Geodesic
Elliptic Biorthogonal Polynomials Connected with Hermite's Continued Fraction
Symmetry, integrability and geometry: methods and applications, Tome 3 (2007) Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

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 Szegő polynomials orthogonal on the unit circle. Relations with associated Legendre polynomials are considered.
Keywords: Laurent biorthogonal polynomials; associated Legendre polynomials; elliptic integrals. 
@article{SIGMA_2007_3_a2,
     author = {Luc Vinet and Alexei Zhedanov},
     title = {Elliptic {Biorthogonal} {Polynomials} {Connected} with {Hermite's} {Continued} {Fraction}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2007},
     volume = {3},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2007_3_a2/}
}
TY  - JOUR
AU  - Luc Vinet
AU  - Alexei Zhedanov
TI  - Elliptic Biorthogonal Polynomials Connected with Hermite's Continued Fraction
JO  - Symmetry, integrability and geometry: methods and applications
PY  - 2007
VL  - 3
UR  - http://geodesic.mathdoc.fr/item/SIGMA_2007_3_a2/
LA  - en
ID  - SIGMA_2007_3_a2
ER  - 
%0 Journal Article
%A Luc Vinet
%A Alexei Zhedanov
%T Elliptic Biorthogonal Polynomials Connected with Hermite's Continued Fraction
%J Symmetry, integrability and geometry: methods and applications
%D 2007
%V 3
%U http://geodesic.mathdoc.fr/item/SIGMA_2007_3_a2/
%G en
%F SIGMA_2007_3_a2
Luc Vinet; Alexei Zhedanov. Elliptic Biorthogonal Polynomials Connected with Hermite's Continued Fraction. Symmetry, integrability and geometry: methods and applications, Tome 3 (2007). http://geodesic.mathdoc.fr/item/SIGMA_2007_3_a2/

[1] Barrucand P., Dickinson D., “On the associated Legendre polynomials”, Orthogonal Expansions and Their Continued Analogues, ed. D. T. Haimo, Southern Illinois Press, 1968, 43–50 | MR | Zbl

[2] Bracciali C. F., da Silva A. P., Sri Ranga A., “Szegő polynomials: some relations to $L$-orthogonal and orthogonal polynomials”, J. Comput. Appl. Math., 153 (2003), 79–88 | DOI | MR | Zbl

[3] Chihara T., An introduction to orthogonal polynomials, Gordon and Breach, 1978 | MR | Zbl

[4] Delsarte P., Genin Y., “The split Levinson algorithm”, IEEE Trans. Acoust. Speech Signal Process, 34 (1986), 470–478 | DOI | MR

[5] Geronimus Ya. L., Polynomials orthogonal on a circle and their applications, Amer. Math. Soc. Transl. Ser. 1, 3, Amer. Math. Soc., Providence, 1962, 1–78 | MR

[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 | DOI | MR | Zbl

[7] Hendriksen E., van Rossum H., “Orthogonal Laurent polynomials”, Indag. Math. (Ser. A), 48 (1986), 17–36 | MR | Zbl

[8] Hendriksen E., “Associated Jacobi–Laurent polynomials”, J. Comput. Appl. Math., 32 (1990), 125–141 | DOI | MR | Zbl

[9] Hendriksen E., “A weight function for the associated Jacobi–Laurent polynomials”, J. Comput. Appl. Math., 33 (1990), 171–180 | DOI | MR | Zbl

[10] Hermite C., “Sur la développement en série des intègrales elliptiques de première et de seconde espèce”, Annali di Matematica II. 2 ser., 1868, 97–97 | DOI | Zbl

[11] Hermite C., Oeuvres, Tome II, Paris, 1908, 486–488 | Zbl

[12] Hermite C., Cours d'analyse de la Faculté des Sciences, ed. Andoyer, Hermann, Paris, 1882, Lithographed notes

[13] Ismail M. E. H., Masson D., “Some continued fractions related to elliptic functions”, Continued fractions: from analytic number theory to constructive approximation, Contemp. Math., 236, 1999, 149–166 | MR | Zbl

[14] Jones W. B., Thron W. J., “Survey of continued fraction methods of solving moment problems”, Analytic Theory of Continued Fractions, Lecture Notes in Math., 932, Springer, Berlin – Heidelberg – New York, 1981 | MR

[15] Koekoek R., Swarttouw R. F., The Askey scheme of hypergeometric orthogonal polynomials and its qanalogue, 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 | MR | Zbl

[17] Magnus W., Oberhettinger F., Formeln und Sätze für die speziellen Functionen der mathematischen Physik, Springer, Berlin, 1948 | MR | Zbl

[18] Pollaczek F., “Sur une famille de polynômes orthogonaux à quatre paramitrès”, C. R. Acad. Sci. Paris, 230 (1950), 2254–2256 | MR | Zbl

[19] Rees C. J., “Elliptic orthogonal polynomials”, Duke Math. J., 12 (1945), 173–187 | DOI | MR | Zbl

[20] Spiridonov V., Vinet L., Zhedanov A., “Spectral transformations, self-similar reductions and orthogonal polynomials”, J. Phys. A: Math. Gen., 30 (1997), 7621–7637 | DOI | MR | Zbl

[21] Szegő G., Orthogonal polynomials, AMS, 1959 | MR | Zbl

[22] Vinet L., Zhedanov A., “An integrable chain and bi-orthogonal polynomials”, Lett. Math. Phys., 46 (1998), 233–245 | DOI | MR | Zbl

[23] Vinet L., Zhedanov A., “Spectral transformations of the Laurent biorthogonal polynomials. I. $q$-Appel polynomials”, J. Comput. Appl. Math., 131 (2001), 253–266 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[27] Zhedanov A., “The “classical” Laurent biorthogonal polynomials”, J. Comput. Appl. Math., 98 (1998), 121–147 | DOI | MR | Zbl