Abstract
The work of the present paper deals with enveloping series for certain of the special functions of mathematical physics. In many (but not all) cases, the Maclaurin-series expansion of the function concerned envelops the function itself and can be regarded as an asymptotic expansion of the function about the origin.
An important case refers to the hypergeometric function 2F1(a, b; c; x), which has simple upper and/or lower bounds whenever its Maclaurin-series expansion has coefficients with nonconstant sign. A similar result is found for certain other higher-order hypergeometric functions, including the confluent hypergeometric function 1F1(a;b;x). The methods described here have applications to other classes of functions, including those referred to in Truesdell’s “Essay toward a unified theory of special functions.” It is Truesdell’s essay which has prompted the present approach to the subject.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
M. Abramowitz and I. A. Stegun (eds.), Handbook of Mathematical Functions, Dover Publications, Inc., New York, 1965.
R. Askey, One-sided approximation to special functions, Siam Rev. 16 (1974), 545–546;
R. Askey, One-sided approximation to special functions, Siam Rev. 18 (1976), 121–122.
C. V. Durell and A. Robson, Advanced Trigonometry, G. Bell and Sons, Ltd., 1949.
T. Erber, Inequalities for hypergeometric functions, Arch. Rational Mech. and Anal. 4 (1959–1960), 341–351.
A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions, Vol. II, McGraw-Hill, New York, 1953.
T. M. Flett, Some inequalities for a hypergeometric integral, Edin. Math. Soc. Proc. 18 (1972), 31–34.
G. Gasper, Positive integrals of Bessel functions, Siam J. Math. Anal. 6 (1975), 868–881.
L. Gerber, An extension of Bernoulli’s inequality, Amer. Math. Monthly 75 (1968), 875–876.
E. Makai, On a monotonic property of certain Sturm-Liouville functions, Acta. Math. Acad. Sci. Hungar. 3 (1952), 165–172.
G. Pólya and G. Szegö, Problems and Theorems in Analysis, Vol. I, Springer-Verlag, New York, 1972.
D. K. Ross, A note on a generalisation of Bernoulli’s inequality for the binomial theorem, to appear.
J. Steinig, The sign of Lommel’s function, Trans. Amer. Math. Soc. 163 (1972), 123–129.
G. Szegö, Orthogonal Polynomials, Amer. Math. Soc. Colloq. Publ. 23, Providence, R. I., 1967.
E. C. Titchmarsh, The Theory of Functions, Oxford University Press, 1950.
C. Truesdell, An Essay Toward a Unified Theory of Special Functions, Princeton University Press, 1948.
R. E. Williamson, Multiply monotone functions and their Laplace transforms, Duke Math. J. 23 (1956), 189–207.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1980 Springer Basel AG
About this chapter
Cite this chapter
Ross, D.K., Mahajan, A. (1980). On Enveloping Series for Some of the Special Functions, and on Integral Inequalities Involving Them. In: Beckenbach, E.F. (eds) General Inequalities 2. International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série Internationale d’Analyse Numérique, vol 47. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-6324-7_17
Download citation
DOI: https://doi.org/10.1007/978-3-0348-6324-7_17
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-7643-1056-1
Online ISBN: 978-3-0348-6324-7
eBook Packages: Springer Book Archive