The generating function for the associated Legendre polynomials is a mathematical tool that, when expanded as a power series, produces associated Legendre polynomials as its coefficients in terms of a variable.
can be derived from the generating function for the Legendre polynomials. It begins with differentiating eq345 times using Leibniz’s theorem to give
Substituting eq363 in eq365 yields
The derivatives on the RHS have the following pattern:
Derivatives | |
Since
Eq366 is the generating function for the associated Legendre polynomials. It can also be expressed as:
which is obtained by letting .
Question
How do we use eq366 to generate and ?
Answer
For ,
Expanding using the binomial series gives
The polynomial corresponds to the coefficient of on the LHS of eq368. Comparing the coefficients of on both sides of eq368, . Using the same logic, .