The Convergence of Two Series Representations for Associated Legendre Functions of the First and Second Kind

摘要

The regions of convergence and divergence of two series representations for associated Legendre functions of the first and second kind degree nu and order mu, are investigated. The series are shown to be absolutely convergent when the real part of the complex order mu is less than zero, conditionally convergent when the real part of mu is greater than or equal to zero and less than one-half, and divergent when the real part of mu is greater than or equal to one-half. Numerical calculations for several real values of nu and mu are used to demonstrate the behavior of the partial sums of the series in the regions of convergence and divergence.