Proceedings of the Edinburgh Mathematical Society (Series 2)

Research Article

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

R. B. Parisa1

a1 Association Euratom—Cea, Centre d'Etudes Nucleaires, 92260 Fontenay-aux-Roses, France

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

S0013091500016503_eqn1

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

S0013091500016503_eqn2

The first demonstrations that θn lies between ½ and S0013091500016503_inline1 and that θn decreases monotoni-cally to the value S0013091500016503_inline1 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

S0013091500016503_eqn3

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

S0013091500016503_eqn4

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

S0013091500016503_eqn5

(Received June 06 1980)