Chapter 1.7, Problem 9E

(a) The formula

$\begin{array}{rll}h\left(1\right)& =& 1,\\ (\ast )h\left(2\right)& =& 2,\\ h\left(n\right)& =& {\left[h\left(n+1\right)\right]}^2-{\left[h\left(n-1\right)\right]}^{\text{2}}\text{ for }n\ge 2\end{array}$

is not one to which the principle of recursive definition applies. Show that nevertheless there does exist a function $h:{\mathbb{Z}}_{+}\to ℝ$ satisfying this formula. [Hint: Reformulate $(\ast )$ so that the principle will apply and require $h$ to be positive.]

(b) Show that the formula $(\ast )$ of part (a) does not determine $h$ uniquely. [Hint: If $h$ is a positive function satisfying $(\ast )$, let $f\left(i\right)=h\left(i\right)$ for $i\ne 3$, and let $f\left(3\right)=-h\left(3\right)$.]

(c) Show that there is no function $h:{\mathbb{Z}}_{+}\to ℝ$ satisfying the formula

$\begin{array}{rll}h\left(1\right)& =& 1,\\ h\left(2\right)& =& 2,\\ h\left(n\right)& =& {\left[h\left(n+1\right)\right]}^2+{\left[h\left(n-1\right)\right]}^{\text{2}}\text{ for }n\ge 2\end{array}$.