Continuous q-Hermite Polynomials When q > 1
The continuous q-Hermite polynomials have a reasonably well known orthogonality relation when — 1 < q < 1. When q > 1 the change of variables x → ix lends to a set of orthogonal polynomials. The same happens for the continuous q-ultraspherical polynomials, but now only finitely many polynomials are orthogonal with respect to a positive measure. A positive measure is found in each of these cases.
Unable to display preview. Download preview PDF.
- W. Allaway, The identification of a class of orthogonal polynomials, Ph.D. thesis, University of Alberta, Canada, 1972.Google Scholar
- R. Askey, A integral of Ram an uj an and orthogonal polynomials, J. Indian Math. Soc. (to appear).Google Scholar
- R. Askey AND M. Ismail, The Rogers q-ultraspherical polynomials, in Approximation Theory III, ed. E.W. Cheney, Academic Press, New York, 1980, pp. 175–182.Google Scholar
- R. Askey AND M. Ismail, A generalization of ultraspherical polynomials, in Studies in Pure Mathematics, ed. P. Erdös, Birkhäuser, Basel, 1983, pp. 55–78.Google Scholar
- G. Freud, Orthogonal Polynomials, Pergamon Press, Oxford, 1971.Google Scholar
- M. Ismail AND D. Stanton, On the Askey-Wilson and Rogers polynomials, Canadian J. Math. (to appear).Google Scholar
- E.G. Kalnins AND W. Miller, Symmetry techniques for q-series: the Askey-Wilson polynomials, Rocky Mountain J. Math. (to appear).Google Scholar
- W. Miller, Symmetry techniques and orthogonality for q-series, this volume.Google Scholar
- V.R. Thiruvenkalachar AND K. Venkatachaliengar, Ramanujan at Elementary Levels; Glimses. unpublished manuscript.Google Scholar
- S. Wigert, Sur les polynomes orthogonaux et l’approximation des fonctions continues, Arkiv för Mat., Astron och Fysik, 17(18) (1923), pp. 1–15.Google Scholar