HMMT 2011 T8

(a) [5] Find $a_{1}, a_{2}$, and $a_{3}$, so that the following equation holds for all $m \geq 1$ :

$$p_{m}(0)=a_{1} p_{m-1}(0)+a_{2} p_{m-1}(2)+a_{3} p_{m-1}(4)$$

(b) [5] Find $b_{1}, b_{2}, b_{3}$, and $b_{4}$, so that the following equation holds for all $m \geq 1$ :

$$p_{m}(2)=b_{1} p_{m-1}(0)+b_{2} p_{m-1}(2)+b_{3} p_{m-1}(4)+b_{4} p_{m-1}(6)$$

(c) [5] Find $c_{1}, c_{2}, c_{3}$, and $c_{4}$, so that the following equation holds for all $m \geq 1$ and $j \geq 2$ :

$$p_{m}(2 j)=c_{1} p_{m-1}(2 j-2)+c_{2} p_{m-1}(2 j)+c_{3} p_{m-1}(2 j+2)+c_{4} p_{m-1}(2 j+4)$$