# Correct expressions for one-center Gaussian radial two-electron integrals

I am working on a new atomic solver library, for which I need expressions for the one-center two-electron integrals. Pitzer has furnished the necessary expressions for Slater-type and Gaussian-type orbitals in a book chapter (doi:10.1007/978-94-009-7921-5_11) and a Computer Physics Communications article (doi:10.1016/j.cpc.2005.04.003).

While the Slater-type orbital expressions appear to be correct, when evaluated against Maple, the Gaussian-type expressions on page 245 of the CPC article

$$R_{mn \nu}(x,y) = \int_0^\infty \int_0^\infty r_1^m e^{-xr_1^2} (r_<^\nu / r_>^{\nu+1}) r_2^n e^{-y r_2^2} {\rm d}r_1 {\rm d}r_2$$

appear to be wrong. The article gives the result

$$R_{mn \nu}(x,y) = \frac {\frac 1 2 (m+n-3)!} {x y (x+y)^{\frac 1 2 (m+n-3)} } \left[ 1 + E^{\frac 1 2 (m+n-3)}_{\frac 1 2 (n-\nu-2)}(y/x) + E^{\frac 1 2 (m+n-3)}_{\frac 1 2 (m-\nu-2)}(x/y) \right]$$

where

$$E_k^n (x) = \left[ \sum_{j=0}^{k-1} {n \choose j} x^j \right] / \left[ {n \choose k} x^k \right]$$

The problem is quite coarse in nature: the indices of $$E$$ are half-integer!

Does anyone happen to know what the correct equations are? The code of the article dates back to the 1960s and is basically unintelligible: the variable names are 4 characters and there are no comments whatsoever.

Upon further examination, although the recursion relation used to evaluate the $$E$$ functions given in the paper are wrong, it turns out that the equation in terms of the binomial coefficients is actually correct. There are numerous other typos in the paper, as well.