Sequences and Series – Can the coefficients for an infinite sum of Sinh and Cosh be determined?

I am interested in trying to represent a function as a series of the form:

$ f (x) = a_0 + sum_ {n = 1} ^ infty b_nn ^ x + c_nn ^ {- x} $

This probably already has a name and was already working with him, but I could not find any literature about the series. Does anyone know where I can read more on the topic or when a series like this can be useful? I realize that this is similar to a Fourier series, but given the unperiodic nature of the real $ n ^ x $ and $ n ^ {- x} $I am not sure how I would begin to derive the coefficients.

I suspect this $ n ^ {- x} $ The term must be included in the summation to ensure that the summation remains finite when x goes to infinity. But if I were to define the limits of my function between negative infinity and zeroes, would that be necessary? How would I know if I defined a series that could represent any function as a Fourier series? Maybe if I can prove it $ b_n = c_n $this will be similar to a hyperbolic Fourier series, if there is such a thing.


I have succeeded in transforming the above equation into a new form. Maybe it's easier to find the coefficients for the equation:

$ f (x) = a_0 + sum_ {n = 1} ^ infty b_n sinh (x) + c_n cosh (x) $

However, since sinh and cosh are divergent, my attempts to find the coefficients are always infinite.