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  • Proceedings of the Edinburgh Mathematical Society, Volume 24, Issue 3
  • October 1981, pp. 179-195

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

  • R. B. Paris (a1)
  • DOI: http://dx.doi.org/10.1017/S0013091500016503
  • Published online: 01 January 2009
Abstract

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

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This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

(2)J. D. Buckholtz , Concerning an approximation of Copson, Proc. Amer. Math. Soc. 14 (1963), 564568.

(3)L. Carlitz , The coefficients in an asymptotic expansion, Proc. Amer. Math. Soc. 16 (1965), 248252.

(15)R. Wong , On uniform asymptotic expansion of definite integrals, J. Approx. Theory 7 (1973), 7686.

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Proceedings of the Edinburgh Mathematical Society
  • ISSN: 0013-0915
  • EISSN: 1464-3839
  • URL: /core/journals/proceedings-of-the-edinburgh-mathematical-society
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