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In attempting to solve a recursion relation I have used a generating function method. This resulted in a differential equation to which I have the solution, and now I need to calculate the Taylor series around $z=0$. The solution involves Bessel functions of $1/z$, and hence I am not sure how to calculate the complete Taylor Series (or if it is even possible). The function is

$$f(z) = \frac{z}{2}+\frac{I_{-\frac23}(\frac2{3z}) + I_{\frac43}(\frac2{3z}) }{2 I_{\frac13}(\frac2{3z})}$$

where $I_{\alpha}(x)$ is the modified Bessel function of the first kind. I have attempted to calculate the Taylor coefficients numerically and it appears that they do exist (and are roughly what they recursion relation gives).

Is it possible to calculate the complete Taylor series of this function, and if so how would you go about it?

Edit: My apologies, I've made a typo in the function! There should be a 2 in the bessel function denominator. (Which there now is)

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    You can use double dollar signs to get displayed equations; they're a lot easier to read.2011-12-12
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    What recursion relation did you start from, and what was the series you said you got?2011-12-12
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    As a tiny note: Bessel functions of one-third order are expressible in terms of Airy functions.2011-12-12
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    The recursion relation is quite long, and difficult to explain. I do not have a closed form for the series (which is what I'm looking to get), but I can easily calculate using the recursion relation. This is a problem that arises from the WKB method of solving the 1d Schrodinger equation. Here is the paper that I'm working with - http://arxiv.org/PS_cache/nlin/pdf/0003/0003069v1.pdf . However it will contain a lot of extra information. The above problem is from a special case of equation 18, the differential equation being given in equation 31.2011-12-12

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