Tbilisi Mathematical Journal

Some properties of the pseudo-Chebyshev polynomials of half-integer degree

Primo Brandi and Paolo Emilio Ricci

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New sets of orthogonal functions, derived from the first and second kind Chebyshev polynomials, considering half-integer indexes, have been recently introduced. In this article several properties of these new sets are considered and the links with the classical Chebyshev polynomials are underlined.

Article information

Tbilisi Math. J., Volume 12, Issue 4 (2019), 111-121.

Received: 29 May 2019
Accepted: 20 October 2019
First available in Project Euclid: 3 January 2020

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Mathematical Reviews number (MathSciNet)

Primary: 33C45: Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) [See also 42C05 for general orthogonal polynomials and functions]
Secondary: 33D45: Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.) 42C05: Orthogonal functions and polynomials, general theory [See also 33C45, 33C50, 33D45] 42A20: Convergence and absolute convergence of Fourier and trigonometric series

Chebyshev polynomials pseudo-Chebyshev polynomials recurrence relations composition properties orthogonality property


Brandi, Primo; Ricci, Paolo Emilio. Some properties of the pseudo-Chebyshev polynomials of half-integer degree. Tbilisi Math. J. 12 (2019), no. 4, 111--121. doi:10.32513/tbilisi/1578020571. https://projecteuclid.org/euclid.tbilisi/1578020571

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  • K. Aghigh, M. Masjed-Jamei and M. Dehghan, A survey on third and fourth kind of Chebyshev polynomials and their applications, Appl. Math. Comput., 199 (1), (2008), 2–12.
  • J. P. Boyd, Chebyshev and Fourier Spectral Methods, 2nd ed., Dover, Mineola, 2001.
  • L. Carleson, On convergence and growth of partial sums of Fourier series, Acta Math., 116 (1966), 135–157.
  • C. Cesarano, S. Pinelas and P. E. Ricci, The third and fourth kind pseudo-Chebyshev polynomials of half-integer degree, Symmetry (2019), 11, 274; \hfill doi: 10.3390/sym11020274
  • E. H. Doha, W. M. Abd-Elhameed and M. M. Alsuyuti, On using third and fourth kinds Chebyshev polynomials for solving the integrated forms of high odd-order linear boundary value problems, J. Egypt. Math. Soc., (2014), (to appear).
  • Ravelo J. García, R. Cuevas, A. Queijeiro, J. J. Pe$\rm \tilde n$a and J. Morales, Chebyshev functions of half-integer order, Integral Transforms Spec. Funct., 18, (2007), 743–749.
  • J. Gielis, The Geometrical Beauty of Plants, Atlantis Press, Springer Nature, (2017).
  • T. Kim, D. S. Kim, D. V. Dolgy and J. Kwon, Sums of finite products of Chebyshev polynomials of the third and fourth kinds, Adv. Difference Equ., (2018), paper no. 283, 17 pp.
  • W. Koepf and D. Schmersau, Representations of orthogonal polynomials, J. Comput. Appl. Math., 90 (1998,) 57–94.
  • J. C. Mason and D. C. Handscomb, Chebyshev Polynomials, Chapman and Hall, New York, NY, CRC, Boca Raton, 2003.
  • P. E. Ricci, Alcune osservazioni sulle potenze delle matrici del secondo ordine e sui polinomi di Tchebycheff di seconda specie, Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur., 109 (1975), 405–410.
  • P. E. Ricci, Sulle potenze di una matrice, Rend. Mat. (6) 9, (1976), 179–194.
  • P. E. Ricci, Una proprietà iterativa dei polinomi di Chebyshev di prima specie in più variabili, Rend. Mat. Appl., (7) 6, (1986), 555–563.
  • P. E. Ricci, Complex spirals and pseudo-Chebyshev polynomials of fractional degree, Symmetry (2018), 10, 671.
  • T. J. Rivlin, The Chebyshev polynomials, J. Wiley, 1974.
  • H. M. Srivastava and H. L. Manocha, A Treatise on Generating Functions, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbane and Toronto, 1984.
  • H. M. Srivastava, P. E. Ricci and P. Natalini, A Family of Complex Appell Polynomial Sets, Rev. Real Acad. Sci. Exact. Fis Nat., Ser. A Math. (2018), https://doi.org/10.1007/s13398-018-00622-z