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Is there an accepted terminology for a function that's like a polynomial function (or a formal expression that's like a polynomial) except that the exponents don't have to be natural numbers? That is, something of the form $$ \sum _ { k = 1 } ^ n a _ k x ^ { e _ k } = a _ 1 x ^ { e _ 1 } + a _ 2 x ^ { e _ 2 } + \cdots + a _ n x ^ { e _ n } \text , $$ where $ x $ is the argument of the function (or an indeterminate), $ n $ is a natural number, each $ a _ k $ is a real number, and each $ e _ k $ is a real number? (A variation in another number system may be useful, but it would need to be more general than just integers; ‘Laurent polynomial’ is not the answer.)

I know that such functions are not of great mathematical importance, although they do form a commutative algebra (at least if you treat them as formal expressions so as not to worry too much about their domains). But I get tired of telling my Calculus students, when we first start learning to differentiate and integrate, that we know how to handle ‘polynomials and things that are like polynomials except that the exponents don't have to be whole numbers’, and it would be handy to have a shorthand for that.

Toby Bartels
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  • (It was my first Google hit for polynomials with noninteger exponents.) – Mark S. Apr 10 '23 at 00:27
  • @MarkS. : That sounds promising, thank you! (The algorithm suggested a different question for me that didn't work, and at first I thought that you were linking to the same one, but you didn't.) – Toby Bartels Apr 10 '23 at 00:32
  • https://en.wikipedia.org/wiki/Signomial gives an answer, but .... – kimchi lover Apr 10 '23 at 00:56
  • @MarkS. : On a closer look, exponential polynomials aren't quite right, since they're generalized polynomials in $ \ln x $ instead of in $ x $ (at least the exponential part is). As I think you just noticed in that other thread. Although this will be a useful term in my Differential Equations course! – Toby Bartels Apr 10 '23 at 01:55
  • @kimchilover : That seems to work, other than the restriction on the domain (although I can understand why they did that). But it's definitely what I want as far as the formal expressions go. – Toby Bartels Apr 10 '23 at 01:58
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    @TobyBartels How about linear combination of power functions. – dxiv Apr 10 '23 at 02:15
  • @dxiv : Not very snappy, but better than what I've been saying, so thanks! – Toby Bartels Apr 10 '23 at 05:11

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I kept looking around and found a partial answer: if the exponents can be arbitrary rational numbers, then this is a Puiseux polynomial (also called a Newton–Puiseux polynomial). This is not quite a good answer, since I want arbitrary real exponents, and I can't fix this by saying something like ‘real Puiseux polynomials’ (since that would just mean that the coefficients are real).

I therefore suggest the term transcendental Puiseux polynomial, because by allowing arbitrary real exponents, we're allowing the functions to be transcendental. And this is one of the simplest ways to make them transcendental, since every transcendental Puiseux polynomial is a limit of Puiseux polynomials. (Although there are other things that are limits of Puiseux polynomials, such as Puiseux series.)

But this is a term that I just made up, and maybe somebody knows a better answer.

Toby Bartels
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