Empirically, the Legendre functions of second kind, $Q\_n(x)$, appear to be of form 
$$
Q\_n(x)=\frac{P\_n(x)}{2} \cdot\ln(\frac{1+x}{1-x})+p\_n(x),
$$
with $P\_n(x)$ the Legendre polynomials of first kind and $p\_n(x)$ some rational polynomial of degree $n-1$.


This observation came up with my current reimplementation of $Q\_n(x)$ for the Sage CAS. Probably it has to do with the $Q\_n(x)$ satisfying the same recurrence as the $P\_n(x)$? Would it be possible to give the $p\_n(x)$ in terms of $P\_n(x)$? It would make this implementation much faster. Also, I haven't seen such a form in the standard literature, so this may be even new (though unsurprising).


Update: as the answer shows the form is not new. However, the $p\_n=W\_n$ satisfy the recurrence
$$ nW\_n = (2nx-x)W\_{n-1} - (n-1)W\_{n-2}, W\_0=0, W\_1=1, $$
which is not in the NIST handbook.



Legendre Q(n,x) function coefficients in terms of P(n,x) coefficients
The formular you search for is really known. It is on p. 360 of the NIST Handbook of Mathematical Functions, formulas (14.7.2)--(14.7.7). $p\_n(x)$ is in fact an explicit polynomial (not rational) with coefficients depending on $\psi$--function. 


