# Orthogonality of Associated Legendre Functions for Noninteger Order and Index

Orthogonality of Associated Legendre Functions for Noninteger Order and Index

This Demonstration shows an unusual orthogonality relation for Legendre functions of the first kind . In contrast to the well-known situation of integer degree and integer order (associated Legendre polynomials), in this case both degree and order are allowed to take noninteger real values. For fixed , these functions satisfy an orthogonality relation, according to which (x)(x)x is 0 whenever is equal to 2 if and , and is given by the formula in any other case.

α

P

μ

μ

α

α

1

∫

-1

α

P

μ

α

P

ν

μ≠ν

μ=ν=-1/2

α=1/2

2(μ+α)!

(2μ+α)Γ(μ-α+1)