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Mathematics
General Math
Associated Legendre Function of Second Kind
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[QUOTE="Vick, post: 6312346, member: 657093"] [B]TL;DR Summary:[/B] 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: [URL='https://en.wikipedia.org/wiki/Associated_Legendre_polynomials#Recurrence_formula']Associated Legendre function of 1st kind[/URL] 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? [/QUOTE]
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Mathematics
General Math
Associated Legendre Function of Second Kind
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