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I'm successively integrating $x^{n} \cos{k x}$ for increasing values of positive integer n. I'm finding:

$\frac{\sin{kx}}{k}$,

$\frac{\cos{kx}}{k^2}+\frac{x\sin{kx}}{k}$,

$\frac{2 x \cos{kx}}{k^2}+\frac{\left(-2+k^2 x^2\right)sin{kx}}{k^3}$,

$\frac{3 \left(-2+k^2 x^2\right) \cos{kx}}{k^4}+\frac{x \left(-6+k^2 x^2\right) \sin(kx)}{k^3}$

Is there a name for the sequence of polynomials: $x$, $2x$, $k^2x^2-2$, $3(k^2x^2-2)$, $x(k^2x^2-6)$ ... ?

Here is more:

$\frac{\sin{kx}}{k}$

$\frac{\cos{kx}}{k^2}+\frac{x \sin{kx}}{k}$

$\frac{2 x \cos{kx}}{k^2}+\frac{\left(-2+k^2 x^2\right) \sin{kx}}{k^3}$

$\frac{3 \left(-2+k^2 x^2\right) \cos{kx}}{k^4}+\frac{x \left(-6+k^2 x^2\right) \sin{kx}}{k^3}$

$\frac{4 x \left(-6+k^2 x^2\right) \cos{kx}}{k^4}+\frac{\left(24-12 k^2 x^2+k^4 x^4\right) \sin{kx}}{k^5}$

$\frac{5 \left(24-12 k^2 x^2+k^4 x^4\right) \cos{kx}}{k^6}+\frac{x \left(120-20 k^2 x^2+k^4 x^4\right) \sin{kx}}{k^5}$

$\frac{6 x \left(120-20 k^2 x^2+k^4 x^4\right) \cos{kx}}{k^6}+\frac{\left(-720+360 k^2 x^2-30 k^4 x^4+k^6 x^6\right) \sin{kx}}{k^7}$

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    They look a bit like Hermite polynomials.2012-06-04
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    They don't seem to be exactly Hermite polynomials :-)2012-06-04
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    Indeed, but for $k=2$ it might be a good place to start your search.2012-06-04
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    Is there a website for sequences of polynomials like OEIS: http://oeis.org/?2012-06-04
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    I did some search but didn't turn up anything. It doesn't seem to be an interesting sequence of polynomials, even for k = 2.2012-06-04

2 Answers 2