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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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I want to change the name of this thread since polynomials by definition have only nonnegative integer powers of the variable, but I can't think of a better title... :( – J. M. Oct 24 '10 at 23:46
@J.M.: "linear combinations of real powers of $x$"? – Arturo Magidin Oct 25 '10 at 0:53
I stated it as a question... so perhaps it is okay. "is it a polynomial of $x$?" – normvector Oct 25 '10 at 12:46
Is $\alpha<1$ a typo for $\alpha>0$ or $\alpha\geq 0$? – user23211 Feb 2 '12 at 19:30

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

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Thanks. From the title of the paper, don't you think it is called "polynomial with variable exponents?" – normvector Oct 24 '10 at 23:17
(though, it seems like this name was rarely used, according to google.) – normvector Oct 24 '10 at 23:18

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 $1$.

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