Let $X\in\mathbb{Z}.$ Let $F_n$ be a sequence of positive integers given by
$$F_{i+1}=F_i+F_{i-1}$$
$$F_2=X*F_1+F_0$$
I am trying to find an upper bound or (sharp) inequality of $F_i$ in terms of $F_1$ and $F_0$.
If $F_{i+1}=F_i+F_{i-1}$ for all $i$, then substituting $F_i=q^i$ gives $F_i=(\frac{1+\sqrt{5}}{2})^i$.
In our case, $$F_j= (\frac{1+\sqrt{5}}{2})^j+(XF_1+F_0)-(F_1+F_0)= (\frac{1+\sqrt{5}}{2})^j+(X-1)F_1 \,\,(*)$$
Is the first equality in $(*)$ correct?
On
Just to add a formula in terms of Fibonacci numbers: Let $u_k$ denote the $k$th Fibonacci number, for $k=1,2,3,4$ the sequence $1,1,2,3,...$, and extend backwards to define $u_0=0$ and $u_{-1}=1$ (so the same recursion works on all the $u_k$). Then your sequence of polynomials is, using $b=F_0,\ a=F_1,$ $$F_k=au_kx+bu_{k-1},$$ which satisfies the recurrence (since the $u_k$ satisfy it), and gives $F_0$ and $xF_1$ for the first two values, thereafter matching the sequence generated by the recursion. So bounds on the $F_i$ would involve knowing the initial $F_0,F_1$, as well as the range of values $x$ may take on, since if $x$ is not restricted there is no bound.
To fill this out and get a bound, use the fact that the Fibonacci number $u_n$ is alternately slightly above and below the value of $\phi^n/\sqrt{5}$ (above when $n$ is even), so that you may safely use the upper bound $B_n=(\phi^n/\sqrt{5}+1)$ for $u_n.$ Then if $0\le x \le d$ you have the bound $$F_k \le aB_kd+bB_{k-1}.$$
On
Any fibbonaccci series starting with (a,b) can be written as $a(1,0)+b(0,1)$. These two are the fibonacci series begining at $F(-1)$ and $F(0)$.
Counting these terms of the series start as $0$ and $1$, we have
$$T(n) = \phi^{n} (a /\phi +b) / \sqrt{5}$$
derived from adding two fibonacci convergences together.
The general solution of $F_{i+1}=F_i+F_{i-1}$ is $F_j=A\phi^j+B\overline\phi\,^{j}$ where $\phi=(1+\sqrt5)/2$ and $\overline\phi=(1-\sqrt5)/2$, and $A$ and $B$ are determined by the initial conditions. In your case the initial conditions are $F_1=F_1$ and $F_2=XF_1+F_0$. So you can substitute in $i=1$ and $i=2$ to get two linear equations for the two unknowns $A$ and $B$, and then you'll have your equation for $F_j$.