On Convolved Fibonacci Polynomials
This work delves deeply into convolved Fibonacci polynomials (CFPs) that are considered generalizations of the standard Fibonacci polynomials. We present new formulas for these polynomials. An expression for the repeated integrals of the CFPs in terms of their original polynomials is given. A new ap...
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MDPI AG
2024-12-01
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| Series: | Mathematics |
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| author | Waleed Mohamed Abd-Elhameed Omar Mazen Alqubori Anna Napoli |
| author_facet | Waleed Mohamed Abd-Elhameed Omar Mazen Alqubori Anna Napoli |
| author_sort | Waleed Mohamed Abd-Elhameed |
| collection | DOAJ |
| description | This work delves deeply into convolved Fibonacci polynomials (CFPs) that are considered generalizations of the standard Fibonacci polynomials. We present new formulas for these polynomials. An expression for the repeated integrals of the CFPs in terms of their original polynomials is given. A new approach is followed to obtain the higher-order derivatives of these polynomials from the repeated integrals formula. The inversion and moment formulas for these polynomials, which we find, are the keys to developing further formulas for these polynomials. The derivatives of the moments of the CFPs in terms of their original polynomials and different symmetric and non-symmetric polynomials are also derived. New product formulas of these polynomials with some polynomials, including the linearization formulas of these polynomials, are also deduced. Some closed forms for definite and weighted definite integrals involving the CFPs are found as consequences of some of the introduced formulas. |
| format | Article |
| id | doaj-art-500618ca9bcb4c6088c6f51fdf1fe6f3 |
| institution | Kabale University |
| issn | 2227-7390 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj-art-500618ca9bcb4c6088c6f51fdf1fe6f32025-01-10T13:17:59ZengMDPI AGMathematics2227-73902024-12-011312210.3390/math13010022On Convolved Fibonacci PolynomialsWaleed Mohamed Abd-Elhameed0Omar Mazen Alqubori1Anna Napoli2Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, EgyptDepartment of Mathematics and Statistics, College of Science, University of Jeddah, Jeddah 23831, Saudi ArabiaDepartment of Mathematics and Computer Science, University of Calabria, 87036 Rende, ItalyThis work delves deeply into convolved Fibonacci polynomials (CFPs) that are considered generalizations of the standard Fibonacci polynomials. We present new formulas for these polynomials. An expression for the repeated integrals of the CFPs in terms of their original polynomials is given. A new approach is followed to obtain the higher-order derivatives of these polynomials from the repeated integrals formula. The inversion and moment formulas for these polynomials, which we find, are the keys to developing further formulas for these polynomials. The derivatives of the moments of the CFPs in terms of their original polynomials and different symmetric and non-symmetric polynomials are also derived. New product formulas of these polynomials with some polynomials, including the linearization formulas of these polynomials, are also deduced. Some closed forms for definite and weighted definite integrals involving the CFPs are found as consequences of some of the introduced formulas.https://www.mdpi.com/2227-7390/13/1/22Fibonacci polynomialsgeneralized polynomialsrecurrence formulaslinearization and connection coefficientsdefinite integrals |
| spellingShingle | Waleed Mohamed Abd-Elhameed Omar Mazen Alqubori Anna Napoli On Convolved Fibonacci Polynomials Mathematics Fibonacci polynomials generalized polynomials recurrence formulas linearization and connection coefficients definite integrals |
| title | On Convolved Fibonacci Polynomials |
| title_full | On Convolved Fibonacci Polynomials |
| title_fullStr | On Convolved Fibonacci Polynomials |
| title_full_unstemmed | On Convolved Fibonacci Polynomials |
| title_short | On Convolved Fibonacci Polynomials |
| title_sort | on convolved fibonacci polynomials |
| topic | Fibonacci polynomials generalized polynomials recurrence formulas linearization and connection coefficients definite integrals |
| url | https://www.mdpi.com/2227-7390/13/1/22 |
| work_keys_str_mv | AT waleedmohamedabdelhameed onconvolvedfibonaccipolynomials AT omarmazenalqubori onconvolvedfibonaccipolynomials AT annanapoli onconvolvedfibonaccipolynomials |