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.
One term for this is signomial, where the argument is restricted to the positive reals, to avoid definition problems of things like $x^{\sqrt 2}$ for negative $x$, and so on. I hope this term is not in widespread use, but the Wikipedia page does refer to a textbook that uses it.
