Analytical solution of an infinite series

[Matheco]

How to find the value of an infinite series. for e.g.Ʃ_{n=1}^{\infty} (β^{n-1}y^{R^{n}}e^{A(1-R^{2n})})

where β<1, R<1, y>1, and A>0?

Note that this series is covergent by Ratio test. I already have the numerical solution of the above. However, I am interested in analytical solution (approximation) of the value of infinite series (in terms of β,R, y, A).

 

[chiro]

Hey Matheco and welcome to the forums.

One suggestion I have is to see if you can find a taylor series or DE series (like the Hypergeometric or Bessel functions) and see if you can get a form that is similar.

Google should give you some results for tables of series in mathematics somewhere.

Other than that, you might try looking at the structure of the differential equations and integrals and see if you can construct a DE system that matches your equation.

Depending on how badly you want to solve it, it could be an interesting research problem.

 

[Matheco]

Thanks chiro for your reply. At the moment I am clueless about how to deal with this series. Could you recommend a reference where series are approximated by methods suggested by you?

 

[jackmell]

At the moment I am clueless about how to deal with this series.

Just one. That's all I'd be interested in at this point. That is, can I find just one series that I can compute it's analytic expression?

The term y^{R^n} is indicative of a lacunary function:

http://en.wikipedia.org/wiki/Lacunary_function

So we have a start. How much can we simplify your series, keep the lacunary term, and find an analytic expression for the resulting series?

How about we let \beta=1/2, R=1/2, A=1

Then we get:

\sum_{n=1}^{\infty} \frac{1}{2^{n-1}} y^{\frac{1}{2^n}}e^{(1-\frac{1}{4^n})}

Can we find an analytic expression for that one? If not, then can we come up with another set of non-trivial values of the parameters and find the analytic expression for the resulting series? Suppose that was your goal: find just one non-trivial series that you can compute any kind of analytic expression for including an analytic expression containing special functions like Bessel, Erf, hypergeometric, Lambert W, or other functions.

Could you do that?