# $n$-th derivative of the prolate spheroidal function

For a given real number $c>0$ define functions $\left(\psi_{k,c}(\cdot)\right)_{k\ge0}$, as an eigenfunctions of the Sturm-Liouville operators $L_c$ defined $$L_c(\psi)=(1-x^2)\frac{d^2\psi}{dx^2}-2x \frac{d\psi}{dx}-c^2x^2\psi$$ ( in fact, $\psi_{k,c}^{(n)}$ is called the prolate spheroidal wave function).

I would like to find the $n$-th derivative of the prolate spheroidal function $\psi_{k,c}^{(n)}(x)$ .

Any referense r ideas will be very helpful.

Thank you very much.

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You need to be a bit more specific about the context you are after. For instance, knowing that $L_c(\psi)=0$ already tells you what $\psi''$ is in terms of $\psi$ and $\psi'$. Similarly, taking higher derivatives of the differential equation, $\frac{d^n}{dx^n} L_c(\psi)=0$, lets you obtain the higher order derivatives in terms of the lower order ones. This is very similar to solving the ODE by power series. However, you need to know $\psi$ and $\psi'$ to get anywhere with this method. If this is not what you need, please make your question more precise. –  Igor Khavkine May 31 '12 at 10:48
BTW, if you do not already know it, here's a useful reference on spheroidal functions: dlmf.nist.gov/30 –  Igor Khavkine May 31 '12 at 10:48
Crossposted to MSE: math.stackexchange.com/questions/151683/… where it is tagged as homework –  Yemon Choi Jun 8 '12 at 21:23