The Legendre function of the second kind, $Q_n(z)$, along with the usual Legendre polynomial $P_n(z)$, are the two linearly independent solutions of the Legendre differential equation.

$Q_n(z)$ can be expressed in the following form:

$$Q_n(z)=P_n(z)\mathrm{artanh}\,z-W_{n-1}(z)$$

where $W_{n-1}(z)$ can be expressed either as

$$W_{n-1}(z)=\sum_{k=1}^n \frac{P_{k-1}(z) P_{n-k}(z)}{k}$$

or as

$$W_{n-1}(z)=\sum_{k=0}^{n-1} \frac{(H_n-H_k)(n+k)!}{2^k (n-k)! (k!)^2} (z-1)^k$$

where $H_k$ is the $k$-th harmonic number, $H_k=\sum_{j=1}^k \frac1{j}$.

My questions:

1. Mathematica returns a rather complicated expression for $W_{n-1}(z)$ involving the unknown solution of a certain recurrence (i.e. DifferenceRoot[]). Is there possibly a simpler form for this polynomial?

2. Might there be a (hopefully simple) $n$-term recurrence that generates these polynomials?