Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-26T23:40:12.364Z Has data issue: false hasContentIssue false
  • English
  • Français

Contractions of rotation groups and their representations

Published online by Cambridge University Press:  24 October 2008

A. H. Dooley  and
J. W. Rice
A. H. Dooley
Affiliation:
School of Mathematics, University of New South Wales
J. W. Rice
Affiliation:
School of Mathematical Sciences, Flinders University
Get access Rights & Permissions [Opens in a new window]

Extract

It is a classical result in the theory of special functions that Bessel functions are limits in an appropriate sense of Legendre polynomials. For example in (11), § 17.4, the following result is attributed to Heine:

such limiting formulae are also known for certain other special functions (cf. (1), (5)). Apart from their intrinsic interest, these formulae have been used in the theory of special functions to obtain product formulae, etc. for the limit function from those of the approximating sequence.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 94 , Issue 3 , November 1983 , pp. 509 - 517
Copyright
Copyright © Cambridge Philosophical Society 1983

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

(1)Clerc, J.-L.Une formule asymptotique du type Melker–Heine pour les zonales d'un espace riemannien symétrique. Studia Math. 57 (1976), 27–32.Google Scholar
(2)Dooley, A. H. and Gaudry, G. I.An extension of de Leeuw's theorem to the n-dimensional rotation group. Ann. Inst. Fourier (Grenoble). (In the Press.)Google Scholar
(3)Helgason, S.Differential geometry and symmetric spaces (Academic Press, 1978).Google Scholar
(4)Inönu, E. and Wigner, E. P.. Contractions of groups and their representations. Proc. Nat. Acad. Sci. U.S.A. 39 (1953), 510–524.CrossRefGoogle ScholarPubMed
(5)Miller, W.Lie theory and special functions (Academic Press, 1968).Google Scholar
(6)Rubin, R. L.Harmonic analysis on the group of rigid motions of the Euclidean plane. Studia Math. 57 (1978), 125–141.CrossRefGoogle Scholar
(7)Saletan, E. J.Contraction of Lie groups. J. Math. Phys. 2 (1961), 1–21.CrossRefGoogle Scholar
(8)Sugiura, M.Unitary representations and harmonic analysis (Kodansha, Tokyo, 1975).Google Scholar
(9)Vilenkin, N. J.Special functions and the theory of group representations. Amer. Math. Soc. 1968.Google Scholar
(10)Wallach, N. R.Harmonic analysis on homogeneous spaces (Marcel Dekker, 1973).Google Scholar
(11)Whittaker, E. T. and Watson, G. N.A course of modern analysis, 4th ed. (Cambridge University Press, 1958).Google Scholar