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Results old and new on the hyper-Bessel equation

Published online by Cambridge University Press:  14 November 2011

R. B. Paris  and
A. D. Wood
R. B. Paris
Affiliation:
Association Euratom-CEA, Centre d'Etudes Nucléaires de Cadarache, 13108 St. Paul-lez-Durance, France
A. D. Wood
Affiliation:
National Institute for Higher Education, Glasnevin, Dublin 9, Ireland
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Synopsis

We consider a variety of integral representations, single and multiple, old and new, for solutions of the hyper-Bessel equation u(n) – zmu =0. In particular, we show how a very early multiple Laplace integral solution of Molins (1876) may be related to recent Mellin–Barnes integral representations given by the present authors by way of multiple integral solutions given by Saxton and the second author for an associated equation. Although both these multiple integral solutions may be found by elementary methods, it is not easy to find their asymptotic expansions for large z, and we show how these may conveniently be obtained from our earlier results [10].

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 106 , Issue 3-4 , 1987 , pp. 259 - 265
Copyright
Copyright © Royal Society of Edinburgh 1987

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References

1Bleistein, N. and Handelsman, R. A.. The asymptotic expansion of integrals (New York: Holt, Rinehart and Winston, 1975).Google Scholar
2Braaksma, B. L. J.. Asymptotic analysis of a differential equation of Turrittin. SIAM J. Math. Anal. 2 (1971), 1–16.CrossRefGoogle Scholar
3Delerue, P.. Sur l'utilisation des fonctions hyperbesseliennes á la résolution d'une équation différentielle. C.R. Acad. Sci. Paris 230 (1950), 912–914.Google Scholar
4Heading, J.. The Stokes phenomenon and certain nsth order differential equations I, II. Proc. Camb. philos. Soc. Math. phys. Sci. 53 (1957), 399–441.CrossRefGoogle Scholar
5Humbert, P.. Les fonctions de Bessel du troisième ordre. Atti Accad. pontif. Nuovi Lincei 83 (1930), 128–146.Google Scholar
6Lobatto, R.. Sur l'intégration des équations d ny/dx n – xy =0, d 2y/dx 2 + abx ny = 0 par des intégrates déflnies. J. Math. (Crelle) 17 (1837), 363–371.Google Scholar
7Molins, M. H.. Sur l'intégration de l'équation différentielle d ky/dx k = ax my. Mém. Acad. Sci. Inscript. Toulouse 1 (VIII) (1876), 167–189.Google Scholar
8Paris, R. B.. On the asymptotic expansions of solutions of an nth order linear differential equation. Proc. Roy. Soc. Edinburgh Sect. A 85 (1980), 15–57.CrossRefGoogle Scholar
9Paris, R. B. and Wood, A. D.. Asymptotics of high order differential equations. Pitman Research Notes in Mathematics Series 129 (London: Longman 1986).Google Scholar
10Paris, R. B. and Wood, A. D.. On the asymptotic expansion of solutions of an nth order linear differential equation with power coefficients. Proc. Roy. Irish Acad. Sect. A 85A (1985), 201–220.Google Scholar
11Saxton, R. A.. Asymptotics of integral representation solutions for a class of higher order linear ordinary differential equation. Thesis, Cranfield Institute of Technology, 1977.Google Scholar
12Scherk, H. F.. Uber die Integration der Gleichung d ny/dx n = (a + bx)y. J. Math. (Crelle) 10 (1833), 92–97.Google Scholar
13Spitzer, S.. Integration der linearen Differentialgleichung y(n) = Ax 2y n + Bxy' + Cy mittelst bestimmter Integrate. Math. Ann. 3 (1871), 453–455.CrossRefGoogle Scholar
14Turrittin, H. L.. Stokes multipliers for asymptotic solutions of a certain differential equation. Trans. Amer. Math. Soc. 68 (1950), 304–329.CrossRefGoogle Scholar
15Watson, G. N.. A treatise on the theory of Bessel functions (Edinburgh: Cambridge University Press, 1944).Google Scholar