Patch against git
The recurrence relation given by orthopoly_recur() for legendre_{p,q}() is inconsistent with relations for other polynomials in that it gives the Legendre polynomial of degree (n) in terms of polynomials of degree (n-1) and (n-2) instead of giving the (n+1) polynomial in terms of (n) and (n-1) like the others. This patch makes it consistent with the other results (such as assoc_legendre_{p,q}) and with A&S p782.
This patch also corrects two recurrence relations (ultraspherical and hermite) in the comments. orthopoly_recur() gives the same relations as A&S, but there were sign errors in the comments.
To avoid possibly garbled text, I'll also attach a file showing the previous result for legendre_p(n,x), the result for assoc_legendre_p(n,0,x) and the new result for legendre_p(n,x), showing that the latter two are in the same form. (I guess I can upload other files after I initially submit this).
Thanks,
Kris Katterjohn
I just noticed that the description for orthopoly_recur() shows an example of its usage with legendre_p(). I'm not sure how the doc system works, but I attached a patch to update the output in doc/info/orthopoly.texi. I see that files in subdirectories of doc/info also contain this text, but I'm not sure if it just requires substituting in each file separately or if there is some generation process that takes care of it. Hopefully this helps either way.
Applied as commit e5a8abaa23ca and commit ff248ac1ee83a. Thanks, Kris.