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I am using a generating function method to try and solve a recurrence. I have solved the resulting differential equation to find the generating function takes the form:

$$A(z) = \frac{\, _1F_1\left(\frac{3}{2}+\frac{1}{n+2};2+\frac{2}{n+2};-\frac{4}{n z+2 z}\right)}{z \, _1F_1\left(\frac{1}{2}+\frac{1}{n+2};\frac{n+4}{n+2};-\frac{4}{n z+2 z}\right)}-1$$

Notably the bottom $_1F_1$ can be converted to a bessel function (which has an asymptotic series)

So to complete the solution of the recurrence I need to convert this expression into a taylor series. When faced with a similar problem, recently, where I had a ratio of Bessel functions of (1/z) I used the asymptotic series to get an expression for the taylor coefficients.

That is, I expect (though only motivated by previous experience) the taylor coefficients around zero of this function to be related to the coefficients in the asymptotic series of ${}_1F_1$. However, I cannot find an asymptotic series for ${}_1F_1$. Is there a way to compute the coefficients? Or is there a better way to find the series coefficients for this generating function?

Thanks.

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