International Journal of Mathematics and Mathematical Sciences
Volume 11 (1988), Issue 4, Pages 815-822
doi:10.1155/S0161171288000997
Research Notes
On Legendre numbers of the second kind
Department of Mathematics, East Carollna University, Greenville 27858, NC, USA
Received 22 March 1988
Copyright © 1988 Paul W. Haggard. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
The Legendre numbers of the second kind, an infinite set of rational numbers, are defined from the associated Legendre functions. An explicit formula and a partial table for these numbers are given and many elementary properties are presented. A connection is shown between Legendre numbers of the first and second kinds. Extended Legendre numbers of the first and second kind are defined in a natural way and these are expressed in terms of those of the second and first kind, respectively. Two other sets of rational numbers are defined from the associated Legendre functions by taking derivatives and evaluating these at x=0. One of these sets is connected to Legendre numbers of the first find while the other is connected to Legendre numbers of the second kind. Some series are also discussed.