In the reflection formula for the polygamma function $$ \psi^{(n)}(1-z) + (-1)^{n+1}\psi^{(n)}(z) % = (-1)^{n} \pi \frac{d^{n}}{d z^{n}} \cot (\pi z) $$ the right hand side is a polynomial $(-1)^{n}\pi^{n+1} P_{n+1}(\cot \pi z)$. With $x=\cot \pi z$, the first few polynomials are: $$ x\\ -x^2-1\\ 2 x^3 + 2 x\\ -6 x^4 - 8 x^2 - 2\\ 24 x^5 +40 x^3 + 16 x\\ -120 x^6-240 x^4-136 x^2-16\\ $$ For a numerical library function I have already computed and stored coefficients up to $n=12$. The Wikipedia article gives a recursion formula for (slightly different) $P_{n+1}$ in terms of $P_n$ and $P_n'$. Now my question: Is there a known recurrence formula for computing the values $P_n(x)$ of the form like those for orthogonal polynomials \begin{align} P_n(a) &= a(n,x) P_{n-1}(x) + b(n,x) P_{n-2}(x), \quad\text{or maybe}\\ P_n(a) &= a(n,x) P_{n-1}(x) + b(n,x) P_{n-2}(x)+ c(n,x) P_{n-3}(x) \end{align} with (relative) simple expressions $a, b, c$? I do not want to compute and cache the coefficients for higher degrees; and I am open to other or better numerical algorithms.


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