Hypergeometric origins of Diophantine properties associated with the Askey scheme
HTML articles powered by AMS MathViewer
- by Yang Chen and Mourad E. H. Ismail PDF
- Proc. Amer. Math. Soc. 138 (2010), 943-951 Request permission
Abstract:
The “Diophantine” properties of the zeros of certain polynomials in the Askey scheme, recently discovered by Calogero and his collaborators, are explained, with suitably chosen parameter values, in terms of the summation theorem of hypergeometric series. Here the Diophantine property refers to integer valued zeros. It turns out that the same procedure can also be applied to polynomials arising from the basic hypergeometric series. We found, with suitably chosen parameters and certain $q$-analogues of the summation theorems, zeros of these polynomials explicitly which are no longer integer valued. This goes beyond the results obtained by the authors previously mentioned.References
- V. È. Adler and A. B. Shabat, On a class of Toda chains, Teoret. Mat. Fiz. 111 (1997), no. 3, 323–334 (Russian, with English and Russian summaries); English transl., Theoret. and Math. Phys. 111 (1997), no. 3, 647–657. MR 1472211, DOI 10.1007/BF02634053
- George E. Andrews, Richard Askey, and Ranjan Roy, Special functions, Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge University Press, Cambridge, 1999. MR 1688958, DOI 10.1017/CBO9781107325937
- M. Bruschi, F. Calogero, and R. Droghei, Proof of certain Diophantine conjectures and identification of remarkable classes of orthogonal polynomials, J. Phys. A 40 (2007), no. 14, 3815–3829. MR 2325078, DOI 10.1088/1751-8113/40/14/005
- M. Bruschi, F. Calogero, and R. Droghei, Tridiagonal matrices, orthogonal polynomials and Diophantine relations. I, J. Phys. A 40 (2007), no. 32, 9793–9817. MR 2370544, DOI 10.1088/1751-8113/40/32/006
- M. Bruschi, F. Calogero, and R. Droghei, Tridiagonal matrices, orthogonal polynomials and Diophantine relations. II, J. Phys. A 40 (2007), no. 49, 14759–14772. MR 2441873, DOI 10.1088/1751-8113/40/49/010
- M. Bruschi, F. Calogero, and R. Droghei, Additional recursion relations, factorizations, and Diophantine properties associated with the polynomials of the Askey scheme, Adv. Math. Phys. , posted on (2009), Art. ID 268134, 43. MR 2500948, DOI 10.1155/2009/268134
- Francesco Calogero, Isochronous systems, Oxford University Press, Oxford, 2008. MR 2383111, DOI 10.1093/acprof:oso/9780199535286.001.0001
- Jerry L. Fields and Jet Wimp, Expansions of hypergeometric functions in hypergeometric functions, Math. Comp. 15 (1961), 390–395. MR 125992, DOI 10.1090/S0025-5718-1961-0125992-3
- George Gasper and Mizan Rahman, Basic hypergeometric series, 2nd ed., Encyclopedia of Mathematics and its Applications, vol. 96, Cambridge University Press, Cambridge, 2004. With a foreword by Richard Askey. MR 2128719, DOI 10.1017/CBO9780511526251
- Mourad E. H. Ismail, Classical and quantum orthogonal polynomials in one variable, Encyclopedia of Mathematics and its Applications, vol. 98, Cambridge University Press, Cambridge, 2005. With two chapters by Walter Van Assche; With a foreword by Richard A. Askey. MR 2191786, DOI 10.1017/CBO9781107325982
- Mourad E. H. Ismail and Dennis Stanton, $q$-Taylor theorems, polynomial expansions, and interpolation of entire functions, J. Approx. Theory 123 (2003), no. 1, 125–146. MR 1985020, DOI 10.1016/S0021-9045(03)00076-5
- R. Koekoek and R. Swarttouw, The Askey-scheme of hypergeometric orthogonal polynomials and its $q$-analogues, Reports of the Faculty of Technical Mathematics and Informatics, no. 98-17, Delft University of Technology, Delft, 1998.
- A. B. Shabat and R. I. Yamilov, To a transformation theory of two-dimensional integrable systems, Phys. Lett. A 227 (1997), no. 1-2, 15–23. MR 1435916, DOI 10.1016/S0375-9601(96)00922-X
- L. J. Slater, Generalized Hypergeometric Series, Cambridge University Press, Cambridge, 1964.
- Gábor Szegő, Orthogonal polynomials, 4th ed., American Mathematical Society Colloquium Publications, Vol. XXIII, American Mathematical Society, Providence, R.I., 1975. MR 0372517
- R. I. Yamilov, Classification of Toda-type scalar lattices, in: Non-linear Evolution Equations and Dynamical Systems, World Scientific, Singapore, 1993.
Additional Information
- Yang Chen
- Affiliation: Department of Mathematics, Imperial College, 180 Queen’s Gate, London SW7 2BZ, United Kingdom
- Email: ychen@ic.ac.uk
- Mourad E. H. Ismail
- Affiliation: Department of Mathematics, University of Central Florida, Orlando, Florida 32816
- MR Author ID: 91855
- Email: ismail@math.ucf.edu
- Received by editor(s): March 7, 2009
- Received by editor(s) in revised form: June 12, 2009
- Published electronically: October 23, 2009
- Communicated by: Walter Van Assche
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 138 (2010), 943-951
- MSC (2000): Primary 33C20, 33C45
- DOI: https://doi.org/10.1090/S0002-9939-09-10106-5
- MathSciNet review: 2566561