International Journal of Mathematics and Mathematical Sciences
Volume 11 (1988), Issue 2, Pages 405-412
Some applications of Legendre numbers
Department of Mathematics, East Carolina University, Greenville 27858, North Carolina, USA
Received 26 August 1986
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.
The associated Legendre functions are defined using the Legendre numbers. From these the associated Legendre polynomials are obtained and the derivatives of these polynomials at are derived by using properties of the Legendre numbers. These derivatives are then used to expand the associated Legendre polynomials and in series of Legendre polynomials. Other applications include evaluating certain integrals, expressing polynomials as linear combinations of Legendre polynomials, and expressing linear combinations of Legendre polynomials as polynomials. A connection between Legendre and Pascal numbers is also given.