Let $P_n(x)$ be Legendre polynomials: $$\frac{1}{\sqrt{1-2tx+t^2}}=\sum\limits_{n=0}^{\infty}P_n(x)t^n.$$ Usual arguments from the theory of formal groups allow to prove that for any $n$ $$P_n(x)=Q_n(P_1(x),P_2(x),P_4(x), \ldots,P_{2^k}(x),\ldots),$$ where $Q_n$ is a polynomial with integer coefficients, depending on $P_{2^k}(x)$ such that $2^k\le n$. (Over $\mathbb Q$ every $P_n$ is a polynomial of $P_1$ and $P_2$.) For example $$P_1=x,\quad P_2=\frac{1}{2}(3x^2-1),\quad P_3=\frac{1}{2}(5x^3-3x)=P_1(3P_2-2P_1^2),$$ $$P_4=\frac{1}{8}(35x^4-30x^2+3),\quad P_5=\frac{x}{8}(63x^4-70x^2+15)=P_1(5P_4+4P_1^2(P_1^2-5P_2)).$$ Here we have binary expasion in leading terms: $$P_3=3P_2P_1+\ldots,\quad P_5=5P_4P_1+\ldots$$ Probably this property (existence of $Q_n$) is known. Can you give any references concerning this fact?

For any $n$ the polynomial $Q_n$ is unique if (thanks to Victor Kleptsyn) we consider symbols $P_1$, $P_2$, $\ldots$ as coefficients of the formal group law. I don't know whether it is interesting to study properties of $Q_n$ but I'll be greatful for any additional information.