# When on interval there is smooth measure with integral moments?

Let $$a. When there is a smooth, strictly positive function $$f$$ on $$[a,b]$$ such that for any $$n\in\mathbb Z, n\geq 0$$ we have:

$$m_n(d\mu):=\int_{a}^bx^nd\mu \in\mathbb Z,$$

where $$d\mu=f(x)dx$$.

When $$a=-1, b=1$$ this is clearly impossible as $$0 and $$m_n(d\mu)\to0$$.

When $$a=-2, b=2$$ there is non-trivial example

$$$$\dfrac 1{2\pi}\int\limits_{-2}^2\sqrt{4-x^2}x^k= \begin{cases} C_{k/2}, & \text{if}\ k\ \text{even} \\ 0, & \text{otherwise} \end{cases}$$$$ Here $$C_k$$ is Catalan numbers.

When $$a=0, b=+\infty$$ we can take $$f(x)=e^{-x}$$.

Can we construct such a measure when $$b-a < 4$$?

• Is the key point of this that you want all the $m_n$ to be integers? Sep 18 at 12:55
• Yes, you are correct. Sep 18 at 13:12
• For what it is worth, stackoverflow.com/questions/65975300/… has an example with $a=0,b=4$ (in fact all the moments are Catalan numbers) and there may be simpler examples Sep 18 at 13:16
• Uniform distribution doesn't work! Because integral of $x^k$ has $k$ as denominator. Sep 18 at 19:58