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

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

Probably, this already has a name and has been worked before, but I could not find any literature on the series. Does anyone know where I can find more readings on the subject, or know when a series like this can be useful? I realize that this looks like a Fourier Series, but given the non-periodic nature of the real $ n ^ x $ Y $ n ^ {- x} $, I'm not sure how I would start deriving the coefficients.

I suppose the $ n ^ {- x} $ it is necessary to include a term in the sum to make sure that the sum remains finite as x goes to infinity; but if I defined the limits of my function between negative infinity and zeros, would it still be necessary? How would I know if I had defined a series that can represent any function as a Fourier series can? Maybe if I can try $ b_n = c_n $, this will resemble a series of hyperbolic Fourier, if such a thing exists.