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Spherical curves and quadratic relationships for special functions

Published online by Cambridge University Press:  17 February 2009

Even Mehlum  and
Jet Wimp
Even Mehlum
Affiliation:
Central Institute for Industrial Research, Oslo, Norway.
Jet Wimp
Affiliation:
Department of Mathematical Sciences, Drexel University, Philadelphia, Pennsylvania 19104, U.S.A.
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Abstract

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We show that the position vector of any 3-space curve lying on a sphere satisfies a third-order linear (vector) differential equation whose coefficients involve a single arbitrary function A(s). By making various identifications of A(s), we are led to nonlinear identities for a number of higher transcendental functions: Bessel functions, Horn functions, generalized hypergeometric functions, etc. These can be considered natural geometrical generalizations of sin2t + cos2t = 1. We conclude with some applications to the theory of splines.

Type
Research Article
Information
The ANZIAM Journal , Volume 27 , Issue 1 , July 1985 , pp. 111 - 124
Copyright
Copyright © Australian Mathematical Society 1985

References

[1]Askey, R., “Jacobi polynomials, I.”, SIAM J. Math. Anal. 5 (1974), 119–124.CrossRefGoogle Scholar
[2]Askey, R. (ed.), Theory and application of special functions (Academic Press, New York, 1975).Google Scholar
[3]Eisenhart, L. P., An introduction to differential geometry (Princeton University Press, 1947).Google Scholar
[4]Erdélyi, A., “Some confluent hypergeometric functions of two variables”, Proc. Roy. Soc. Edinburgh 60 (1939–1940), 341–361.Google Scholar
[5]Erdélyi, A. et al. , Higher transcendental functions, 3 volumes (McGraw-Hill, New York, 1953).Google Scholar
[6]Fields, J. L. and Ismail, M. E., “On the positivity of some 1F2's”, SIAM J. Math. Anal. 6 (1975), 551–559.CrossRefGoogle Scholar
[7]Ince, E. L., Ordinary differential equations (Dover, New York, 1956).Google Scholar
[8]Luke, Y. L., Mathematical functions and their approximations (Academic Press, New York, 1975).Google Scholar
[9]Mehlum, E., “Nonlinear splines”, in Computer aided geometric design (eds. Barnhill, R. E. and Riesenfield, R. F.), (Academic Press, New York, 1974).Google Scholar
[10]Mehlum, E., “Appell and the apple (Nonlinear splines in space)”, Report, Sentralinstitutt for industriell forskning, Oslo. (This report is obtained by writing to SI, Forskningsveien 1, Oslo 3, Norway.)Google Scholar
[11]Struik, D. J., Lectures on classical differential geometry (Addison-Wesley, Reading, Massachusetts, 1950).Google Scholar
[12]Watson, G. N., A treatise on the theory of Bessel functions, 2nd ed. (Cambridge University Press, 1962).Google Scholar