Let $P_n (x) , n=0,1,...$ be the monic orthogonal polynomials associated to the weight function $w(x)$. Let $Q(x), n=0,1,2,...$ be the monic orthogonal polynomials associated to the weight function $u(x)$. Suppose $P_m(x)=Q_m(x), P_{m-1}(x)=Q_{m-1}(x)$ for some positive integer $m$, prove that $P_i(x)=Q_i(x)$ for all $i \leq m.$
I have one theorem in hand, namely the orthogonal polynomials $P_n(x)$ satisfy the three terms recurrence relation: for all $n \geq 0$, $xP_n(x)=P_{n+1}(x)+b_{n}P_{n}(x)+a_n^2P_{n-1}(x)$ for unique positive real numbers $a_n^2$ and real numbers $b_n.$
But I have no idea this theorem helps. Thanks in advance.