# $n$-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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In which form are you seeking to express the derivatives? A power series? Maybe a series of Legendre functions? Perhaps letting $\psi=z^{-m}\sum_{k=0}^{\infty}c_{k}z^{-k}$ could get you started? –  Valentin May 30 '12 at 21:58
Any form would be good for me. What is $c_k$ in your formula? Thank you. –  David May 30 '12 at 22:56
this is a standard beginning of derivation of a solution to Legendre's equation. $c_k$ are the coefficients to be determined. You may consider also expansion in associated Legendre functions, though this leads to a quite complicated recurrence relation –  Valentin May 30 '12 at 23:01
cross-posted to MO: mathoverflow.net/questions/98442/… –  user16299 Jun 8 '12 at 21:23
Nothing listed here, sadly; IIRC the derivatives are not expressible in terms of the spheroidal functions, and are thus treated as new functions in their own right. –  Guess who it is. Jun 9 '12 at 14:29