Hostname: page-component-6b989bf9dc-g5k2d Total loading time: 0 Render date: 2024-04-14T20:50:01.476Z Has data issue: false hasContentIssue false
  • English
  • Français

Special functions, infinite divisibility and transcendental equations

Published online by Cambridge University Press:  24 October 2008

Mourad E. H. Ismail  and
C. Ping May
Mourad E. H. Ismail
Affiliation:
McMaster University, Hamilton, Ontario, Canada
C. Ping May
Affiliation:
University of Alberta, Edmonton, Alberta, Canada
Get access Rights & Permissions [Opens in a new window]

Abstract

We establish integral representations for quotients of Tricomi ψ functions and of several quotients of modified Bessel functions and of linear combinations of them. These integral representations are used to prove the complete monotonicity of the functions considered and to prove the infinite divisibility of a three parameter probability distribution. Limiting cases of this distribution are the hitting time distributions considered recently by Kent and Wendel. We also derive explicit formulas for the Kent–Wendel probability density functions.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 85 , Issue 3 , May 1979 , pp. 453 - 464
Copyright
Copyright © Cambridge Philosophical Society 1979

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

(1)Boersma, J. Note on the explicit solution of the transcendental equation .Google Scholar
(2)Bondesson, L.A general result in infinite divisibility. Ann. Prob. 7 (1979) (To appear.)CrossRefGoogle Scholar
(3)Buchholz, H.The confluent hypergeometric function (New York, Springer-Verlag, 1969).CrossRefGoogle Scholar
(4)Durand, L.Nicholson-type integrals for products of Jacobi functions I. SIAM J. Math. Anal. 9, (1978), 76–86.CrossRefGoogle Scholar
(5)Ebdélyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F. G.Higher transcendental functions, vol. I (New York, McGraw-Hill, 1953).Google Scholar
(6)Erdélyt, A., Magnus, W., Oberhettinger, F. and Tricomi, F. G.Higher transcendental functions, vol. II (New York, McGraw-Hill, 1953).Google Scholar
(7)Erdélyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F. G.Tables of integral transforms, vol. II (New York, McGraw-Hill, 1954).Google Scholar
(8)Feller, W.An introduction to probability theory and its applications, vol. II (New York, John Wiley, 1966).Google Scholar
(9)Grosswald, E.The Student t-distribution of any degrees of freedom is infinitely divisible. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 36, (1976), 103–109.CrossRefGoogle Scholar
(10)Hirschman, I. I. and Widder, D. V.The convolution transform (Princeton University Press, 1955).Google Scholar
(11)Ismail, M. E. H.Bessel functions and the infinite divisibility of the Student t-distribution. Ann. Prob. 5, (1977), 582–585.CrossRefGoogle Scholar
(12)Ismail, M. E. H.Integral representations and complete monotonicity of various quotients of Bessel functions. Canadian J. Math. 29, (1977), 1198–1207.CrossRefGoogle Scholar
(13)Ismail, M. E. H. and Kelker, D. H.Special functions, Stieltjes transforms and infinite divisibility. SIAM J. Math. Anal. 10 (1979) (To appear.)CrossRefGoogle Scholar
(14)Kent, J.Some probabilistic properties of Bessel functions. Ann. Prob. 6, (1978), 760–770.CrossRefGoogle Scholar
(15)Loéve, M.Probability theory, foundations, random sequences, 4th ed. (Van Nostrand, 1976).Google Scholar
(16)Rainville, R. E.Special functions (New York, MacMillan, 1960).Google Scholar
(17)Siewert, G. E. and Burniston, E. E.An exact analytical solution of x coth x = αx 2 + 1. J. Comp. Appl. Math. 2 (1976), 19–21.CrossRefGoogle Scholar
(18)Smart, J. S.Effective field theories of magnetism (Philadelphia, Saunders, 1966).CrossRefGoogle Scholar
(19)Thorin, O.On the infinite divisibility of the Pareto distribution. Scand. Actuarial J. (1977), 31–40.CrossRefGoogle Scholar
(20)Thorin, O.On the infinite divisibility of the log normal distribution. Scand. Actuarial J. (1977), 121–148.CrossRefGoogle Scholar
(21)Tricomi, F. G.Über die Abzählung der Nullstellen der konfluenten hypergeometrischen Funktionen. Math. Zeit. 52, (1950), 669–675.CrossRefGoogle Scholar
(22)Watson, G. N.A treatise on the theory of Bessel functions, 2nd ed. (Cambridge, Cambridge University Press, 1944).Google Scholar
(23)Wendel, J. Hitting spheres (submitted).Google Scholar
(24)Widder, D. V.The Laplace transform (Princeton, Princeton University Press, 1941).Google Scholar