Editing Associated Legendre polynomials (section)

==Gaunt's formula==
<!-- This section is linked from [[Gaunt's formula]]. See [[WP:MOS#Section management]] -->

The integral over the product of three associated Legendre polynomials (with orders matching as shown below) is a necessary ingredient when developing products of Legendre polynomials into a series linear in the Legendre polynomials.  For instance, this turns out to be necessary when doing atomic calculations of the [[Hartree–Fock]] variety where matrix elements of the [[Coulomb operator]] are needed.  For this we have Gaunt's formula <ref>From John C. Slater ''Quantum Theory of Atomic Structure'', McGraw-Hill (New York, 1960), Volume I, page 309, which cites the original work of J. A. Gaunt, ''Philosophical Transactions of the Royal Society of London'', A228:151 (1929)</ref><ref>{{cite journal|first1=Yu-Lin|last1=Xu|title=Fast evaluation of the Gaunt coefficients|journal=Math. Comp.|year=1996|volume=65|number=216|pages=1601-1612|doi=10.1090/S0025-5718-96-00774-0}}</ref>
<math display="block">\begin{align}
\frac{1}{2} \int_{-1}^1 P_l^u(x) P_m^v(x) P_n^w(x) dx
={}&{}(-1)^{s-m-w}\frac{(m+v)!(n+w)!(2s-2n)!s!}{(m-v)!(s-l)!(s-m)!(s-n)!(2s+1)!} \\
   &{}\times \ \sum_{t=p}^q (-1)^t \frac{(l+u+t)!(m+n-u-t)!}{t!(l-u-t)!(m-n+u+t)!(n-w-t)!}
\end{align}</math>
This formula is to be used under the following assumptions: 
# the degrees are non-negative integers <math>l,m,n\ge0</math>
# all three orders are non-negative integers <math>u,v,w\ge 0</math>
# <math>u</math> is the largest of the three orders
# the orders sum up <math>u=v+w</math>
# the degrees obey <math> m\ge n</math>

Other quantities appearing in the formula are defined as
<math display="block"> 2s = l+m+n </math>
<math display="block"> p = \max(0,\,n-m-u) </math>
<math display="block"> q = \min(m+n-u,\,l-u,\,n-w) </math>

The integral is zero unless
# the sum of degrees is even so that <math>s</math> is an integer
# the triangular condition is satisfied <math>m+n\ge l \ge m-n</math>

Dong and Lemus (2002)<ref>Dong S.H., Lemus R., (2002), [http://www.sciencedirect.com/science/article/pii/S0893965902800040 "The overlap integral of three associated Legendre polynomials"], Appl. Math. Lett. 15, 541-546.</ref> generalized the derivation of this formula to integrals over a product of an arbitrary number of associated Legendre polynomials.