let $P_0(x)=0$;$P_1(x)=1$ Let $\forall n $ integer $ \geq 2$, $\forall x$ real, $$P_n(x)=\displaystyle \sum_{k=0}^{n-1} C_{n+k}^n (-x)^k \alpha_{n,k}$$ and where $\forall k$ such that $0 \leq k \leq n-1 $ $$ \alpha_{n,k}= \displaystyle \sum_{p=1}^{n-k} \displaystyle C_{n}^{n-k-p} \frac{(-1)^{p+1}}{p}$$ So it's easy to check that $ \forall n $ integer $ \geq 1$, degree of $P_n$ is $n-1$.

I have found the following using Mapple: $ \forall 0 \leq n \leq 20,$

$(n+2).P_{n+2}(x)-(2.n+3).(1-2.x).P_{n+1}(x)+(n+1).P_{n}(x)=0$

I think i can proove that this recurrence relation is true $ \forall n $ integer with working on the coefficient of the polynom $P_n$.

What interess me is to know the weight $w$ such that $\int_{0}^1 P_n(x)w(x)x^idx=0$, ( PS:i'm not sure for the existence of $w$)

and i need to proove that $ \forall n \geq 1, \forall x \in [0,1], |P_n(x)| \leq |P_n(0)|$.

Thanks for your help

Ps: the familiy of legendre polynomial $L_n(x)=\displaystyle \frac{1}{n!}(x^n (1-x)^n)^{(n)}$ satisfy exactly the same recurrence relation but obviously with not with others initials conditions.