|
SIGMA 5 (2009), 038, 12 pages arXiv:0903.4803
https://doi.org/10.3842/SIGMA.2009.038
Contribution to the Proceedings of the Workshop “Elliptic Integrable Systems, Isomonodromy Problems, and Hypergeometric Functions”
Elliptic Hypergeometric Solutions to Elliptic Difference Equations
Alphonse P. Magnus
Université catholique de Louvain,
Institut mathématique, 2 Chemin du Cyclotron,
B-1348 Louvain-La-Neuve, Belgium
Received December 01, 2008, in final form March 20, 2009; Published online March 27, 2009
Abstract
It is shown how to define difference
equations on particular lattices {xn}, n Î Z, made of
values of an elliptic function at a sequence of arguments in
arithmetic progression (elliptic lattice). Solutions to special
difference equations have remarkable
simple interpolatory expansions.
Only linear difference equations of first order are considered here.
Key words:
elliptic difference equations; elliptic hypergeometric expansions.
pdf (270 kb)
ps (193 kb)
tex (17 kb)
References
- Andrews G.E., Askey R., Roy R., Special functions,
Encyclopedia of Mathematics and its Applications, Vol. 71, Cambridge University Press, Cambridge, 1999.
- Askey R., Wilson J., Some basic hypergeometric
orthogonal polynomials that generalize Jacobi polynomials,
Mem. Amer. Math. Soc. 54 (1985), no. 319.
- Bagby T., On interpolation by rational functions, Duke Math. J. 36 (1969), 95-104.
- Burskii V.P., Zhedanov A.S., The Dirichlet and the Poncelet problems, RIAM Symposium No.16ME-S1 "Physics and Mathematical Structures of Nonlinear Waves" (November 15-17, 2004, Kyushu University, Kasuga, Fukuoka, Japan), 2004, 22-26, available at http://www.riam.kyushu-u.ac.jp/fluid/meeting/16ME-S1/papers/Article_No_24.pdf.
- Burskii V.P., Zhedanov A.S.,
Dirichlet and Neumann Problems for string equation,
Poncelet problem and Pell-Abel equation,
SIGMA 2 (2006), 041, 5 pages, math.AP/0604278.
- Burskii V.P., Zhedanov A.S.,
On Dirichlet, Poncelet and Abel problems, arXiv:0903.2531.
- Ganelius T., Degree of rational approximation,
in Lectures on
Approximation and Value Distribution, Editors T. Ganelius et al., Sém. Math. Sup., Vol. 79, Presses Univ. Montréal, Montreal, Que., 1982, 9-78.
- Goncar A.A., The problems of E. I. Zolotarev which are connected with rational functions, Mat. Sb. 78 (120) (1969), 640-654 (English transl.: Math. USSR-Sb. 7 (1969), 623-635).
- Gonchar A.A.,
Rational approximations of analytic functions,
Sovrem. Probl. Mat. Current Problems in Mathematics, no. 1,
Ross. Akad. Nauk, Inst. Mat. im. V.A. Steklova, Moscow, 2003, 83-106 (in Russian).
- Gonchar A.A., Rakhmanov E.A., Equilibrium distributions and the rate of rational approximation of analytic functions,
Mat. Sb. 134 (176) (1987), 306-352 (English transl.:
Math. USSR-Sb. 62 (1989), 305-348).
- Gonchar A.A., Rakhmanov E.A., Suetin S.P., On the rate of convergence of Padé approximants of orthogonal expansions, in Progress in Approximation Theory (Tampa, FL, 1990), Springer Ser. Comput. Math., Vol. 19, Springer, New York, 1992, 169-190.
- Gonchar A.A., Suetin S.P.,
On Padé approximants of meromorphic functions of Markov type,
Current Problems in Mathematics, no. 5,
Ross. Akad. Nauk, Inst. Mat. im. V.A. Steklova, Moscow, 2004, 68 pages (in Russian),
available at
http://www.mi.ras.ru/spm/pdf/005.pdf.
- Grünbaum F.A., Haine L.,
On a q-analogue of Gauss equation and some q-Riccati equations, in
Special Functions, q-Series
and Related Topics (Toronto, ON, 1995), Editors M.E.H. Ismail et al., Fields Inst. Commun., Vol. 14, Amer. Math. Soc., Providence, RI, 1997, 77-81.
- Hardy G.H., Littlewood J.E., Notes on the theory of series. XXIV. A curious
power-series, Proc. Cambridge Philos. Soc. 42 (1946), 85-90.
- Ismail M.E.H., Classical and quantum orthogonal polynomials in one variable, with two chapters by Walter Van Assche,
Encyclopedia of Mathematics and its Applications, Vol. 98, Cambridge University Press, Cambridge, 2005.
- Koekoek R., Swarttouw R.F.,
The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue,
Delft University of Technology,
Faculty of Information Technology and Systems,
Department of Technical Mathematics and Informatics,
Report no. 98-17, 1998, math.CA/9602214.
- Koornwinder T.H.,
Compact quantum groups and q-special functions,
in Representations of Lie Groups and Quantum Groups (Trento, 1993),
Editors V. Baldoni and M.A. Picardello,
Pitman Res. Notes Math. Ser., Vol. 311,
Longman Sci. Tech., Harlow, 1994,
46-128,
Chapters 1, 2: General compact quantum groups, a tutorial, hep-th/9401114,
Chapters 3, 4: q-special functions, a tutorial, math.CA/9403216.
- Loutsenko I., Spiridonov V., Spectral self-similarity, one-dimensional Ising
chains and random matrices,
Nuclear Phys. B 538 (1999), 731-758.
- Loutsenko I., Spiridonov V., Soliton solutions of integrable hierarchies and
Coulomb plasmas,
J. Statist. Phys. 99 (2000), 751-767, cond-mat/9909308.
- Loutsenko I., Spiridonov V., A critical phenomenon in
solitonic Ising chains, SIGMA 3 (2007), 059, 11 pages, arXiv:0704.3173.
- Lubinsky D.S., Rogers-Ramanujan and the
Baker-Gammel-Wills (Padé) conjecture,
Ann. of Math. (2) 157 (2003), 847-889,
math.CA/0402305.
- Magnus A.P., Associated Askey-Wilson polynomials as
Laguerre-Hahn orthogonal polynomials, in
Orthogonal Polynomials and
their Applications (Segovia, 1986), Editors M. Alfaro et al.,
Lecture Notes in Math., Vol. 1329, Springer, Berlin, 1988, 261-278.
- Magnus A.P.,
Special non uniform lattice (snul) orthogonal polynomials
on discrete dense sets of points,
J. Comp. Appl. Math. 65 (1995), 253-265, math.CA/9502228.
- Magnus A.P., Rational interpolation to solutions of Riccati
difference equations on elliptic lattices,
J. Comp. Appl. Math., 2009, to appear,
https://doi.org/10.1016/j.cam.2009.02.047,
preprint available at
http://perso.uclouvain.be/alphonse.magnus/num3/MagnusLuminy2007.pdf.
- Meinguet J., An electrostatic approach to the determination
of extremal measures, Math. Phys. Anal.
Geom. 3 (2000), 323-337.
- Milne-Thomson L.M., The calculus of finite differences, Macmillan and Co., Ltd., London, 1951,
available at
http://www.archive.org/details/calculusoffinite032017mbp.
- Nikiforov A.F., Suslov S.K., Classical orthogonal
polynomials of a discrete variable on nonuniform lattices,
Lett. Math. Phys. 11 (1986), 27-34.
- Nikiforov A.F., Suslov S.K., Uvarov V.B., Classical
orthogonal polynomials of a discrete variable, Series in Computational Physics, Springer-Verlag, Berlin, 1991.
- Saff E.B., Totik V., Logarithmic potentials with external fields. Appendix B by Thomas Bloom, Grundlehren der Mathematischen Wissenschaften, Vol. 316, Springer-Verlag, Berlin, 1997.
- Spiridonov V.P., Essays on the theory of elliptic hypergeometric functions, Uspekhi Mat. Nauk 63 (2008), no. 3, 3-72
(English transl.: Russ. Math. Surv. 63 (2008), 405-472), arXiv:0805.3135
- Spiridonov V.P., On the analytical properties of infinite elliptic hypergeometric series, talk presented at the workshop "Elliptic Integrable Systems, Isomonodromy Problems, and Hypergeometric Functions",
Hausdorff Center for Mathematics, Bonn, July 2008.
- Spiridonov V.P., Zhedanov A.S.,
Generalized eigenvalue problem and a new family of rational functions biorthogonal on elliptic grids, in
Special Functions 2000: Current Perspective and Future Directions (Tempe, AZ, 2000), Editors J. Bustoz et al.,
NATO Sci. Ser. II, Math. Phys. Chem., Vol. 30, Kluwer Acad. Publ., Dordrecht, 2001, 365-388.
- Spiridonov V.P., Zhedanov A.S.,
Private communication to the author, Wednesday, July 23, 2008, on the Rhine river,
en route towards Königswinter.
- Spiridonov V.P., Zhedanov A.S., Elliptic grids, rational functions, and the Padé interpolation
Ramanujan J. 13 (2007),
285-310.
- Stahl H., Convergence of rational interpolants, in Numerical Analysis,
(Louvain-la-Neuve, 1995), Bull. Belg. Math. Soc. Simon Stevin 1996 (1996), suppl., 11-32.
- Walsh J.L., Interpolation and approximation by
rational functions in the complex domain, 4th
ed., American Mathematical Society Colloquium Publications, Vol. 20, American Mathematical Society, Providence, R.I., 1965.
|
|