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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
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 en, 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
Copyright
Copyright © Edinburgh Mathematical Society 1981

References

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