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AIMS Mathematics

2024, Volume 9, Issue 1: 565-593. doi: 10.3934/math.2024030
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New formulas of convolved Pell polynomials

  • 1.
    Department of Mathematics and Statistics, College of Science, University of Jeddah, Jeddah, Saudi Arabia
  • 2.
    Department of Mathematics and Computer Science, University of Calabria, Rende (Cs), Italy
  • Received: 13 October 2023 Revised: 11 November 2023 Accepted: 14 November 2023 Published: 01 December 2023

  • MSC : 33C20, 33C45

  • The article investigates a class of polynomials known as convolved Pell polynomials. This class generalizes the standard class of Pell polynomials. New formulas related to convolved Pell polynomials are established. These formulas may be useful in different applications, in particular in numerical analysis. New expressions are derived for the high-order derivatives of these polynomials, both in terms of their original polynomials and in terms of various well-known polynomials. As special cases, connection formulas linking the convolved Pell polynomials with some other polynomials can be deduced. The new moments formula of the convolved Pell polynomials that involves a terminating hypergeometric function of the unit argument is given. Then, some reduced specific moment formulas are deduced based on the reduction formulas of some hypergeometric functions. Some applications, including new specific definite and weighted definite integrals, are deduced based on some of the developed formulas. Finally, a matrix approach for this kind of polynomial is presented.

    Citation: Waleed Mohamed Abd-Elhameed, Anna Napoli. New formulas of convolved Pell polynomials[J]. AIMS Mathematics, 2024, 9(1): 565-593. doi: 10.3934/math.2024030

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  • The article investigates a class of polynomials known as convolved Pell polynomials. This class generalizes the standard class of Pell polynomials. New formulas related to convolved Pell polynomials are established. These formulas may be useful in different applications, in particular in numerical analysis. New expressions are derived for the high-order derivatives of these polynomials, both in terms of their original polynomials and in terms of various well-known polynomials. As special cases, connection formulas linking the convolved Pell polynomials with some other polynomials can be deduced. The new moments formula of the convolved Pell polynomials that involves a terminating hypergeometric function of the unit argument is given. Then, some reduced specific moment formulas are deduced based on the reduction formulas of some hypergeometric functions. Some applications, including new specific definite and weighted definite integrals, are deduced based on some of the developed formulas. Finally, a matrix approach for this kind of polynomial is presented.


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