Geodesic
Elliptic Hypergeometric Laurent Biorthogonal Polynomials with a Dense Point Spectrum on the Unit Circle
Symmetry, integrability and geometry: methods and applications, Tome 5 (2009) Cet article a éte moissonné depuis la source Math-Net.Ru

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Using the technique of the elliptic Frobenius determinant, we construct new elliptic solutions of the $QD$-algorithm. These solutions can be interpreted as elliptic solutions of the discrete-time Toda chain as well. As a by-product, we obtain new explicit orthogonal and biorthogonal polynomials in terms of the elliptic hypergeometric function ${_3}E_2(z)$. Their recurrence coefficients are expressed in terms of the elliptic functions. In the degenerate case we obtain the Krall–Jacobi polynomials and their biorthogonal analogs.
Keywords: elliptic Frobenius determinant; $QD$-algorithm; orthogonal and biorthogonal polynomials on the unit circle; dense point spectrum; elliptic hypergeometric functions; Krall–Jacobi orthogonal polynomials; quadratic operator pencils. 
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     author = {Satoshi Tsujimoto and Alexei Zhedanov},
     title = {Elliptic {Hypergeometric} {Laurent} {Biorthogonal} {Polynomials} with {a~Dense} {Point} {Spectrum} on the {Unit} {Circle}},
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}
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Satoshi Tsujimoto; Alexei Zhedanov. Elliptic Hypergeometric Laurent Biorthogonal Polynomials with a Dense Point Spectrum on the Unit Circle. Symmetry, integrability and geometry: methods and applications, Tome 5 (2009). http://geodesic.mathdoc.fr/item/SIGMA_2009_5_a32/

[1] Translations of Mathematical Monographs, 79, American Mathematical Society, Providence, R.I., 1990 | MR | MR | Zbl | Zbl

[2] Allouche H., Cuyt A., Reliable pole detection using a deflated $qd$-algorithm: when Bernoulli, Hadamard and Rutishauser cooperate, Numer. Math. to appear

[3] Amdeberhan T., “A determinant of the Chudnovskys generalizing the elliptic Frobenius–Stickelberger–Cauchy determinantal identity”, Electron. J. Combin., 7:1 (2000), Note 6, 3 pp., ages | MR

[4] Baxter G., “Polynomials defined by a difference system”, J. Math. Anal. Appl., 2 (1961), 223–263 | DOI | MR | Zbl

[5] de Andrade X. L., McCabe J. H., “On the two point Padé table for a distribution”, Rocky Mountain J. Math, 33 (2003), 545–566 | DOI | MR | Zbl

[6] Derevyagin M., Zhedanov A., “An operator approach to multipoint Padé approximations”, J. Approx. Theory, 157 (2009), 70–88 ; arXiv:0802.3432 | DOI | MR | Zbl

[7] Faddeev D. K., Faddeeva V. N., Computational methods of linear algebra, W. H. Freeman and Co., San Francisco – London, 1963 | MR

[8] Frobenius G., Stickelberger L., “Über die Addition und Multiplication der elliptischen Functionen”, J. Reine Angew. Math., 88 (1880), 146–184; reprinted in Frobenius F. G., Gesammelte Abhandlungen, Vol. 1, ed. J.-P. Serre, Springer, Berlin, 1968, 612–650 | MR | Zbl

[9] Frobenius G., “Über die elliptischen Functionen zweiter Art”, J. Reine Angew. Math., 93 (1882), 53–68; reprinted in Ferdinand F. G., Gesammelte Abhandlungen, Vol. 2, ed. J.-P. Serre, Springer, Berlin, 1968, 81–96 | MR

[10] Gasper G., Rahman M., Basic hypergeometric series, 2nd ed., Encyclopedia of Mathematics and its Applications, 96, Cambridge University Press, Cambridge, 2004 | MR | Zbl

[11] Geronimus Ya. L., Polynomials orthogonal on a circle and their applications, Amer. Math. Soc. Translation, 1954 no. 104, 1954 | MR

[12] Hendriksen E., van Rossum H., “Orthogonal Laurent polynomials”, Nederl. Akad. Wetensch. Indag. Math., 48 (1986), 17–36 | MR | Zbl

[13] Henrici P., Applied and computational complex analysis, John Wiley Sons, New York, 1974 | MR | Zbl

[14] Ismail M. E. H., Masson D. R., “Generalized orthogonality and continued fractions”, J. Approx. Theory, 83 (1995), 1–40 ; math.CA/9407213 | DOI | MR | Zbl

[15] Jones W. B., Thron W. J., “Survey of continued fraction methods of solving moment problems”, Analytic Theory of Continued Fractions (Loen, 1981), Lecture Notes in Mathematics, 932, Springer, Berlin – New York, 1982, 4–37 | MR

[16] Kharchev S., Mironov A., Zhedanov A., “Faces of relativistic Toda chain”, Internat. J. Modern Phys. A, 12 (1997), 2675–2724 ; hep-th/9606144 | DOI | MR | Zbl

[17] Koekoek R., Swarttouw R. F., The Askey scheme of hypergeometric orthogonal polynomials and its $q$-analogue, Report 94-05, Faculty of Technical Mathematics and Informatics, Delft University of Technology, 1994; math.CA/9602214

[18] Krattenthaler C., “Advanced determinant calculus”, Sém. Lothar. Combin., 1999, Art. B42q, 67 pp., ages ; math.CO/9902004 | MR

[19] Littlejohn L. L., “The Krall polynomials: a new class of orthogonal polynomials”, Quaestiones Math., 5 (1982), 255–265 | MR | Zbl

[20] Littlejohn L. L., “On the classification of differential equations having orthogonal polynomial solutions”, Ann. Mat. Pura Appl. (4), 138 (1984), 35–53 | DOI | MR | Zbl

[21] Magnus A. P., “Special nonuniform lattice (snul) orthogonal polynomials on discrete dense sets of points”, J. Comp. Appl. Math., 65 (1995), 253–265 | DOI | MR | Zbl

[22] Pastro P. I., “Orthogonal polynomials and some $q$-beta integrals of Ramanujan”, J. Math. Anal. Appl., 112 (1985), 517–540 | DOI | MR | Zbl

[23] Ruijsenaars S. N. M., “Relativistic Toda systems”, Comm. Math. Phys., 133 (1990), 217–247 | DOI | MR | Zbl

[24] Simon B., Orthogonal polynomials on the unit circle, American Mathematical Society Colloquium Publications, 51, American Mathematical Society, Providence, R.I., 2005 | Zbl

[25] Spiridonov V. P., Zhedanov A. A., “To the theory of biorthogonal rational functions”, RIMS Kokyuroku, no. 1302 (2003), 172–192 | MR

[26] Spiridonov V. P., “An elliptic incarnation of the Bailey chain”, Int. Math. Res. Not., 2002:37 (2002), 1945–1977 | DOI | MR | Zbl

[27] Spiridonov V. P., “Essays on the theory of elliptic hypergeometric functions”, Russ. Math. Surv., 63 (2008), 405–472 ; arXiv:0805.3135 | DOI | MR | Zbl

[28] Suris Yu. B., “A discrete-time relativistic Toda lattice”, J. Phys. A: Math. Gen., 29 (1996), 451–465 ; solv-int/9510007 | DOI | MR | Zbl

[29] Szegő G., Orthogonal polynomials, 4th ed., American Mathematical Society Colloquium Publications, 23, American Mathematical Society, Providence, R.I., 1975 | MR

[30] Whittaker E. T., Watson G. N., A course of modern analysis. An introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions, reprint of 4th ed. (1927), Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1996 | MR | Zbl

[31] Wilson J. A., “Orthogonal functions from Gram determinants”, SIAM J. Math. Anal., 22 (1991), 1147–1155 | DOI | MR | Zbl

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

[33] Zhedanov A., “Elliptic polynomials orthogonal on the unit circle with a dense point spectrum”, Ramanujan J., 19:3 (2009) ; arXiv:0711.4696 | DOI | MR | Zbl | MR