On Legendre numbers of the second kind

• Main Author: Paul W. Haggard
• Format: Article
• Language: English
• Published: Hindawi Limited 1988-01-01
• Series: International Journal of Mathematics and Mathematical Sciences
• Subjects: associated Legendre functions; extended Legendre numbers of the first and second kinds; gamma function; infinite products; Legendre numbers of the first and second kinds; Legendre's associated differential equation; Legendre's differential equation; spherical functions.
• Online Access: http://dx.doi.org/10.1155/S0161171288000997

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.