A Method for Performing the Symmetric Anti-Difference Equations in Quantum Fractional Calculus
In this paper, we develop theorems on finite and infinite summation formulas by utilizing the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">q</mi></semantics><...
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-12-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/14/12/2604 |
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| author | V. Rexma Sherine T. G. Gerly P. Chellamani Esmail Hassan Abdullatif Al-Sabri Rashad Ismail G. Britto Antony Xavier N. Avinash |
| author_facet | V. Rexma Sherine T. G. Gerly P. Chellamani Esmail Hassan Abdullatif Al-Sabri Rashad Ismail G. Britto Antony Xavier N. Avinash |
| author_sort | V. Rexma Sherine |
| collection | DOAJ |
| description | In this paper, we develop theorems on finite and infinite summation formulas by utilizing the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">q</mi></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi mathvariant="fraktur">q</mi><mo>,</mo><mi mathvariant="fraktur">h</mi><mo>)</mo></mrow></semantics></math></inline-formula> anti-difference operators, and also we extend these core theorems to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="fraktur">q</mi><mrow><mo>(</mo><mi>α</mi><mo>)</mo></mrow></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow><mo>(</mo><mi mathvariant="fraktur">q</mi><mo>,</mo><mi mathvariant="fraktur">h</mi><mo>)</mo></mrow><mi>α</mi></msub></semantics></math></inline-formula> difference operators. Several integer order theorems based on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">q</mi></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="fraktur">q</mi><mrow><mo>(</mo><mi>α</mi><mo>)</mo></mrow></msub></semantics></math></inline-formula> difference operator have been published, which gave us the idea to derive the fractional order anti-difference equations for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">q</mi></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="fraktur">q</mi><mrow><mo>(</mo><mi>α</mi><mo>)</mo></mrow></msub></semantics></math></inline-formula> difference operators. In order to develop the fractional order anti-difference equations for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">q</mi></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="fraktur">q</mi><mrow><mo>(</mo><mi>α</mi><mo>)</mo></mrow></msub></semantics></math></inline-formula> difference operators, we construct a function known as the quantum geometric and alpha-quantum geometric function, which behaves as the class of geometric series. We can use this function to convert an infinite summation to a limited summation. Using this concept and by the gamma function, we derive the fractional order anti-difference equations for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">q</mi></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="fraktur">q</mi><mrow><mo>(</mo><mi>α</mi><mo>)</mo></mrow></msub></semantics></math></inline-formula> difference operators for polynomials, polynomial factorials, and logarithmic functions that provide solutions for symmetric difference operator. We provide appropriate examples to support our results. In addition, we extend these concepts to the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi mathvariant="fraktur">q</mi><mo>,</mo><mi mathvariant="fraktur">h</mi><mo>)</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow><mo>(</mo><mi mathvariant="fraktur">q</mi><mo>,</mo><mi mathvariant="fraktur">h</mi><mo>)</mo></mrow><mi>α</mi></msub></semantics></math></inline-formula> difference operators, and we derive several integer and fractional order theorems that give solutions for the mixed symmetric difference operator. Finally, we plot the diagrams to analyze the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi mathvariant="fraktur">q</mi><mo>,</mo><mi mathvariant="fraktur">h</mi><mo>)</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow><mo>(</mo><mi mathvariant="fraktur">q</mi><mo>,</mo><mi mathvariant="fraktur">h</mi><mo>)</mo></mrow><mi>α</mi></msub></semantics></math></inline-formula> difference operators for verification. |
| first_indexed | 2024-03-09T15:47:39Z |
| format | Article |
| id | doaj.art-5b3701e2cc6840a3a55bf6ec51c7b6e8 |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-09T15:47:39Z |
| publishDate | 2022-12-01 |
| publisher | MDPI AG |
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| series | Symmetry |
| spelling | doaj.art-5b3701e2cc6840a3a55bf6ec51c7b6e82023-11-24T18:19:55ZengMDPI AGSymmetry2073-89942022-12-011412260410.3390/sym14122604A Method for Performing the Symmetric Anti-Difference Equations in Quantum Fractional CalculusV. Rexma Sherine0T. G. Gerly1P. Chellamani2Esmail Hassan Abdullatif Al-Sabri3Rashad Ismail4G. Britto Antony Xavier5N. Avinash6Department of Mathematics, Sacred Heart College (Autonomous), Tirupattur 635601, Tamil Nadu, IndiaDepartment of Mathematics, Sacred Heart College (Autonomous), Tirupattur 635601, Tamil Nadu, IndiaDepartment of Mathematics, St. Joseph’s College of Engineering, OMR, Chennai 600119, Tamil Nadu, IndiaDepartment of Mathematics, College of Science and Arts (Muhayl Assir), King Khalid Universiy, Abha 62529, Saudi ArabiaDepartment of Mathematics, College of Science and Arts (Muhayl Assir), King Khalid Universiy, Abha 62529, Saudi ArabiaDepartment of Mathematics, Sacred Heart College (Autonomous), Tirupattur 635601, Tamil Nadu, IndiaDepartment of Mathematics, Sacred Heart College (Autonomous), Tirupattur 635601, Tamil Nadu, IndiaIn this paper, we develop theorems on finite and infinite summation formulas by utilizing the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">q</mi></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi mathvariant="fraktur">q</mi><mo>,</mo><mi mathvariant="fraktur">h</mi><mo>)</mo></mrow></semantics></math></inline-formula> anti-difference operators, and also we extend these core theorems to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="fraktur">q</mi><mrow><mo>(</mo><mi>α</mi><mo>)</mo></mrow></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow><mo>(</mo><mi mathvariant="fraktur">q</mi><mo>,</mo><mi mathvariant="fraktur">h</mi><mo>)</mo></mrow><mi>α</mi></msub></semantics></math></inline-formula> difference operators. Several integer order theorems based on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">q</mi></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="fraktur">q</mi><mrow><mo>(</mo><mi>α</mi><mo>)</mo></mrow></msub></semantics></math></inline-formula> difference operator have been published, which gave us the idea to derive the fractional order anti-difference equations for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">q</mi></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="fraktur">q</mi><mrow><mo>(</mo><mi>α</mi><mo>)</mo></mrow></msub></semantics></math></inline-formula> difference operators. In order to develop the fractional order anti-difference equations for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">q</mi></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="fraktur">q</mi><mrow><mo>(</mo><mi>α</mi><mo>)</mo></mrow></msub></semantics></math></inline-formula> difference operators, we construct a function known as the quantum geometric and alpha-quantum geometric function, which behaves as the class of geometric series. We can use this function to convert an infinite summation to a limited summation. Using this concept and by the gamma function, we derive the fractional order anti-difference equations for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">q</mi></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="fraktur">q</mi><mrow><mo>(</mo><mi>α</mi><mo>)</mo></mrow></msub></semantics></math></inline-formula> difference operators for polynomials, polynomial factorials, and logarithmic functions that provide solutions for symmetric difference operator. We provide appropriate examples to support our results. In addition, we extend these concepts to the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi mathvariant="fraktur">q</mi><mo>,</mo><mi mathvariant="fraktur">h</mi><mo>)</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow><mo>(</mo><mi mathvariant="fraktur">q</mi><mo>,</mo><mi mathvariant="fraktur">h</mi><mo>)</mo></mrow><mi>α</mi></msub></semantics></math></inline-formula> difference operators, and we derive several integer and fractional order theorems that give solutions for the mixed symmetric difference operator. Finally, we plot the diagrams to analyze the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi mathvariant="fraktur">q</mi><mo>,</mo><mi mathvariant="fraktur">h</mi><mo>)</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow><mo>(</mo><mi mathvariant="fraktur">q</mi><mo>,</mo><mi mathvariant="fraktur">h</mi><mo>)</mo></mrow><mi>α</mi></msub></semantics></math></inline-formula> difference operators for verification.https://www.mdpi.com/2073-8994/14/12/2604<named-content content-type="inline-formula"><inline-formula> <mml:math display="block" id="mm1111"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="fraktur">q</mml:mi> </mml:mrow> </mml:semantics> </mml:math> </inline-formula></named-content> and <named-content content-type="inline-formula"><inline-formula> <mml:math id="mm16111111"> <mml:semantics> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi mathvariant="fraktur">q</mml:mi> <mml:mo>,</mml:mo> <mml:mi mathvariant="fraktur">h</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:semantics> </mml:math> </inline-formula></named-content> difference operatorsquantum geometric functionalpha quantum geometric functiongamma functions and fractional order sum |
| spellingShingle | V. Rexma Sherine T. G. Gerly P. Chellamani Esmail Hassan Abdullatif Al-Sabri Rashad Ismail G. Britto Antony Xavier N. Avinash A Method for Performing the Symmetric Anti-Difference Equations in Quantum Fractional Calculus Symmetry <named-content content-type="inline-formula"><inline-formula> <mml:math display="block" id="mm1111"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="fraktur">q</mml:mi> </mml:mrow> </mml:semantics> </mml:math> </inline-formula></named-content> and <named-content content-type="inline-formula"><inline-formula> <mml:math id="mm16111111"> <mml:semantics> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi mathvariant="fraktur">q</mml:mi> <mml:mo>,</mml:mo> <mml:mi mathvariant="fraktur">h</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:semantics> </mml:math> </inline-formula></named-content> difference operators quantum geometric function alpha quantum geometric function gamma functions and fractional order sum |
| title | A Method for Performing the Symmetric Anti-Difference Equations in Quantum Fractional Calculus |
| title_full | A Method for Performing the Symmetric Anti-Difference Equations in Quantum Fractional Calculus |
| title_fullStr | A Method for Performing the Symmetric Anti-Difference Equations in Quantum Fractional Calculus |
| title_full_unstemmed | A Method for Performing the Symmetric Anti-Difference Equations in Quantum Fractional Calculus |
| title_short | A Method for Performing the Symmetric Anti-Difference Equations in Quantum Fractional Calculus |
| title_sort | method for performing the symmetric anti difference equations in quantum fractional calculus |
| topic | <named-content content-type="inline-formula"><inline-formula> <mml:math display="block" id="mm1111"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="fraktur">q</mml:mi> </mml:mrow> </mml:semantics> </mml:math> </inline-formula></named-content> and <named-content content-type="inline-formula"><inline-formula> <mml:math id="mm16111111"> <mml:semantics> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi mathvariant="fraktur">q</mml:mi> <mml:mo>,</mml:mo> <mml:mi mathvariant="fraktur">h</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:semantics> </mml:math> </inline-formula></named-content> difference operators quantum geometric function alpha quantum geometric function gamma functions and fractional order sum |
| url | https://www.mdpi.com/2073-8994/14/12/2604 |
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