>Yes, the MathieuC and MathieuS functions are derived as Fourier coefficients, but they are recursively established by the +/- perturbation of the $\omega$ frequency on the wave equation. It does take lots of computations to get the Mathieu functions to converge properly ( see here ).

I don't understand what you mean by "+/- perturbation of the $\omega$ frequency on the wave equation" and I have troubles to understand this recursion in the link you gave, finally m is supposed to go to infinity in the fourier expansion, but then in the recursion explanation it says m goes until $a_{2n}$, where $a_{2n}$ [seems](http://mathworld.wolfram.com/MathieuFunction.html) to be a characteristic value.

>It definitely is not something as simple as the Taylor series expansion of a Sine function, which is what I meant by not “easily understood”.

Yes. Taylor expensions are funny, aren't they?

>and unless one has a long-enough time series to deal with, one might not pick up the repeat sequence.

I don't understand what you mean with repeat sequence.

But apart from that Mathieu function discussion, I am not so sure wether there exists at all a mathematical function, which

would describe such a "forced to periodicity" behaviour as it seems to be the case for the QBO (at least if one believes that blog post diagram)

Did you check the QBO diagram?

I don't understand what you mean by "+/- perturbation of the $\omega$ frequency on the wave equation" and I have troubles to understand this recursion in the link you gave, finally m is supposed to go to infinity in the fourier expansion, but then in the recursion explanation it says m goes until $a_{2n}$, where $a_{2n}$ [seems](http://mathworld.wolfram.com/MathieuFunction.html) to be a characteristic value.

>It definitely is not something as simple as the Taylor series expansion of a Sine function, which is what I meant by not “easily understood”.

Yes. Taylor expensions are funny, aren't they?

>and unless one has a long-enough time series to deal with, one might not pick up the repeat sequence.

I don't understand what you mean with repeat sequence.

But apart from that Mathieu function discussion, I am not so sure wether there exists at all a mathematical function, which

would describe such a "forced to periodicity" behaviour as it seems to be the case for the QBO (at least if one believes that blog post diagram)

Did you check the QBO diagram?