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A unified matrix approach to the Legendre–Sheffer and certain hybrid polynomial sequences

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Abstract

The aim of this article is to study certain properties of the Legendre–Sheffer polynomials by making effective use of matrix algebra. The recursive formulas and differential equation for the Legendre–Sheffer polynomials are established by using the properties and relationships between the Pascal functional and Wronskian matrices. The corresponding results for the Legendre-associated Sheffer and Legendre–Appell families are derived. Moreover, certain examples of these polynomials are also considered which serve as a focal theme of the study. This approach provides a powerful tool for investigating the properties of the hybrid polynomial sequences.

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References

  1. Andrews, L.C. 1985. Special functions for engineers and applied mathematicians. New York: Macmillan Publishing Company.

    Google Scholar 

  2. Appell, P. 1880. Sur une classe de polynômes. Annales scientifiques de l'École normale supérieure 9: 119–144.

    Article  MathSciNet  Google Scholar 

  3. Bell, E.T. 1934. Exponential polynomials. Annals of Mathematics 35: 258–277.

    Article  MathSciNet  Google Scholar 

  4. Bretti, G., C. Cesarano, and P.E. Ricci. 2004. Laguerre-type exponentials and generalized Appell polynomials. Computers and Mathematics with Applications 48: 833–839.

    Article  MathSciNet  Google Scholar 

  5. Dattoli, G., and P.E. Ricci. 2001. A note on Legendre polynomials. International Journal of Nonlinear Sciences and Numerical Simulation 2: 365–370.

    Article  MathSciNet  Google Scholar 

  6. Dattoli, G., M. Migliorati, and H.M. Srivastava. 2007. Sheffer polynomials, monomiality principle, algebraic methods and the theory of classical polynomials. Mathematical and Computer Modelling 45 (9–10): 1033–1041.

    Article  MathSciNet  Google Scholar 

  7. Subuhi Khan, M.W., W. Al-Saad, and R. Khan. 2010. Laguerre-based Appell polynomials: Properties and applications. Mathematical and Computer Modelling 52 (1–2): 247–259.

    Article  MathSciNet  Google Scholar 

  8. Khan, S., and N. Raza. 2012. Monomiality principle, operational methods and family of Laguerre-Sheffer polynomials. Journal of Mathematical Analysis and Applications 387: 90–102.

    Article  MathSciNet  Google Scholar 

  9. Khan, S., and N. Raza. 2012. Family of Legendre-Sheffer polynomials. Mathematical and Computer Modelling 55: 969–982.

    Article  MathSciNet  Google Scholar 

  10. Kim, D.S., T., Kim, H.I. Kwon, 2016. Mansour, T. Powers under umbral composition and degeneration for Sheffer sequences, Adv. Difference Equ. 2016, Paper No. 66, 11 pp

  11. Kim, D.S., and T. Kim. 2021. Degenerate Sheffer sequences and \(\lambda \)-Sheffer sequences. Journal of Mathematical Analysis and Applications 49 (1): 21 (Paper No. 124521).

    MathSciNet  Google Scholar 

  12. Kim, T., and D.S. Kim. 2022. Degenerate Whitney numbers of first and second kind of Dowling lattices. Russian Journal of Mathematical Physics 29 (3): 358–377.

    Article  ADS  MathSciNet  Google Scholar 

  13. Kim, D.S., and T. Kim. 2015. A matrix approach to some identities involving Sheffer polynomial sequences. Applied Mathematics and Computation 253: 83–101.

    Article  MathSciNet  Google Scholar 

  14. Lahiri, M. 1971. On a generalisation of Hermite polynomials. Proceedings of the American Mathematical Society 27: 117–121.

    Article  MathSciNet  Google Scholar 

  15. Rainville, E.D. 1971. Special functions, Reprint of 1960, 1st ed. Bronx, New York: Chelsea Publishig Co.

    Google Scholar 

  16. Roman, S. 1984. The umbral calculus. New york: Academic Press.

    Google Scholar 

  17. Sheffer, I.M. 1939. Some properties of polynomial sets of type zero. Duke Mathematical Journal 5: 590–622.

    Article  MathSciNet  Google Scholar 

  18. Yang, Y., and C. Micek. 2007. Generalized Pascal functional matrix and its applications. Linear Algebra and its Applications 423: 230–245.

    Article  MathSciNet  Google Scholar 

  19. Youn, H., Y. Yang. 2011. Differential equation and recursive formulas of Sheffer polynomial sequences, ISRN Discrete Mathematics, pp. 1–16 (Article ID 476462).

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Authors and Affiliations

  1. Department of Applied Sciences, Symbiosis Institute of Technology, Symbiosis International University, Pune, 412115, India

    Shahid Ahmad Wani

  2. Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore, India

    Tabinda Nahid

  3. Department of CSE, North Campus University of Kashmir, Srinagar, India

    Khalid Hussain

  4. Department of Mathematics, College of Science, Imaam Mohammad Bin Saud University, Riyadh, Saudi Arabia

    Mdi Begum Jeelani

Authors
  1. Shahid Ahmad Wani
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  2. Tabinda Nahid
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  3. Khalid Hussain
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  4. Mdi Begum Jeelani
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Correspondence to Shahid Ahmad Wani.

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Communicated by S. Ponnusamy.

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Wani, S.A., Nahid, T., Hussain, K. et al. A unified matrix approach to the Legendre–Sheffer and certain hybrid polynomial sequences. J Anal 32, 843–867 (2024). https://doi.org/10.1007/s41478-023-00648-6

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  • DOI: https://doi.org/10.1007/s41478-023-00648-6

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