I am currently presented with a Legendre polynomial of the second kind with an argument greater than 1, and I'm wondering if there exists any recursions for them? I am currently using the hypergeometric definition
$ Q_n (z) = \frac{\Gamma(n+1)\Gamma\left(\frac{1}{2}\right)}{2^{n+1}\Gamma\left(n+\frac{3}{2}\right)}z^{-(n+1)} \, _2F_1 \left(\frac{n+2}{2}, \frac{n+1}{2}; n+\frac{3}{2}; \frac{1}{z^2} \right) $
to evaluate it due to the nature of it's argument. The reason I ask is I have to calculate the first $k$ of these polynomials all with a common argument, but have no really efficient way to do this besides repeatedly using this definition. I have tried searching in various table / handbook texts but nothing really applies to the $Q$ polynomials over the interval [1, $\infty$).
The recursion formulas are the same, but for $|x|>1$ the forward recurrence is not stable and the backward recursion should be performed to compute $Q_l(x)$ \begin{align*} Q_{k-1}(x) &= \Big( (2k+1)xQ_k(x) - (k+1)Q_{k+1}(x)\Big) / k,\\ Q_1(x) &= \frac{x}{2}\ln\frac{x+1}{x-1}-1,\\ Q_0(x) &= \frac{1}{2}\ln\frac{x+1}{x-1}, \end{align*} with the starting values $Q_{l+2} = t,\; Q_{l+1} = f(l,x)t$, where $t$ is an arbitrary tiny value and $f(l,x)$ is the continued fraction from http://dlmf.nist.gov/14.14:
$$ l \frac{Q_l(x)}{Q_{l-1}(x)} = \frac{x_0}{y_0 -} \, \frac{x_1}{y_1 -} \, \frac{x_2}{y_2-} \cdots \quad \text{with} \quad x_k = (l+k)^{2}, \;y_k = (2l+2k+1)x. $$ The result of the recurrence is normalised with the true value $Q_0(x)$ (or $Q_1(x),$ dependent on the ratio of the last two values). For $x$ values close to 1, the continued fraction may not converge fast enough, in this case the hypergeometric representation can be used.