Orthogonality of Associated Legendre Functions for Noninteger Order and Index

α

-5

μ + α

0

ν + α

0

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positive area part

5.01042×

-8

10

negative area part

0.

sum of both

5.01042×

-8

10

theoretical prediction

5.01042×

-8

10

This Demonstration shows an unusual orthogonality relation for Legendre functions of the first kind

α

P

μ

. In contrast to the well-known situation of integer degree

μ

and integer order

α

(associated Legendre polynomials), in this case both degree and order are allowed to take noninteger real values. For fixed

α

, these functions satisfy an orthogonality relation, according to which

1

∫

-1

α

P

μ

(x)

α

P

ν

(x)x

is 0 whenever

μ≠ν

is equal to 2 if

μ=ν=-1/2

and

α=1/2

, and is given by the formula

2(μ+α)!

(2μ+α)Γ(μ-α+1)

in any other case.