Skip to main content
SpringerLink
Log in

New Explicitly Diagonalizable Hankel Matrices Related to the Stieltjes–Carlitz Polynomials

  • Published:
Integral Equations and Operator Theory Aims and scope Submit manuscript

Abstract

Four new examples of explicitly diagonalizable Hankel matrices depending on a parameter \(k\in (0,1)\) are presented. The Hankel matrices are regarded as matrix operators on the Hilbert space \(\ell ^{2}(\mathbb {N}_{0})\) and the solution of the spectral problem is based on an application of the commutator method. Each of the Hankel matrices commutes with a Jacobi matrix which is related to a particular family of the Stieltjes–Carlitz polynomials. More examples of explicitly diagonalizable structured matrix operators are obtained when taking into account also weighted Hankel matrices.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover Publications, New York (1972)

    MATH  Google Scholar 

  2. Akhiezer, N.I.: Elements of the Theory of Elliptic Functions, American Mathematical Society, Providence, RI (1990). Translated from the second Russian edition by H.H. McFaden

  3. Beckermann, B.: Complex Jacobi matrices. J. Comput. Appl. Math. 127, 17–65 (2001)

    Article  MathSciNet  Google Scholar 

  4. Carlitz, L.: Some orthogonal polynomials related to elliptic functions. Duke Math. J. 27, 443–459 (1960)

    MathSciNet  MATH  Google Scholar 

  5. Chihara, T.S.: An Introduction to Orthogonal Polynomials. Gordon and Breach Science Publishers, New York (1978)

    MATH  Google Scholar 

  6. Farid Khwaja, S., Olde Daalhuis, A.B.: Uniform asymptotic expansions for hypergeometric functions with large parameters IV. Anal. Appl. (Singap.) 12, 667-710 (2014)

  7. Ismail, M.E.H.: Classical and quantum orthogonal polynomials in one variable, Encyclopedia of Mathematics and its Applications vol. 98, (Cambridge University Press, Cambridge, 2009). With two chapters by Walter Van Assche, With a foreword by Richard A. Askey, Reprint of the 2005 original

  8. Ismail, M.E.H., Valent, G.: On a family of orthogonal polynomials related to elliptic functions. Illinois J. Math. 42, 294–312 (1998)

    Article  MathSciNet  Google Scholar 

  9. Ismail, M.E.H., Valent, G., Yoon, G.J.: Some orthogonal polynomials related to elliptic functions. J. Approx. Theory 112, 251–278 (2001)

    Article  MathSciNet  Google Scholar 

  10. Kalvoda, T., Šťovíček, P.: A family of explicitly diagonalizable weighted Hankel matrices generalizing the Hilbert matrix. Linear Multilinear Algebra 64, 870–884 (2016)

    Article  MathSciNet  Google Scholar 

  11. Kiper, A.: Fourier series coefficients for powers of the Jacobian elliptic functions. Math. Comp. 43, 247–259 (1984)

    Article  MathSciNet  Google Scholar 

  12. Lawden, D.F.: Elliptic Functions and Applications. Springer, New York (1989)

    Book  Google Scholar 

  13. Magnus, W.: On the spectrum of Hilberts matrix. Am. J. Math. 72, 699–704 (1950)

  14. NIST Digital Library of Mathematical Functions. http://dlmf.nist.gov/, Release 1.0.17 of 2017-12-22. F.W.J. Olver, A.B. Olde Daalhuis, D.W. Lozier, B.I. Schneider, R.F. Boisvert, C.W. Clark, B.R. Miller, B.V. Saunders, eds

  15. Olver, F.W.J.: Asymptotics and Special Functions. A. K. Peters Ltd., Wellesley (1997)

    Book  Google Scholar 

  16. Rosenblum, M.: On the Hilbert matrix. II. Proc. Am. Math. Soc. 9 (1958), 581-585

  17. Štampach, F., Šťovíček, P.: Spectral representation of some weighted Hankel matrices and orthogonal polynomials from the Askey scheme. J. Math. Anal. Appl. 472, 483–509 (2019)

    Article  MathSciNet  Google Scholar 

  18. Štampach, F., Šťovíček, P.: On Hankel matrices commuting with Jacobi matrices from the Askey scheme. Linear Alg. Appl. 591, 235–267 (2020)

    Article  MathSciNet  Google Scholar 

  19. Koekoek, R., Lesky, P.A., Swarttouw, R.F.: Hypergeometric Orthogonal Polynomials and Their q-Analogues. Springer, Berlin (2010)

  20. Yafaev, D.R.: A commutator method for the diagonalization of Hankel operators. Funct. Anal. Appl. 44, 295–306 (2010)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors acknowledge financial support by the Ministry of Education, Youth and Sports of the Czech Republic Project Number CZ.02.1.01/0.0/0.0/ 16_019/0000778.

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, Faculty of Information Technology Czech Technical University in Prague, Thákurova 9, 160 00, Praha, Czech Republic

    František Štampach

  2. Department of Mathematics, Faculty of Nuclear Sciences and Physical Engineering Czech Technical University in Prague, Trojanova 13, 120 00, Praha, Czech Republic

    Pavel Šťovíček

Authors
  1. František Štampach
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Pavel Šťovíček
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to František Štampach.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Štampach, F., Šťovíček, P. New Explicitly Diagonalizable Hankel Matrices Related to the Stieltjes–Carlitz Polynomials. Integr. Equ. Oper. Theory 93, 29 (2021). https://doi.org/10.1007/s00020-021-02638-4

Download citation

  • Received:

  • Revised:

  • Published:

  • DOI: https://doi.org/10.1007/s00020-021-02638-4

Keywords

Mathematics Subject Classification

Search

Navigation