It's possible we have the formula mixed up, although it appears to follow
the convention for alpha,beta listed at the top of the doc/pair_eam.html page,
for the standard EAM (not FS) formula. The safest thing to do is look
in the code pair_eam_fs.cpp where the EAM/FS file is read and see what
arrays are assigned to what variables, and then which array is used when
the computation is done for that term. If you find something inconsistent,
please let us know. Also Tim Lau at MIT implemented this FS variant,
so see if you can find his email and ask him to clarify.

Also, I took a look to source code of pair_eam.cpp, line 193-194 where rho is calculated:
coeff = rhor_spline[type2rhor[jtype][itype]][m];
rho[i] += ((coeff[3]*p + coeff[4])*p + coeff[5])*p + coeff[6];

This is what I expected: rho[ j-type ][ i-type ], where i-type corresponds to E_i (energy of i-atom).
Like rho_H_Fe means induced electron density by H-atom to Fe-atom, and F_Fe(rho) should be used.
So, it seems code is consistent with papers, and only in formula alpha and beta should be flipped in rho notation (see, please, pic attached).

ok - looks like it is backwards in the formula - can you check that
the type2rhor array is read-in from the FS file in a way consistent
with what you are explaining and how the doc page says the
array inputs are ordered in the potential file? And let me know
if you contact Tim and he has any input on this. Else I will change
the formula.