- #1

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- 2

- Summary:
- Recurrence Identities for the Associated Legendre Function of Second Kind

The associated Legendre Function of Second kind is related to the Legendre Function of Second kind as such:

$$

Q_{n}^m(z)= (-1)^m (1-z^2)^{m/2} \frac{d^m}{dz^m}(Q_{n}(z))

$$

The recurrence relations for the former are the same as those of the first kind, for which one of the relations is:

$$

(n-m+1)Q_{n+1}^m(z)=(2n+1)zQ_{n}^m(z)-(n+m)Q_{n-1}^m(z)

$$

The recurrence relations and identities of the associated legendre function of the first kind are here: Associated Legendre function of 1st kind

However, none of the literature show anything about some identities I'm looking for!

For example one of the identities is when ##m=0##, the result is the same as such:

$$

Q_{n}^0(z)=Q_{n}(z)

$$

What I'm looking for are the identities when:

##1)## ##n## and ##m## are the same, that is

$$

Q_{n}^n(z)=?

$$

and ##2)## the identity when ##n = 0##, that is

$$

Q_{0}^m(z)=?

$$

Do those two identities exist?

$$

Q_{n}^m(z)= (-1)^m (1-z^2)^{m/2} \frac{d^m}{dz^m}(Q_{n}(z))

$$

The recurrence relations for the former are the same as those of the first kind, for which one of the relations is:

$$

(n-m+1)Q_{n+1}^m(z)=(2n+1)zQ_{n}^m(z)-(n+m)Q_{n-1}^m(z)

$$

The recurrence relations and identities of the associated legendre function of the first kind are here: Associated Legendre function of 1st kind

However, none of the literature show anything about some identities I'm looking for!

For example one of the identities is when ##m=0##, the result is the same as such:

$$

Q_{n}^0(z)=Q_{n}(z)

$$

What I'm looking for are the identities when:

##1)## ##n## and ##m## are the same, that is

$$

Q_{n}^n(z)=?

$$

and ##2)## the identity when ##n = 0##, that is

$$

Q_{0}^m(z)=?

$$

Do those two identities exist?