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I want to refer to functions of the form $f(x) = \sum_{i=1}^n a_i x^{\alpha_i}$ where $\alpha_i < 1$.

This is not a polynomial, because $\alpha_i$ could be just real arbitrary numbers (though positive ones). Is there a name for this class of functions that I can refer to? Wikipedia does claim these are not polynomials.

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    Is \alpha<1 a typo for \alpha>0 or $\alpha\geq 0$?2012-02-02

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In 1983 they still had no official name, see for example this paper by Peter Borwein.

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    (though, it seems like this name was rarely used, according to google.)2010-10-24
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Such a function is called "sum of power functions". I don't know of any shorter name. If $\alpha_i<1$ is not a typo, then it's a sum of power functions with exponents less than $1$.