Let $n$ integer $\geq 2$ ( or suffisemment big) fixed and for $ x$ real, let $$P_n(x)=\displaystyle \sum_{k=0}^{n-1} C_{n+k}^n (-x)^k \alpha_{n,k}$$ and  where $\forall k$ such that $  0 \leq k \leq n-1 $ $$ \alpha_{n,k}= \displaystyle \sum_{p=1}^{n-k} \displaystyle C_{n}^{n-k-p} \frac{(-1)^{p+1}}{p}$$.

NUMERICALY i have found the following:   $ \forall 2 \leq n \leq 10 $ ,

$$\max_{[0,1]} |P_n(x)|=|P_n(0)|=\sum_{j=1}^{n} \frac{1}{j}$$

 Is there an article about $P_n$ or similar polynom?

i need  a proof that it's true $\forall n \geq 2 $.



i have noticed NUMERICALLY too  that $P_n(x)=(-1)^{n+1}P_{n}(1-x)$ and $P_n$ have exactelly $n-1$ zeros over $]0,1[$ . ALL this properties are very similar to those of Legendre polynomial: $L_n(x)=\displaystyle \frac{1}{n!}(x^n(1-x)^n)^{(n)}$ 

I suppose that the family of $P_n$ are orthogonal but i don't know to what, and may be satisfy a recurrence relation...

if any one suggest an idea for (the proof that i suppose true) thanks for his help 


i'm looking too  for an integral representation  of type $ \int_{0}^1 f^n(t).g(t)dt$ for $P_n(x), 0<x<1$ bur it's not very important for me, what interess me the most is the maximum od$ P_n$ over $[0,1]$