Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 3 (2007), 003, 18 pages      math.CA/0701135      https://doi.org/10.3842/SIGMA.2007.003
Contribution to the Vadim Kuznetsov Memorial Issue

Elliptic Biorthogonal Polynomials Connected with Hermite's Continued Fraction

Luc Vinet a and Alexei Zhedanov b
a) Université de Montréal, PO Box 6128, Station Centre-ville, Montréal QC H3C 3J7, Canada
b) Donetsk Institute for Physics and Technology, Donetsk 83114, Ukraine

Received October 07, 2006, in final form December 12, 2006; Published online January 04, 2007

Abstract
We study a family of the Laurent biorthogonal polynomials arising from the Hermite continued fraction for a ratio of two complete elliptic integrals. Recurrence coefficients, explicit expression and the weight function for these polynomials are obtained. We construct also a new explicit example of the Szegö polynomials orthogonal on the unit circle. Relations with associated Legendre polynomials are considered.

Key words: Laurent biorthogonal polynomials; associated Legendre polynomials; elliptic integrals.

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References

  1. Barrucand P., Dickinson D., On the associated Legendre polynomials, in Orthogonal Expansions and Their Continued Analogues, Editor D.T. Haimo, Southern Illinois Press, 1968, 43-50.
  2. Bracciali C.F., da Silva A.P., Sri Ranga A., Szegö polynomials: some relations to L-orthogonal and orthogonal polynomials, J. Comput. Appl. Math. 153 (2003), 79-88.
  3. Chihara T., An introduction to orthogonal polynomials, Gordon and Breach, 1978.
  4. Delsarte P., Genin Y., The split Levinson algorithm, IEEE Trans. Acoust. Speech Signal Process 34 (1986), 470-478.
  5. Geronimus Ya.L., Polynomials orthogonal on a circle and their applications, in Amer. Math. Soc. Transl. Ser. 1, Vol. 3, Amer. Math. Soc., Providence, 1962, 1-78.
  6. Grünbaum F.A., Vinet L., Zhedanov A., Linear operator pencils on Lie algebras and Laurent biorthogonal polynomials, J. Phys. A: Math. Gen. 37 (2004), 7711-7725.
  7. Hendriksen E., van Rossum H., Orthogonal Laurent polynomials, Indag. Math. (Ser. A) 48 (1986), 17-36.
  8. Hendriksen E., Associated Jacobi-Laurent polynomials, J. Comput. Appl. Math. 32 (1990), 125-141.
  9. Hendriksen E., A weight function for the associated Jacobi-Laurent polynomials, J. Comput. Appl. Math. 33 (1990), 171-180.
  10. Hermite C., Sur la développement en série des integrales elliptiques de premiere et de seconde espece, Annali di Matematica II (1868), 2 ser., 97-97.
  11. Hermite C., Oeuvres, Tome II, Paris, 1908, 486-488.
  12. Hermite C., Cours d'analyse de la Faculté des Sciences, Editor Andoyer, Hermann, Paris, 1882 (Lithographed notes).
  13. Ismail M.E.H., Masson D., Some continued fractions related to elliptic functions, Contemp. Math. 236 (1999), 149-166.
  14. Jones W.B., Thron W.J., Survey of continued fraction methods of solving moment problems, in Analytic Theory of Continued Fractions, Lecture Notes in Math., Vol. 932, Springer, Berlin - Heidelberg - New York, 1981.
  15. Koekoek R., Swarttouw R.F., The Askey scheme of hypergeometric orthogonal polynomials and its q-analogue, Report 94-05, Delft University of Technology, Faculty of Technical Mathematics and Informatics, 1994.
  16. Lomont J.S., Brillhart J., Elliptic polynomials, Chapman & Hall/CRC, Boca Raton, FL, 2001.
  17. Magnus W., Oberhettinger F., Formeln und Sätze fur die speziellen Functionen der mathematischen Physik, Springer, Berlin, 1948.
  18. Pollaczek F., Sur une famille de polynômes orthogonaux à quatre paramitres, C. R. Acad. Sci. Paris 230 (1950), 2254-2256.
  19. Rees C.J., Elliptic orthogonal polynomials, Duke Math. J. 12 (1945), 173-187.
  20. Spiridonov V., Vinet L., Zhedanov A., Spectral transformations, self-similar reductions and orthogonal polynomials, J. Phys. A: Math. Gen. 30 (1997), 7621-7637.
  21. Szegö G., Orthogonal polynomials, AMS, 1959.
  22. Vinet L., Zhedanov A., An integrable chain and bi-orthogonal polynomials, Lett. Math. Phys. 46 (1998), 233-245.
  23. Vinet L., Zhedanov A., Spectral transformations of the Laurent biorthogonal polynomials. I. q-Appel polynomials, J. Comput. Appl. Math. 131 (2001), 253-266.
  24. Whittacker E.T., Watson G.N., A course of modern analysis, 4th ed., Cambridge University Press, 1927.
  25. Zhedanov A., Rational spectral transformations and orthogonal polynomials, J. Comput. Appl. Math. 85 (1997), 67-86.
  26. Zhedanov A., On some classes of polynomials orthogonal on arcs of the unit circle connected with symmetric orthogonal polynomials on an interval, J. Approx. Theory 94 (1998), 73-106.
  27. Zhedanov A., The "classical" Laurent biorthogonal polynomials, J. Comput. Appl. Math. 98 (1998), 121-147.

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