Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-19T06:08:58.605Z Has data issue: false hasContentIssue false
  • English
  • Français

VI.—Generating Functions of Certain Continuous Orthogonal Systems

Published online by Cambridge University Press:  14 February 2012

A. Erdélyi
A. Erdélyi
Affiliation:
Mathematical Institute, University of Edinburgh
Get access Rights & Permissions [Opens in a new window]

Summary

10. Generating functions and bilinear generating functions (of the type of Mehler's celebrated formula) are known to be of great importance in the formal theory of orthogonal sequences. The present paper contains analogous formulae for a number of continuous orthogonal systems as well as “mixed” systems (which have a point spectrum as well as a continuous one). Four systems of the hypergeometric type have been selected as examples which are thought to be of some importance because of their presenting themselves in certain problems of Mathematical Physics.

My thanks are due to the Carnegie Trust for the Universities of Scotland for grants towards the printing of this paper and my paper in Proceedings, vol. lx, no. 26, 1940.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 61 , Issue 1 , 1941 , pp. 61 - 70
Copyright
Copyright © Royal Society of Edinburgh 1941

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

References to Literature

Hobson, E. W., 1931. The Theory of Spherical and Ellipsoidal Harmonics, Cambridge.Google Scholar
Husimi, K., 1940. “Some Formal Properties of the Density Matrix,” Proc. Phys.-Math. Soc. Japan (3), vol. xxii, pp. 264–314.Google Scholar
Mehler, F. G., 1868. “Ueber die Vertheilung der statischen Elektricität in einem von zwei Kugelkalotten begrenzten Korper,” Journ. fur Math., vol. lxviii, pp. 134–150.Google Scholar
Mehler, F. G., 1881. “Ueber eine mit den Kugel- und Cylinderfunctionen verwandte Function und ihre Anwendung in der Theorie der Elektricitatsvertheilung,” Math. Ann., vol. xviii, pp. 161–194.CrossRefGoogle Scholar
Meijer, C. S., 1934. “Einige Integraldarstellungen für Whittakersche und Besselsche Funktionen,” Proc. Akad. Amsterdam, vol. xxxvii, pp. 805–812.Google Scholar
Poole, E. G. C, 1936. Theory of Linear Differential Equations, Oxford.Google Scholar
Pöschl, G., and Teller, E., 1933. “Bemerkungen zur Quantenmechanik des anharmonischen Oszillators,” Zeits. für Physik, vol. Ixxxiii, pp. 143–151.CrossRefGoogle Scholar
Temple, G., 1930. “Doubly Orthogonal Systems of Functions,” Proc. London Math. Soc. (2), vol. xxxi, pp. 231–252.CrossRefGoogle Scholar
Titchmarsh, E. C, 1937. Introduction to the Theory of Fourier Integrals” Oxford.Google Scholar
WATSON, G. N., 1922. A Treatise on the Theory of Bessel Functions, Cambridge.Google Scholar
Titchmarsh, E. C, 1933-1934. “Notes on Generating Functions of Polynomials. I. Laguerre Polynomials. II. Hermite Polynomials. III. Polynomials of Legendre and Gegenbauer. IV. Jacobi Polynomials,” Journ. London Math. Soc, vol. viii, pp. 189–192, 194-199, 289-292, and vol. ix, pp. 22-28.Google Scholar
Weyl, H., 1910a. “Ueber gewöhnliche Differentialgleichungen mit Singularitäten und die zugehörigen Entwicklungen willkürlicher Funktionen,” Math. Ann., vol. lxviii, pp. 220–269.CrossRefGoogle Scholar
Weyl, H., 1910b. “Ueber gewöhnliche lineare Differentialgleichungen mit singulären Stellen und ihre Eigenfunktionen,” Nachr. der Göttinger Ges. Wiss., 1910, pp. 442–467.Google Scholar
Whittaker, E. T., 1941. “On Hamilton's Principal Function in Quantum Mechanics,” Proc. Roy. Soc. Edin., vol. lxi, A, pp. 1–19.Google Scholar
Whittaker, E. T., and Watson, G. N., 1927. A Course of Modern Analysis, Cambridge.Google Scholar