Exploring the Characteristics of Δ<sub><i>h</i></sub> Bivariate Appell Polynomials: An In-Depth Investigation and Extension through Fractional Operators
The objective of this article is to introduce the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mo>∆</mo><mi>h</mi></msub></semantics></math></inline-formula...
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| Format: | Article |
| Language: | English |
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MDPI AG
2024-01-01
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| Series: | Fractal and Fractional |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2504-3110/8/1/67 |
| _version_ | 1827372292459462656 |
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| author | Musawa Yahya Almusawa |
| author_facet | Musawa Yahya Almusawa |
| author_sort | Musawa Yahya Almusawa |
| collection | DOAJ |
| description | The objective of this article is to introduce the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mo>∆</mo><mi>h</mi></msub></semantics></math></inline-formula> bivariate Appell polynomials <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow></mrow><msub><mo>∆</mo><mi>h</mi></msub></msub><msubsup><mi mathvariant="double-struck">A</mi><mi>s</mi><mrow><mo>[</mo><mi>r</mi><mo>]</mo></mrow></msubsup><mrow><mo>(</mo><mi>λ</mi><mo>,</mo><mi>η</mi><mo>;</mo><mi>h</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and their extended form via fractional operators. The study described in this paper follows the line of study in which the monomiality principle is used to develop new results. It is further discovered that these polynomials satisfy various well-known fundamental properties and explicit forms. The explicit series representation of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mo>∆</mo><mi>h</mi></msub></semantics></math></inline-formula> bivariate Gould–Hopper polynomials is first obtained, and, using this outcome, the explicit series representation of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mo>∆</mo><mi>h</mi></msub></semantics></math></inline-formula> bivariate Appell polynomials is further given. The quasimonomial properties fulfilled by bivariate Appell polynomials <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mo>∆</mo><mi>h</mi></msub></semantics></math></inline-formula> are also proved by demonstrating that the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mo>∆</mo><mi>h</mi></msub></semantics></math></inline-formula> bivariate Appell polynomials exhibit certain properties related to their behavior under multiplication and differentiation operators. The determinant form of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mo>∆</mo><mi>h</mi></msub></semantics></math></inline-formula> bivariate Appell polynomials is provided, and symmetric identities for the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mo>∆</mo><mi>h</mi></msub></semantics></math></inline-formula> bivariate Appell polynomials are also exhibited. By employing the concept of the forward difference operator, operational connections are established, and certain applications are derived. Different Appell polynomial members can be generated by using appropriate choices of functions in the generating expression obtained in this study for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mo>∆</mo><mi>h</mi></msub></semantics></math></inline-formula> bivariate Appell polynomials. Additionally, generating relations for the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mo>∆</mo><mi>h</mi></msub></semantics></math></inline-formula> bivariate Bernoulli and Euler polynomials, as well as for Genocchi polynomials, are established, and corresponding results are obtained for those polynomials. |
| first_indexed | 2024-03-08T10:55:29Z |
| format | Article |
| id | doaj.art-9bdcf224a6a9490bb6b2e6b5a88b566c |
| institution | Directory Open Access Journal |
| issn | 2504-3110 |
| language | English |
| last_indexed | 2024-03-08T10:55:29Z |
| publishDate | 2024-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj.art-9bdcf224a6a9490bb6b2e6b5a88b566c2024-01-26T16:36:38ZengMDPI AGFractal and Fractional2504-31102024-01-01816710.3390/fractalfract8010067Exploring the Characteristics of Δ<sub><i>h</i></sub> Bivariate Appell Polynomials: An In-Depth Investigation and Extension through Fractional OperatorsMusawa Yahya Almusawa0Department of Mathematics, College of Science, Jazan University, P.O. Box. 114, Jazan 45142, Saudi ArabiaThe objective of this article is to introduce the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mo>∆</mo><mi>h</mi></msub></semantics></math></inline-formula> bivariate Appell polynomials <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow></mrow><msub><mo>∆</mo><mi>h</mi></msub></msub><msubsup><mi mathvariant="double-struck">A</mi><mi>s</mi><mrow><mo>[</mo><mi>r</mi><mo>]</mo></mrow></msubsup><mrow><mo>(</mo><mi>λ</mi><mo>,</mo><mi>η</mi><mo>;</mo><mi>h</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and their extended form via fractional operators. The study described in this paper follows the line of study in which the monomiality principle is used to develop new results. It is further discovered that these polynomials satisfy various well-known fundamental properties and explicit forms. The explicit series representation of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mo>∆</mo><mi>h</mi></msub></semantics></math></inline-formula> bivariate Gould–Hopper polynomials is first obtained, and, using this outcome, the explicit series representation of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mo>∆</mo><mi>h</mi></msub></semantics></math></inline-formula> bivariate Appell polynomials is further given. The quasimonomial properties fulfilled by bivariate Appell polynomials <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mo>∆</mo><mi>h</mi></msub></semantics></math></inline-formula> are also proved by demonstrating that the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mo>∆</mo><mi>h</mi></msub></semantics></math></inline-formula> bivariate Appell polynomials exhibit certain properties related to their behavior under multiplication and differentiation operators. The determinant form of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mo>∆</mo><mi>h</mi></msub></semantics></math></inline-formula> bivariate Appell polynomials is provided, and symmetric identities for the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mo>∆</mo><mi>h</mi></msub></semantics></math></inline-formula> bivariate Appell polynomials are also exhibited. By employing the concept of the forward difference operator, operational connections are established, and certain applications are derived. Different Appell polynomial members can be generated by using appropriate choices of functions in the generating expression obtained in this study for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mo>∆</mo><mi>h</mi></msub></semantics></math></inline-formula> bivariate Appell polynomials. Additionally, generating relations for the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mo>∆</mo><mi>h</mi></msub></semantics></math></inline-formula> bivariate Bernoulli and Euler polynomials, as well as for Genocchi polynomials, are established, and corresponding results are obtained for those polynomials.https://www.mdpi.com/2504-3110/8/1/67Δ<sub><i>h</i></sub> special polynomialsfractional operatorsAppell polynomialsmonomiality principleoperational formalismdeterminant form |
| spellingShingle | Musawa Yahya Almusawa Exploring the Characteristics of Δ<sub><i>h</i></sub> Bivariate Appell Polynomials: An In-Depth Investigation and Extension through Fractional Operators Fractal and Fractional Δ<sub><i>h</i></sub> special polynomials fractional operators Appell polynomials monomiality principle operational formalism determinant form |
| title | Exploring the Characteristics of Δ<sub><i>h</i></sub> Bivariate Appell Polynomials: An In-Depth Investigation and Extension through Fractional Operators |
| title_full | Exploring the Characteristics of Δ<sub><i>h</i></sub> Bivariate Appell Polynomials: An In-Depth Investigation and Extension through Fractional Operators |
| title_fullStr | Exploring the Characteristics of Δ<sub><i>h</i></sub> Bivariate Appell Polynomials: An In-Depth Investigation and Extension through Fractional Operators |
| title_full_unstemmed | Exploring the Characteristics of Δ<sub><i>h</i></sub> Bivariate Appell Polynomials: An In-Depth Investigation and Extension through Fractional Operators |
| title_short | Exploring the Characteristics of Δ<sub><i>h</i></sub> Bivariate Appell Polynomials: An In-Depth Investigation and Extension through Fractional Operators |
| title_sort | exploring the characteristics of δ sub i h i sub bivariate appell polynomials an in depth investigation and extension through fractional operators |
| topic | Δ<sub><i>h</i></sub> special polynomials fractional operators Appell polynomials monomiality principle operational formalism determinant form |
| url | https://www.mdpi.com/2504-3110/8/1/67 |
| work_keys_str_mv | AT musawayahyaalmusawa exploringthecharacteristicsofdsubihisubbivariateappellpolynomialsanindepthinvestigationandextensionthroughfractionaloperators |