$n^{th}$ derivative of $\cot x$
What is the $n^{th}$ derivative of $\cot(x)$?
I tried to differentiate it may times:
I can't see a pattern forming. Please help.
$\endgroup$ 21 Answer
$\begingroup$There is a pattern but it is not simple. Apparently the pattern was found only quite recently:
V.S. Adamchik, On the Hurwitz function for rational arguments, Applied Mathematics and Computation, Volume 187, Issue 1, 1 April 2007, Pages 3–12
See Lemma 2.1. The text is available at the author's site: pdf
There is also a recursion formula:
If $\dfrac{d^n}{dx^n} \cot x = (-1)^n P_n(\cot x)$, then $P_0(u)=u$, $P_1(u)=u^2+1$, and $$ P_{n+1}(u) = \sum_{j=0}^n \binom{n}{j} P_j(u) P_{n-j}(u) $$ for $n\ge 1$.
This formula appears in
Michael E. Hoffman, Derivative polynomials for tangent and secant, The American Mathematical Monthly, Vol. 102, No. 1 (Jan., 1995), pp. 23-30
I learned of this formula and paper in
$\endgroup$Kurt Siegfried Kölbig, The polygamma function and the derivatives of the cotangent function for rational arguments, CERN-CN-96-005, 1996.
