Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|improve this question
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.