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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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    [1] D. S. Kim, T. Kim, Representing polynomials by degenerate Bernoulli polynomials, Quaest. Math., 46 (2023), 959–980. https://doi.org/10.2989/16073606.2022.2049912 doi: 10.2989/16073606.2022.2049912
    [2] T. Kim, D. S. Kim, J. Kwon, Some identities related to degenerate r-Bell and degenerate Fubini polynomials, Appl. Math. Sci. Eng., 31 (2023), 2205642. https://doi.org/10.1080/27690911.2023.2205642 doi: 10.1080/27690911.2023.2205642
    [3] T. Kim, D. S. Kim, H. K. Kim, On generalized degenerate Euler-Genocchi polynomials, Appl. Math. Sci. Eng., 31 (2023), 2159958. https://doi.org/10.1080/27690911.2022.2159958 doi: 10.1080/27690911.2022.2159958
    [4] H. M. Ahmed, Computing expansions coefficients for Laguerre polynomials, Integr. Transf. Spec. F., 32 (2021), 271–289. https://doi.org/10.1080/10652469.2020.1815727 doi: 10.1080/10652469.2020.1815727
    [5] W. M. Abd-Elhameed, S. O. Alkhamisi, New results of the fifth-kind orthogonal Chebyshev polynomials, Symmetry, 13 (2021), 2407. https://doi.org/10.3390/sym13122407 doi: 10.3390/sym13122407
    [6] W. M. Abd-Elhameed, New product and linearization formulae of Jacobi polynomials of certain parameters, Integr. Transf. Spec. F., 26 (2015), 586–599. https://doi.org/10.1080/10652469.2015.1029924 doi: 10.1080/10652469.2015.1029924
    [7] F. A. Costabile, M. I. Gualtieri, A. Napoli, Recurrence relations and determinant forms for general polynomial sequences. Application to Genocchi polynomials, Integr. Transf. Spec. F., 30 (2019), 112–127. https://doi.org/10.1080/10652469.2018.1537272 doi: 10.1080/10652469.2018.1537272
    [8] A. S. Alali, S. Ali, N. Hassan, A. M. Mahnashi, Y. Shang, A. Assiry, Algebraic structure graphs over the commutative ring ${Z}_{m}$: Exploring topological indices and entropies using $M$-polynomials, Mathematics, 11 (2023), 3833. https://doi.org/10.3390/math11183833 doi: 10.3390/math11183833
    [9] Y. Shang, A remark on the solvability of Diophantine matrix equation over ${M}_2 ({Q})$, Southeast Asian Bull. Math., 38 (2014), 275–282.
    [10] Y. Shang, A remark on the chromatic polynomials of incomparability graphs of posets, Int. J. Pure Appl. Math., 67 (2011), 159–164.
    [11] Z. Fan, W. Chu, Convolutions involving Chebyshev polynomials, Electron. J. Math., 3 (2022), 38–46. https://doi.org/10.47443/ejm.2022.012 doi: 10.47443/ejm.2022.012
    [12] W. M. Abd-Elhameed, A. Napoli, Some novel formulas of Lucas polynomials via different approaches, Symmetry, 15 (2023), 185. https://doi.org/10.3390/sym15010185 doi: 10.3390/sym15010185
    [13] T. Koshy, Fibonacci and Lucas numbers with applications, New York: John Wiley & Sons, 2019. https://doi.org/10.1002/9781118033067
    [14] W. M. Abd-Elhameed, A. N. Philippou, N. A. Zeyada, Novel results for two generalized classes of Fibonacci and Lucas polynomials and their uses in the reduction of some radicals, Mathematics, 10 (2022), 2342. https://doi.org/10.3390/math10132342 doi: 10.3390/math10132342
    [15] E. Özkan, M. Taştan, On Gauss Fibonacci polynomials, on Gauss Lucas polynomials and their applications, Commun. Algebra, 48 (2020), 952–960. https://doi.org/10.1080/00927872.2019.1670193 doi: 10.1080/00927872.2019.1670193
    [16] E. Özkan, I. Altun, Generalized Lucas polynomials and relationships between the Fibonacci polynomials and Lucas polynomials, Commun. Algebra, 47 (2019), 4020–4030. https://doi.org/10.1080/00927872.2019.1576186 doi: 10.1080/00927872.2019.1576186
    [17] T. Du, Z. Wu, Some identities involving the bi-periodic Fibonacci and Lucas polynomials, AIMS Mathematics, 8 (2023), 5838–5846. https://doi.org/10.3934/math.2023294 doi: 10.3934/math.2023294
    [18] S. Haq, I. Ali, Approximate solution of two-dimensional Sobolev equation using a mixed Lucas and Fibonacci polynomials, Eng. Comput., 38 (2022), 2059–2068. https://doi.org/10.1007/s00366-021-01327-5 doi: 10.1007/s00366-021-01327-5
    [19] A. S. Mohamed, Fibonacci collocation pseudo-spectral method of variable-order space-fractional diffusion equations with error analysis, AIMS Mathematics, 7 (2022), 14323–14337. https://doi.org/10.3934/math.2022789 doi: 10.3934/math.2022789
    [20] T. Kim, D. S. Kim, L. C. Jang, D. V. Dolgy, Representing by several orthogonal polynomials for sums of finite products of Chebyshev polynomials of the first kind and Lucas polynomials, Adv. Differ. Equ., 2019 (2019), 162. https://doi.org/10.1186/s13662-019-2092-6 doi: 10.1186/s13662-019-2092-6
    [21] D. S. Kim, D. V. Dolgy, D. Kim, T. Kim, Representing by orthogonal polynomials for sums of finite products of Fubini polynomials, Mathematics, 7 (2019), 319. https://doi.org/10.3390/math7040319 doi: 10.3390/math7040319
    [22] D. V. Dolgy, D. S. Kim, T. Kim, J. Kwon, Connection problem for sums of finite products of Chebyshev polynomials of the third and fourth kinds, Symmetry, 10 (2018), 617. https://doi.org/10.3390/sym10110617 doi: 10.3390/sym10110617
    [23] T. Kim, D. S. Kim, D. V. Dolgy, J. W. Park, Sums of finite products of Legendre and Laguerre polynomials, Adv. Differ. Equ., 2018 (2018), 277. https://doi.org/10.1186/s13662-018-1740-6 doi: 10.1186/s13662-018-1740-6
    [24] W. M. Abd-Elhameed, A. M. Alkenedri, Spectral solutions of linear and nonlinear BVPs using certain Jacobi polynomials generalizing third-and fourth-kinds of Chebyshev polynomials, CMES Comp. Model. Eng., 126 (2021), 955–989. https://doi.org/10.32604/cmes.2021.013603 doi: 10.32604/cmes.2021.013603
    [25] W. M. Abd-Elhameed, Novel expressions for the derivatives of sixth-kind Chebyshev polynomials: Spectral solution of the non-linear one-dimensional Burgers' equation, Fractal Fract., 5 (2021), 53. https://doi.org/10.3390/fractalfract5020053 doi: 10.3390/fractalfract5020053
    [26] A. Napoli, W. M. Abd-Elhameed, An innovative harmonic numbers operational matrix method for solving initial value problems, Calcolo, 54 (2017), 57–76. https://doi.org/10.1007/s10092-016-0176-1 doi: 10.1007/s10092-016-0176-1
    [27] E. Tohidi, A. H. Bhrawy, K. Erfani, A collocation method based on Bernoulli operational matrix for numerical solution of generalized pantograph equation, Appl. Math. Model., 37 (2013), 4283–4294. https://doi.org/10.1016/j.apm.2012.09.032 doi: 10.1016/j.apm.2012.09.032
    [28] W. M. Abd-Elhameed, H. M. Ahmed, Tau and Galerkin operational matrices of derivatives for treating singular and Emden-Fowler third-order-type equations, Int. J. Mod. Phys. C, 33 (2022), 2250061. https://doi.org/10.1142/S0129183122500619 doi: 10.1142/S0129183122500619
    [29] E. H. Doha, W. M. Abd-Elhameed, On the coefficients of integrated expansions and integrals of Chebyshev polynomials of third and fourth kinds, Bull. Malays. Math. Sci. Soc., 37 (2014), 383–398.
    [30] T. Koshy, Pell and P-Lucas numbers with applications, New York: Springer, 2014. https://doi.org/10.1007/978-1-4614-8489-9
    [31] A. F. Horadam, B. Swita, P. Filipponi, Integration and derivative sequences for Pell and Pell-Lucas polynomials, Fibonacci Quart., 32 (1994) 130–135.
    [32] A. F. Horadam, Chebyshev and Pell connections, Fibonacci Quart., 43 (2005), 108–121.
    [33] W. Wang, H. Wang, Some results on convolved $(p, q)$-Fibonacci polynomials, Integr. Transf. Spec. F., 26 (2015), 340–356. https://doi.org/10.1080/10652469.2015.1007502 doi: 10.1080/10652469.2015.1007502
    [34] P. M. C. Catarino, Diagonal functions of the k-Pell and $k$-Pell-Lucas polynomials and some identities, Acta Math. Univ. Comen., 87 (2018), 147–159.
    [35] D. Tasci, F. Yalcin, Vieta-Pell and Vieta-Pell-Lucas polynomials, Adv. Differ. Equ., 2013 (2013), 244. https://doi.org/10.1186/1687-1847-2013-224 doi: 10.1186/1687-1847-2013-224
    [36] W. Chu, N. N. Li, Power sums of Pell and Pell-Lucas polynomials, Fibonacci Quart., 49 (2011), 139–150.
    [37] J. P. O. Santos, M. Ivkovic, Polynomial generalizations of the Pell sequences and the Fibonacci sequence, Fibonacci Quart., 43 (2005), 328–338.
    [38] J. J. Bravo, J. L. Herrera, F. Luca, Common values of generalized Fibonacci and Pell sequences, J. Number Theory, 226 (2021), 51–71. https://doi.org/10.1016/j.jnt.2021.03.001 doi: 10.1016/j.jnt.2021.03.001
    [39] S. E. Rihane, Y. Akrour, A. E. Habibi, Fibonacci numbers which are products of three Pell numbers and Pell numbers which are products of three Fibonacci numbers, Bol. Soc. Mat. Mex., 26 (2020), 895–910. https://doi.org/10.1007/s40590-020-00296-x doi: 10.1007/s40590-020-00296-x
    [40] S. Yüzbaşı, G. Yıldırım, A collocation method to solve the parabolic-type partial integro-differential equations via Pell-Lucas polynomials, Appl. Math. Comput., 421 (2022), 126956. https://doi.org/10.1016/j.amc.2022.126956 doi: 10.1016/j.amc.2022.126956
    [41] S. Yüzbaşı, G. Yıldırım, Pell-Lucas collocation method for solving a class of second order nonlinear differential equations with variable delays, Turk. J. Math., 47 (2023), 37–55. https://doi.org/10.55730/1300-0098.3344 doi: 10.55730/1300-0098.3344
    [42] A. S. Mohamed, Pell collocation pseudo spectral scheme for one-dimensional time-fractional convection equation with error analysis, J. Funct. Space., 2022 (2022), 9734604 https://doi.org/10.1155/2022/9734604 doi: 10.1155/2022/9734604
    [43] G. E. Andrews, R. Askey, R. Roy, Special functions, Cambridge: Cambridge University Press, 1999.
    [44] W. M. Abd-Elhameed, Novel formulae of certain generalized Jacobi polynomials, Mathematics, 10 (2022), 4237. https://doi.org/10.3390/math10224237 doi: 10.3390/math10224237
    [45] J. C. Mason, D. C. Handscomb, Chebyshev polynomials, New Youk: CRC Press, 2002. https://doi.org/10.1201/9781420036114
    [46] W. M. Abd-Elhameed, H. M. Ahmed, A. Napoli, V. Kowalenko, New formulas involving Fibonacci and certain orthogonal polynomials, Symmetry, 15 (2023), 736. https://doi.org/10.3390/sym15030736 doi: 10.3390/sym15030736
    [47] W. Koepf, Hypergeometric summation: An algorithmic approach to summation and special function identities, 2 Eds., London: Springer, 2014. https://doi.org/10.1007/978-1-4471-6464-7
    [48] M. van Hoeij, Finite singularities and hypergeometric solutions of linear recurrence equations, J. Pure Appl. Algebra, 139 (1999), 109–131. https://doi.org/10.1016/S0022-4049(99)00008-0 doi: 10.1016/S0022-4049(99)00008-0
    [49] J. M. Campbell, A Wilf-Zeilberger-based solution to the Basel problem with applications, Discrete Math. Lett., 10 (2022), 21–27. https://doi.org/10.47443/dml.2022.030 doi: 10.47443/dml.2022.030
    [50] J. M. Campbell, Applications of Zeilberger's algorithm to Ramanujan-inspired series involving harmonic-type numbers, Discrete Math. Lett., 11 (2023), 7–13. https://doi.org/10.47443/dml.2022.050 doi: 10.47443/dml.2022.050
    [51] D. A. Wolfram, Solving recurrences for Legendre-Bernstein basis transformations, Examples and Counterexamples, 4 (2023), 100117. https://doi.org/10.1016/j.exco.2023.100117 doi: 10.1016/j.exco.2023.100117
    [52] G. B. Djordjevic, G. V. Milovanovic, Special classes of polynomials, Leskovac: University of Nis, Faculty of Technology, 2014.
    [53] F. A. Costabile, Modern umbral calculus: An elementary introduction with applications of linear interpolation and operator approximation theory, Berlin, Boston: De Gruyter, 2019. https://doi.org/10.1515/9783110652925
    [54] X. G. Lv, T. Z. Huang, A note on inversion of Toeplitz matrices, Appl. Math. Lett., 20 (2007), 1189–1193. https://doi.org/10.1016/j.aml.2006.10.008 doi: 10.1016/j.aml.2006.10.008
    [55] P. G. Martinsson, V. Rokhlin, M. Tygert, A fast algorithm for the inversion of general Toeplitz matrices, Comput. Math. Appl., 50 (2005), 741–752. https://doi.org/10.1016/j.camwa.2005.03.011 doi: 10.1016/j.camwa.2005.03.011
    [56] M. K. Ng, K. Rost, Y. W. Wen, On inversion of Toeplitz matrices, Linear Algebra Appl., 348 (2002), 145–151. https://doi.org/10.1016/S0024-3795(01)00592-4 doi: 10.1016/S0024-3795(01)00592-4
    [57] F. A. Costabile, M. I. Gualtieri, A. Napoli, Polynomial sequences: Basic methods, special classes, and computational applications, Berlin: De Gruyter, 2023.
    [58] N. D. Cahill, J. R. D'Errico, D. A. Narayan, J. Y. Narayan, Fibonacci determinants, The College Mathematics Journal, 33 (2002), 221–225. https://doi.org/10.1080/07468342.2002.11921945
    [59] N. J. Higham, Accuracy and stability of numerical algorithms, Philadelphia: SIAM, 2002.
    [60] R. T. Farouki, The Bernstein polynomial basis: A centennial retrospective, Comput. Aided Geom. D., 29 (2012), 379–419. https://doi.org/10.1016/j.cagd.2012.03.001 doi: 10.1016/j.cagd.2012.03.001
    [61] H. Zhang, Numerical condition of polynomials in different forms, Electron. T. Numer. Anal., 12 (2001), 66–87.
    [62] R. T. Farouki, V. T. Rajan, On the numerical condition of polynomials in Bernstein form, Comput. Aided Geom. D., 4 (1987), 191–216. https://doi.org/10.1016/0167-8396(87)90012-4 doi: 10.1016/0167-8396(87)90012-4
    [63] W. Gautschi, On the condition of algebraic equations, Numer. Math., 21 (1973), 405–424. https://doi.org/10.1007/BF01436491 doi: 10.1007/BF01436491
    [64] J. H. Wilkinson, The perfidious polynomial, Studies in Numerical Analysis, 24 (1984), 1–28.
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