Is there a general series with a real function exponent for n?

[MathematicalPhysicist]

usually in analysis we are dealing with power series, my question is there coverage of a general series with exponent is any real function of n natural.
i.e something like this:
$$\sum_n a_n x^{b_n}$$
where b_n is some real function of n.

Thanks in advance.

 

[HallsofIvy]

I am not sure what your question is. Certainly one can have a series like that but it is not a power series and would not, in general, have the nice properties, radius of convergence, etc., of power series. I suspect you could show similar properties for specific sequence \(\displaystyle b_n\) but they would then be as restrictive as power series.

 

[MathematicalPhysicist]

I guess I am looking for something dealing with the most general case of a series such as this one, if something like this exists.
obviously a radius of convergence like in power series is not guarnteed, but perhaps something else?