We can recursively define a sequence of polynomials by
$P_0(x) := 1$
and then with the definite integral
$P_n(x) := \int_{c_n}^x P_{n-1}(t) ~\mathrm dt$
where the $c_n$ are to be chosen so that
$\int_0^1 P_n(t)~\mathrm dt = 0$
so for $n = 1$ we have $P_1(x) = x - c_1$ so $c_1 = 1/2$. However already for $n=2$ it becomes difficult since $\int_{c_2}^x (t-1/2)~\mathrm dt = t^2/2 - t/2- c_2^2/2 + c_2/2$ and to have $\int_0^1 P_2(t)~\mathrm dt = 0$ there are two solutions for $c_2$ namely
$c_2 = 1/2 \pm \sqrt 3/6.$
So $P_2(x) = x^2/2 - x/2 + 1/12$. For $n=3$ we get $c_3 = 0$ and $P_3(x) = x^3/6 - x^2/4 + x/12$ however for $n=4$ we have $c_4=1/2 - 1/2\sqrt (1 - 2/15\sqrt 30)$ (and 3 other solutions) for $P_4(x)=x^4/24 - x^3/12 + x^2/24 - 1/720$. For $n=5$ again $c_5=0$ (however there are another 4 solutions). Is there some general Formula for the $c_n$ so we could have a shortcut for calculating the coefficients of $P_n(x)$? Also with an eye to fractional calculus it would be nice to know if for the Riemann-Liouville integral
$\frac1{\Gamma(\alpha)} \int_c^x P(t) (x-t)^{\alpha-1}~\mathrm dt$ and
$\int_0^1 P(t)~\mathrm dt = 0$
if there even is a solution for this - let alone if it would fit in the previously defined sequence. The dream result would of course be not only to have a formula for $c_n$ but a function $c(n)$ that continuously defines such a polynomial sequence. But that seems very remote since even in simple cases with for example $\alpha = 1/2$ and $c>0$ the integral produces complex values...