Matlab doesn't calculate associated Legendre type 3 (z>1) function but does do type 2 (-1<=x<=1) approx. .0007 sec compared to average .00014sec  recNM3(n,m,z) which is a recursive implementation.

Julia file that calculates some associated Legendre functions

On line 46 changed 1 to one(x)
On line 84 changed 1.  to one(z)
for arbitrary precision, etc.

Thanks for editting and making the PR more readable; congratulations on your new position AA.  Thanks to SJ for mentoring and encouragement.  
I can't benchmark against what does not exist: matlab, maxima, scipy.special, gsl do not have associated Legendre type 3 (z > 1).  I have used SLegendreP3 (which I derived and used a long time ago) ; similarly SLegendreP2  gives reliable results compared to scipy.special   for |x| <1 .  function BenchmarkZZLegendreP2 , shows that ZZLegendreP2  is faster than scipy.special.  ZZLegendreP3 comes from a modification of work published by Selezneva, etal. 
If the general direction of the PR is OK; then I can extract all the test... functions to test.jl and do additional work.  All suggestions, comments,etc. are welcome.  Thank you for helping me to learn Julia scientific computing.

As requested, @Btime from BenchmarkTools and @time both gave similar results in that ZZLegendreP2 is light years ahead of scipy.special.lpmv.  Accuracy of ZZLegendreP2 ( |z|<=1) against lpmv and ZZLegendreP3 (z > 1) against Fortran table is good.

This PR is the file entitled associatedLegendre.jl.  The first part with all the comments is like a readme.  The rest is the source code and numerous tests(requested) and exploration of area near |x| ~ 1.  As requested @Btime from BenchmarkTools as well as @time gave similar results in that ZZLegendreP2 is light years ahead of scipy.special.lpmv.  Accuracy of ZZLegendreP2 (|z|=<1.) against lpmv and ZZLegendreP3 (z>1.) against Fortran is  good.

delete