"Polynomials" with non-integer exponents

Are there some books or articles regarding "polynomials" with non-integer (real) exponents, i.e., $$f(x)=a_1x^{e_1}+a_2x^{e_2}+\dots+a_nx^{e_n},$$ where $e_1,e_2,\dots$ are any real numbers (and $x$ being also real)?

I am mostly interested in theorems regarding the location of the roots, number of the roots, bounds, etc., of such "polynomials". Thanks.

I think the keyword is exponential polynomial because $x^\alpha=e^{\alpha \log x}$.

Try these papers:

• J.F. Ritt, On the zeros of exponential polynomials, Trans. Amer. Math. Soc. 31 (1929), 680–686.

• C.J. Moreno, The zeros of exponential polynomials (I). Compositio Mathematica, 26 (1973), 69–78.

• G.J.O. Jameson, Counting zeros of generalized polynomials: Descartes' rule of signs and Laguerre's extensions, Math. Gazette 90,(2006), 223–234.