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On a generalisation of a result of Ramanujan connected with the exponential series

Published online by Cambridge University Press:  20 January 2009

R. B. Paris
R. B. Paris
Affiliation:
Association Euratom—Cea, Centre d'Etudes Nucleaires, 92260 Fontenay-aux-Roses, France
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One of the many interesting problems discussed by Ramanujan is an approximation related to the exponential series for en, when n assumes large positive integer values. If the number θn is defined by

Ramanujan (9) showed that when n is large, θn possesses the asymptotic expansion

The first demonstrations that θn lies between ½ and and that θn decreases monotoni-cally to the value as n increases, were given by Szegö (12) and Watson (13). Analogous results were shown to exist for the function e−n, for positive integer values of n, by Copson (4). If φn is defined by

then πn lies between 1 and ½ and tends monotonically to the value ½ as n increases, with the asymptotic expansion

A generalisation of these results was considered by Buckholtz (2) who defined, in a slightly different notation, for complex z and positive integer n, the function φn(z) by

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 24 , Issue 3 , October 1981 , pp. 179 - 195
Copyright
Copyright © Edinburgh Mathematical Society 1981

References

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