Some applications of Legendre numbers
For n and s nonnegative integers, the associated Legendre functions are defined by \[ P^ s_ n(x)=(1-x^ 2)^{s/2}(d^ s/dx^ s)P_ n(x), \] where \(P_ n(x)\) is the Legendre polynomial of degree n. The Legendre numbers are defined by \(P^ k_ n=P^ k_ n(0)\). From these the associated Legendre polynomials are obtained and the derivatives of these polynomials at \(x=0\) are derived using properties of the Legendre numbers. These derivatives are then used to expand the associated Legendre polynomials and \(x^ n\) 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 given.
scientific article; zbMATH DE number 4044220