Rearranging of terms to determine exist distribution

Question

(1.19) is saying that a markov chain in moving from state x to some state x+1 requires taking into consideration the possible states that may occur during this transition of state and the associated probability.

Between equation (1.19) and (1.20), the author rearranged the terms to obtain $p(h(x+1)-h(x)) = q(h(x)-h(x-1))$. Assuming he is only rearranging the terms, I fail to see how he arrived at the above expression.

Any input is greatly appreciated.

Answer 1

$$h(x)=ph(x+1)+qh(x-1)$$

Note that we have $p+q=1$, hence

$$(p+q)h(x)=ph(x+1)+qh(x-1)$$

$$ph(x)+qh(x)=ph(x+1)+qh(x-1)$$

Try to rearrange the terms to get the desired equation.