I know that the general expression for polynomial with degree $n$ is $a_n x^n + a_{n-1}x^{n-1} + \cdots + a_0$. My question is, is there any specific name to linear combination of $x^{\text{rational number}}$?. For example $2 x^{2.5} + 1.3 x^{1.3}$. Notice that we have a similar question here, but there is not any related answer and it is not asking about the terminology. Also, it is not a rational function (is it?).
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2Puiseux polynomials, perhaps? https://en.wikipedia.org/wiki/Puiseux_series – Gary Apr 15 '20 at 11:48
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1No, those are not rational functions since $f(x)= \sqrt{x}= x^{1/2}$ is such a function. I would call them "radical functions" but I doubt that is standard. – user247327 Apr 15 '20 at 12:16
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@Gary it's a definition of a set of series, not a set of functions. However, it's the nearest guess : ) – OmG Apr 15 '20 at 13:46
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@OmG A series in an unknown is a function (or, more pedantically, may be identified with a function). Though it seems that you may object to the fact that a series has infinitely many terms, while your functions do not. – Xander Henderson Apr 15 '20 at 15:35
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@XanderHenderson exactly. – OmG Apr 15 '20 at 15:43
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2I could call $2 x^{2.5} + 1.3 x^{1.3}$ a univariate signomial. Or a univariate posynomial. – Rodrigo de Azevedo Apr 16 '20 at 03:32
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First and foremost, these are definitely not rational functions. A rational function is a function of the form $$ x \mapsto \frac{p(x)}{q(x)} $$ where both $p$ and $q$ are polynomials.
To the best of my knowledge, the answer to you question is that there is no generally understood term for functions which are of the form $$ x \mapsto \sum_{n=1}^{N} a_n x^{r_n}, \tag{1}$$ where the $a_n \in \mathbb{R}$ and $r_n \in \mathbb{Q}$. However, there are a couple of idea which come to mind:
In many precalculus classes, the term radical function is used to mean any function of the form $$ x \mapsto \sqrt[n]{a(x-h)}+h. $$ While this term doesn't seem to appear much beyond elementary coursework, I think that a function of the form (1) could be called a linear combination of (powers of) radical functions, or perhaps a generalized radical function.
Given a function of the form (1), each of the $r_n$ can be written in terms of a common denominator. Specifically, if $$ r_n = \frac{p_n}{q_n} $$ in lowest terms, then take $q = \operatorname{lcm}(q_1, \dotsc, q_n)$. $$ \sum_{n=1}^{N} a_n x^{r_n} = \sum_{n=1}^{N} a_n (x^{1/q})^{\hat{p}_n}, $$ where $\hat{p}_n$ is the integer satisfying $$ \frac{\hat{p}_n}{q} = r_n. $$ This means that every function of the form (1) can be written as a polynomial in $x^{1/q}$. It therefore seems reasonable to call such a function a radical polynomial.
Alternatively, these functions are given by Puiseux series with only finitely many terms. It therefore seems reasonable to refer to them as truncated Puiseux series.
Finally, if you are seeing these functions in a particular context, it might make sense to name them for that context. For example, I work with functions defined by integrals of the form $$ \int_{0}^{\delta} f(t) t^{-s} \mathrm{d}t. $$ Because of their close relation to Dirichlet series, which are of the form $\sum a_n n^{-s}$, we refer to these as Dirichlet type integrals. Bessel functions are so named because they solve the Bessel equation (which is, in turn, named for one of the early mathematicians to work with such equations).
In any event, whatever you end up calling these functions, clearly define your terms early on, and use them consistently.
Xander Henderson
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