular identities of some formulas presented here. We conclude this paper by highlighting a potential use of the newly presented numbers and posing some problems. |
| first_indexed | 2024-03-10T04:05:54Z |
| format | Article |
| id | doaj.art-0a298525b3734c808956fb823e626403 |
| institution | Directory Open Access Journal |
| issn | 2504-3110 |
| language | English |
| last_indexed | 2024-03-10T04:05:54Z |
| publishDate | 2021-10-01 |
| publisher | MDPI AG |
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| series | Fractal and Fractional |
| spelling | doaj.art-0a298525b3734c808956fb823e6264032023-11-23T08:22:57ZengMDPI AGFractal and Fractional2504-31102021-10-015415010.3390/fractalfract5040150Reduction Formulas for Generalized Hypergeometric Series Associated with New Sequences and ApplicationsJunesang Choi0Mohd Idris Qureshi1Aarif Hussain Bhat2Javid Majid3Department of Mathematics, Dongguk University, Gyeongju 38066, KoreaDepartment 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 Applied Sciences and Humanities, Faculty of Engineering and Technology, Jamia Millia Islamia (A Central University), New Delhi 110025, IndiaIn this paper, by introducing two sequences of <i>new</i> numbers and their derivatives, which are closely related to the Stirling numbers of the first kind, and choosing to employ six known generalized Kummer’s summation formulas for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow></mrow><mn>2</mn></msub><msub><mi>F</mi><mn>1</mn></msub><mrow><mo>(</mo><mo>−</mo><mn>1</mn><mo>)</mo></mrow></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>2</mn></msub><msub><mi>F</mi><mn>1</mn></msub><mrow><mo>(</mo><mn>1</mn><mo>/</mo><mn>2</mn><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, we establish six classes of generalized summation formulas for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow></mrow><mrow><mi>p</mi><mo>+</mo><mn>2</mn></mrow></msub><msub><mi>F</mi><mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></semantics></math></inline-formula> with arguments <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>−</mo><mn>1</mn></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></semantics></math></inline-formula> for any positive integer <i>p</i>. Next, by differentiating both sides of six chosen formulas presented here with respect to a specific parameter, among numerous ones, we demonstrate six identities in connection with finite sums of <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><mo>(</mo><mo>−</mo><mn>1</mn><mo>)</mo></mrow></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>4</mn></msub><msub><mi>F</mi><mn>3</mn></msub><mrow><mo>(</mo><mn>1</mn><mo>/</mo><mn>2</mn><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. Further, we choose to give simple particular identities of some formulas presented here. We conclude this paper by highlighting a potential use of the newly presented numbers and posing some problems.https://www.mdpi.com/2504-3110/5/4/150Gamma functionPsi functionPochhammer symbolhypergeometric function <sub>2</sub><i>F</i><sub>1</sub>generalized hypergeometric functions <sub><i>t</i></sub><i>F</i><i><sub>u</sub></i>Gauss’s summation theorem for <sub>2</sub><i>F</i><sub>1</sub>(1) |
| spellingShingle | Junesang Choi Mohd Idris Qureshi Aarif Hussain Bhat Javid Majid Reduction Formulas for Generalized Hypergeometric Series Associated with New Sequences and Applications Fractal and Fractional Gamma function Psi function Pochhammer symbol hypergeometric function <sub>2</sub><i>F</i><sub>1</sub> generalized hypergeometric functions <sub><i>t</i></sub><i>F</i><i><sub>u</sub></i> Gauss’s summation theorem for <sub>2</sub><i>F</i><sub>1</sub>(1) |
| title | Reduction Formulas for Generalized Hypergeometric Series Associated with New Sequences and Applications |
| title_full | Reduction Formulas for Generalized Hypergeometric Series Associated with New Sequences and Applications |
| title_fullStr | Reduction Formulas for Generalized Hypergeometric Series Associated with New Sequences and Applications |
| title_full_unstemmed | Reduction Formulas for Generalized Hypergeometric Series Associated with New Sequences and Applications |
| title_short | Reduction Formulas for Generalized Hypergeometric Series Associated with New Sequences and Applications |
| title_sort | reduction formulas for generalized hypergeometric series associated with new sequences and applications |
| topic | Gamma function Psi function Pochhammer symbol hypergeometric function <sub>2</sub><i>F</i><sub>1</sub> generalized hypergeometric functions <sub><i>t</i></sub><i>F</i><i><sub>u</sub></i> Gauss’s summation theorem for <sub>2</sub><i>F</i><sub>1</sub>(1) |
| url | https://www.mdpi.com/2504-3110/5/4/150 |
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