Assume we have
$$ \frac{x(s+1)+x(s)}{2} =\alpha\, x(s+\frac{1}{2})+\beta.$$
the general solution is:
- if $\alpha\neq1$ then $x(s)=c_1\, k_1^{2s}+c_2\,k_2^{2s}+c_3$ ,
where $k_1,k_2$ are the roots of the equation $$k^2-2\alpha k+1=0$$ and $c_1,c_2$ are arbitrary functions of period $\frac{1}{2},c_3=\beta/(1-\alpha)$.
- When $\alpha=1$ we get $x(s)=c_1 s^2+c_2s+c_3,$
where $c_1=4\beta;c_2$ and $c_3$ are arbitrary functions of period $\frac{1}{2}$.
I couldn't solve this recurrence relation by using the characteristic root technique. please help