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Integral Representations and Complete Monotonicity of Various Quotients of Bessel Functions

Published online by Cambridge University Press:  20 November 2018

Mourad E. H. Ismail
Mourad E. H. Ismail*
Affiliation:
McMaster University, Hamilton, Ontario
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Complete monotonicity of functions, Definition 3.1, is often proved by showing that their inverse Laplace transforms are nonnegative. There are relatively few simple functions whose inverse Laplace transforms can be expressed in terms of standard higher transcendental functions. Inverting a Laplace transform involves integrating a complex-valued function over a vertical line, and establishing the positivity of the resulting integral can be tricky. Sometimes asymptotic methods are helpful, see for example Fields and Ismail [6].

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 29 , Issue 6 , 01 December 1977 , pp. 1198 - 1207
Copyright
Copyright © Canadian Mathematical Society 1977

References

1. Askey, R., Orthogonal polynomials and special functions, CBMS, Volume 21 (SIAM, Philadelphia, 1975).Google Scholar
2. Carslaw, H. S. and Jaeger, J. C., Some problems in the mathematical theory of the conduction of heat, Philosophical Magazine 26 (1938), 473–495.Google Scholar
3. Erdelyi, A., Magnus, W., F. Oberhettinger and Tricomi, G. F., Higher transcendental functions, Volume 1 (McGraw-Hill, New York, 1953).Google Scholar
4. Erdelyi, A. Higher transcendental functions, Volume 2 (McGraw-Hill, New York, 1953).Google Scholar
5. Feller, W., An introduction to probability theory and its applications, Volume 2 (John Wiley, New York, 1966).Google Scholar
6. Fields, J. L. and Ismail, M. E. H., On the positivity of some iF2's, SIAM J. Math. Anal. (1975), 551–559.Google Scholar
7. Grosswald, E., The student t-distribution of any degrees of freedom is infinitely divisible, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 36 (1976), 103–109.Google Scholar
8. Hartman, P., Completely monotone families of solutions of nth order linear differential equations and infinitely divisible distributions, Ann. Scoula Norm. Sup. Pisa (4) 3 (1976), 267–287.Google Scholar
9. Hartman, P. and Watson, G., “Normal” distribution functions on spheres, Ann. Prob. 2 (1974), 593–607.Google Scholar
10. Hirschman, I. I. and Widder, D. V., The convolution transform (Princeton University Press, Princeton, 1955).Google Scholar
11. Ismail, M. E. H., Bessel functions and the infinite divisibility of the student t-distribution, Ann. Prob. 6 (1977), 582–585.Google Scholar
12. Ismail, M. E. H. and Kelker, D. H., The Bessel polynomials and the student t-distribution, SIAM J. Math. Anal. 7 (1976), 82–91.Google Scholar
13. Ismail, M. E. H. and Muldoon, M. E., Monotonicity of the zeros of a cross-product of Bessel functions, SIAM. J. Math. Anal. 9 (1978), to appear.Google Scholar
14. Lorch, L., Inequalities for some Whittaker functions, Arch. Math. (Brno) 3 (1967), 1–9.Google Scholar
15. Olver, F. W. J., Asymptotics and special functions (Academic Press, New York, 1974).Google Scholar
16. Stone, M. H., Linear transformations in hilbert space and their applications to analysis, American Mathematical Society Colloquium Publications 15 (New York, 1932).Google Scholar
17. Watson, G. N., A treatise on the theory of bessel functions, second edition (Cambridge University Press, Cambridge, 1944).Google Scholar
18. Whittaker, E. T. and Watson, G. N., Modern analysis, fourth edition (Cambridge University Press, Cambridge, 1927).Google Scholar
19. Widder, D. V., The Laplace transform (Princeton University Press, Princeton, 1941).Google Scholar