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There is a mathematical concept of polynomials with non integer degree? for example, something mid way between a linear function and a parabola.

I have interest in a general expression which can continuously vary between polynomials of integer degrees, and of course many interpolation may be defined, but there is anyone which preserves interesting properties of polynomials? Or some "canonical" or generally accepted or generally useful way to interpolate between polynomials?

Maybe there is a canonical way to generalize the product operator? $$\prod_{n=0}^{r\in \mathbb{R}}\left ( x-a_n \right )$$

Possibly a "reasonable" polynomial with the same root $a_n=a_0$ would be $(x-a_0)^r$

Rodrigo de Azevedo
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zexot
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    There are functions of the form $\sum a_ix^{r_i} $ with $r_i \in $\Bbb R $ but one wouldnt call them polynomials – krirkrirk Mar 06 '18 at 13:42
  • I think many properties get lost, if we allow non-integer exponents. – Peter Mar 06 '18 at 13:42
  • Yes, what properties you want to keep is relevant. Function of the form $\sum_i a_i x^{r_i}, r_i \in \mathbb R$ would be closed under $+$ and $\times$, $\frac{\mathrm d}{\mathrm dx}$, but is that enough ? – Lærne Mar 06 '18 at 13:44
  • Not completely answering the questions but is related, every algebraic expression( https://en.m.wikipedia.org/wiki/Algebraic_expression ) can be rewrite as rational fraction ( https://en.m.wikipedia.org/wiki/Rational_function ) – ℋolo Mar 06 '18 at 13:47
  • @zexot Adding to what Peter said, polynomials with non-integer degrees can introduce complex solutions. For example, $x^2-2^2$ only has roots $-2$ and $2$, but $x^{2.5}-2^{2.5}$ has roots $2$, and two other complex roots (according to Wolfram Alpha). – Toby Mak Mar 06 '18 at 13:48
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    @Toby well $x^2+1$ also has complex roots so I'm not sure I get your point? – krirkrirk Mar 06 '18 at 13:50
  • @krirkrirk What I'm saying is the complex roots for similar equations (one with integer degree, one without) are much neater with polynomials. $x^{2.5}+1$'s complex roots are quite nasty in their Cartesian form. – Toby Mak Mar 06 '18 at 13:53

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