Novel Results for Two Generalized Classes of Fibonacci and Lucas Polynomials and Their Uses in the Reduction of Some Radicals
The goal of this study is to develop some new connection formulae between two generalized classes of Fibonacci and Lucas polynomials. Hypergeometric functions of the kind <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics&g...
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MDPI AG
2022-07-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/10/13/2342 |
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| author | Waleed Mohamed Abd-Elhameed Andreas N. Philippou Nasr Anwer Zeyada |
| author_facet | Waleed Mohamed Abd-Elhameed Andreas N. Philippou Nasr Anwer Zeyada |
| author_sort | Waleed Mohamed Abd-Elhameed |
| collection | DOAJ |
| description | The goal of this study is to develop some new connection formulae between two generalized classes of Fibonacci and Lucas polynomials. Hypergeometric functions of the kind <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><mi>z</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> are included in all connection coefficients for a specific <i>z</i>. Several new connection formulae between some famous polynomials, such as Fibonacci, Lucas, Pell, Fermat, Pell–Lucas, and Fermat–Lucas polynomials, are deduced as special cases of the derived connection formulae. Some of the introduced formulae generalize some of those existing in the literature. As two applications of the derived connection formulae, some new formulae linking some celebrated numbers are given and also some newly closed formulae of certain definite weighted integrals are deduced. Based on using the two generalized classes of Fibonacci and Lucas polynomials, some new reduction formulae of certain odd and even radicals are developed. |
| first_indexed | 2024-03-09T10:27:45Z |
| format | Article |
| id | doaj.art-f5bd988fbd074f8a9e873ce538ed9a54 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T10:27:45Z |
| publishDate | 2022-07-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-f5bd988fbd074f8a9e873ce538ed9a542023-12-01T21:35:42ZengMDPI AGMathematics2227-73902022-07-011013234210.3390/math10132342Novel Results for Two Generalized Classes of Fibonacci and Lucas Polynomials and Their Uses in the Reduction of Some RadicalsWaleed Mohamed Abd-Elhameed0Andreas N. Philippou1Nasr Anwer Zeyada2Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, EgyptDepartment of Mathematics, University of Patras, 26504 Patras, GreeceDepartment of Mathematics, Faculty of Science, Cairo University, Giza 12613, EgyptThe goal of this study is to develop some new connection formulae between two generalized classes of Fibonacci and Lucas polynomials. Hypergeometric functions of the kind <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><mi>z</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> are included in all connection coefficients for a specific <i>z</i>. Several new connection formulae between some famous polynomials, such as Fibonacci, Lucas, Pell, Fermat, Pell–Lucas, and Fermat–Lucas polynomials, are deduced as special cases of the derived connection formulae. Some of the introduced formulae generalize some of those existing in the literature. As two applications of the derived connection formulae, some new formulae linking some celebrated numbers are given and also some newly closed formulae of certain definite weighted integrals are deduced. Based on using the two generalized classes of Fibonacci and Lucas polynomials, some new reduction formulae of certain odd and even radicals are developed.https://www.mdpi.com/2227-7390/10/13/2342generalized Fibonacci and generalized Lucas numbersLucas and Fibonacci numbersrecurrence relationradicals reduction |
| spellingShingle | Waleed Mohamed Abd-Elhameed Andreas N. Philippou Nasr Anwer Zeyada Novel Results for Two Generalized Classes of Fibonacci and Lucas Polynomials and Their Uses in the Reduction of Some Radicals Mathematics generalized Fibonacci and generalized Lucas numbers Lucas and Fibonacci numbers recurrence relation radicals reduction |
| title | Novel Results for Two Generalized Classes of Fibonacci and Lucas Polynomials and Their Uses in the Reduction of Some Radicals |
| title_full | Novel Results for Two Generalized Classes of Fibonacci and Lucas Polynomials and Their Uses in the Reduction of Some Radicals |
| title_fullStr | Novel Results for Two Generalized Classes of Fibonacci and Lucas Polynomials and Their Uses in the Reduction of Some Radicals |
| title_full_unstemmed | Novel Results for Two Generalized Classes of Fibonacci and Lucas Polynomials and Their Uses in the Reduction of Some Radicals |
| title_short | Novel Results for Two Generalized Classes of Fibonacci and Lucas Polynomials and Their Uses in the Reduction of Some Radicals |
| title_sort | novel results for two generalized classes of fibonacci and lucas polynomials and their uses in the reduction of some radicals |
| topic | generalized Fibonacci and generalized Lucas numbers Lucas and Fibonacci numbers recurrence relation radicals reduction |
| url | https://www.mdpi.com/2227-7390/10/13/2342 |
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