IJPAM: Volume 43, No. 4 (2008)

ON GENERALIZATIONS OF LAMBERT'S SERIES

Tom Osler$^1$, Abdul Hassen$^2$
$^{1,2}$Department of Mathematics
Rowan University
201 Mullica Hill Road, Glassboro, NJ 08028-1701, USA
$^1$e-mail: hassen@rowan.edu
$^2$e-mail: osler@rowan.edu


Abstract.The classical Lambert's series makes it possible to generate many remarkable transformations of series. These Lambert's series are all constructed from the function $z/(1-z)$. In this paper we show how to generalize these series by using an arbitrary function in place of $z/(1-z)$. Series transformations exhibiting beautiful symmetry are obtained. In addition, a double contour integral is found which represents these series. Our method is compared to a general procedure introduced by MacMahon.

Received: December 6, 2007

AMS Subject Classification: 30B10, 30B50, 30E20, 40A05, 40A30

Key Words and Phrases: Lambert's series, series transformations, contour integrals, numerical functions

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2008
Volume: 43
Issue: 4