Generalization Via Hypergeometric Functions
These functions may actually be defined for general complex parameters and argument:
where is the gamma function and is the hypergeometric function
They are called the Legendre functions when defined in this more general way. They satisfy the same differential equation as before:
Since this is a second order differential equation, it has a second solution, defined as:
and both obey the various recurrence formulas given previously.
Read more about this topic: Associated Legendre Polynomials
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