10.57647/j.jtap.2025.1904.38

A hypergeometric sequence of biorthogonal polynomials and its relationship with Meixner-Pollaczek polynomials

Show Details XML PDF
  • Mohammad Masjed-Jamei*1,
  • Nasser Saad2,
  1. Faculty of Mathematics, K. N. Toosi University of Technology, Tehran, Iran
  2. School of Mathematical and Computational Sciences, University of Prince Edward Island, Charlottetown, Canada

Received: 2025-05-04

Revised: 2025-07-23

Accepted: 2025-08-14

Published in Issue 2025-08-31

Copyright (c) 2025 Mohammad Masjed-Jamei, Nasser Saad (Author)

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

How to Cite

1.
Masjed-Jamei M, Saad N. A hypergeometric sequence of biorthogonal polynomials and its relationship with Meixner-Pollaczek polynomials. J Theor Appl phys. 2025 Aug. 31;19(4). Available from: https://oiccpress.com/jtap/article/view/17651
Crossref
Scopus
Google Scholar
Europe PMC

PDF views: 139

Abstract

Some orthogonal polynomial systems are mapped onto each other by the Fourier transform. In this paper, based on generalized Laguerre polynomials, we introduce a new sequence of biorthogonal hypergeometric polynomials and obtain its orthogonality relation explicitly. In this sense, we also recall some similar results available in the literature as the sequence introduced in this work is the last one for classical cases.

Keywords

  • Generalized Laguerre polynomials,
  • Meixner-Pollaczek polynomials,
  • Fourier transform,
  • Parseval identity,
  • Hypergeometric series
PDF

References

  1. G. E. Andrews, R. Askey, R. Roy, Special Functions. Encyclopedia of Mathematics and its Applications 71, Cambridge University Press, Cambridge, 1999.
  2. R. Askey, Continuous Hahn polynomials, J. Physics A 18, 1985, L1017-L1019.
  3. N. M. Atakishiyev, S. K Suslov, The Hahn and Meixner polynomials of an imaginary argument and some of their applications. J. Physics A 18, 1985, 1583-1596.
  4. W. N. Bailey, Generalized Hypergeometric Series. Cambridge Tracts 32, Cambridge University PFTV, 1935. Reprinted by Hafner Publishing Company, 1972.
  5. K. Castillo, D. Mbouna, J. Petronilho, On the functional equation for classical orthogonal polynomials on lattices, J. Math. Anal. Appl., 515 (2022) 126390.
  6. K. Castillo, J. Petronilho, Classical orthogonal polynomials revisited, Results Math., 78, 155 (2023).
  7. K. Castillo, J. Petronilho, A First Course on Orthogonal Polynomials: Classical Orthogonal Polynomials and Related Topics, CRC Press, Chapman & Hall, New York, 2025.
  8. W. Groenevelt, The Wilson function transform, Int. Math. Res. Not. (2003), no. 52, 2779-2817.
  9. H. T. Koelink, On Jacobi and continuous Hahn polynomials. Proc. Amer. Math. Soc. 124 (1996), 887-898.
  10. T. H. Koornwinder, Meixner-Pollaczek polynomials and the Heisenberg algebra, J. Math. Phys. 30 (1989), 767-769.
  11. T. H. Koornwinder, Special orthogonal polynomial systems mapped onto each other by the Fourier-Jacobi transform, Polynômes Orthogonauxet Applications (C. Brezinski, A. Draux, A. P. Magnus, P. Maroni and A. Ronveaux, Eds.), Lecture Notes Math. 1171, Springer, 1985, 174-183.
  12. W. Koepf, M. Masjed-Jamei, Two classes of special functions using Fourier transforms of some finite classes of classical orthogonal polynomials, Proc. Amer. Math. Soc. 135 (2007), 3599-3606.
  13. M. Masjed-Jamei, Biorthogonal exponential sequences with weight function exp(ax^2+ibx) on the real line and an orthogonal sequence of trigonometric functions, Proc. Amer. Math. Soc. 136 (2008), 409-417.
  14. M. Masjed-Jamei, W. Koepf, Two classes of special functions using Fourier transforms of generalized ultraspherical and generalized Hermite polynomials, Proc. Amer. Math. Soc. 140 (2012), 2053-2063.
  15. M. Masjed-Jamei, F. Marcellan, E. J. Huertas, A finite class of orthogonal functions generated by Routh-Romanovski polynomials, Complex Variables and Elliptic Equations, 59 (2014), 162-171.
  16. M. Masjed-Jamei, An improvement of least squares theory: Theory of least p-variances approximation and p-uncorrelated functions, Mathematics, 13 (2025), Article 2255, 89 pages.
  17. B. Mekuye, B. Abera, Nanomaterials: An overview of synthesis, classification, characterization, and applications. Nano Select. 4, 486 (2023). DOI: 10.1002/nano.202300038
  18. P. Agale, V. Salve, S. Mardikar, S. Patange, P. More, Synthesis and characterization of hierarchical Sr-doped ZnO hexagonal nanodisks as an efficient photocatalyst for the degradation of methylene blue dye under sunlight irradiation. Appl. Surf. Sci. 672, 160795 (2024). DOI: 10.1016/j.apsusc.2024.160795
  19. A. M. Al-Fa'ouri, M. Abu-Kharma and A. Awwad. Green Synthesis of Copper Oxide Nanoparticles Using Bougainvillea Leaves Aqueous Extract and Antibacterial Activity Evaluation, Arabian Journal of Chemistry. 16, 104535 (2023). DOI: 10.5281/zenodo.4899095
  20. M. Al-Fa'ouri, O. A. Lafi, H. H. Abu-Safe and M. Abu-Kharma. Investigation of optical and electrical properties of copper oxide - polyvinyl alcohol nanocomposites for solar cell applications, Arabian Journal of Chemistry. 16, 104535 (2023). DOI: 10.1016/j.arabjc.2022.104535
  21. M. Ahamed, H. A. Alhadlaq, M. A. M. Khan, P. Karuppiah and N. A. Al-Dhabi. Synthesis, Characterization, and Antimicrobial Activity of Copper Oxide Nanoparticles. Journal of Nanomaterials. 2014, 637858 (2014). DOI: 10.1155/2014/637858
  22. S. Sahu, P. K. Samanta, Microstructural study and crystallite size analysis of chemically grown bougainvillea flower-like zinc oxide nanostructures. Materials Today Proceedings. 65 (5) 2502 (2022). DOI: 10.1016/j.matpr.2022.04.474
  23. Matei, A., Craciun, G., Romanitan, C., Pachiu, C., & Tucureanu, V. (2023). Biosynthesis and Characterization of Copper Oxide Nanoparticles. Engineering Proceedings, 37(1), 54. DOI:10.3390/ECP2023-14629
  24. D. M. Nzilu, E. S. Madivoli, D. S. Makhanu, S. I. Wanakai, G. K. Kiprono and P. G. Kareru. Green synthesis of copper oxide nanoparticles and its efficiency in degradation of rifampicin antibiotic. Sci Rep 13, 14030 (2023). DOI: 10.1038/s41598-023-41119-z
  25. P. Bhattacharya, S. Swarnakar , S. Ghosh, S. Majumdar and S. Banerjee , J Environ . Disinfection of drinking water via algae mediated green synthesized copper oxide nanoparticles and its toxicity evaluation. Journal of Environmental Chemical Engineering, 7, 102867 (2019). DOI:10.1016/j.jece.2018.102867
  26. P. Mallick and S. Sahu. Structure, Microstructure and Optical Absorption Analysis of CuO Nanoparticles Synthesized by Sol-Gel Route. Nanoscience and Nanotechnology, 2(3) 71 (2012). DOI: doi: 10.5923/j.nn.20120203.05
  27. A. Bouafia, S. E. Laouini and M. R. Ouahrani. A Review on Green Synthesis of CuO Nanoparticles using Plant Extract and Evaluation of Antimicrobial Activity . Asian J Res Chem 13, 65 (2020). DOI: 10.5958/0974-4150.2020.00014.0
  28. M. Chen, Y. He, Q. Ye, Z. Zhang and Y. Hu Y, Experimental study on convective heat transfer of a novel helical coil heat exchanger with porous media , Int J Heat Mass Transf 130, 1133 (2019). DOI: 10.1016/j.ijheatmasstransfer.2018.10.111J.
  29. J. Zhang, H. Wang, Q. Wang, Z. Sun, B. Buhe, Y. Sun, C. Meng, Facile fabrication of a spindle-like CuO anchored on Graphene nanocomposite with relative high-rate performance and cycle stability for flexible supercapacitor. J Mater Sci: Mater Electron 33, 11143 (2022). DOI: 10.1007/s10854-022-08182-4
  30. D. Cao, X. Shu, D. Zhu, S. Liang, M. Hasan and S. Gong. Green synthesis of zinc oxide nanoparticles and their application in biomedical fields. Nano Convergence 7, 14 (2020). DOI: 10.1186/s40580-020-00226-6
  31. S. V. Gupta, V. V. Kulkarni and M. Ahmaruzzaman . Biosynthesis of Silver Nanoparticles from Bauhinia variegata Linn. Leaf Extract and Their Antibacterial Activity against Multidrug-Resistant Clinical Pathogens. Sci Rep 13, 3009 (2023). DOI: 10.1038/s41598-023-29472-3.
  32. A. Debart,L. Dupont, P. Poizot, J. Leriche and J. Tarascon . A Transmission Electron Microscopy Study of the Reactivity Mechanism of Tailor-Made CuO Particles Toward Lithium. J Electrochem Soc 148, A1266 (2001). DOI: 10.1149/1.1409971.