My question: I have to find a function $g$ fulfilling the equation $$2\frac{t_k \cdot t_0 - 1}{t_k-t_{-1}} = g(t_k) + g(t_{k+1}) + t_{k+1}\cdot g(t_k)g(t_{k+1})$$ Whereby $t_{n+1}=t_n + h$ with $t_0, h \in \mathbb R$. My idea is to derive an recursive definition such as $$g(t_{k+1}) = \frac{1}{1+t_{k+1}\cdot g(t_k)} \left(2\frac{t_k \cdot t_0 - 1}{t_k-t_{-1}} - g(t_k)\right)$$ with a suitable value for $g(t_0)$. Are there better attempts to solve such a problem (also similar ones).
My Motivation: I have a discrete function which I want to approximate. I take the difference quotient $\frac{f(t_{k+1})-f(t_k)}{h}$ between two neighbored function values. When I can find a function $\gamma(t,y)$ such that $$f(t_{k+1}) = f(t_k) + \tfrac h2 \gamma(t_k, f(t_k)) + \tfrac h2 \gamma(t_{k+1}, f(t_k) + h \gamma(t_k, f(t_k)))$$ I can say that the function $f(x_k)$ are obtained by the Heun's method of the ODE $$y^\prime(t) = \gamma(t,y)$$. I hope, that the solution $y(t)$ of this ODE (for a suitable initial value) is a good approximation of $f(x_k)$. The above problem is one problem in getting $\gamma(t,y)$. I appreciate also any reference to work where similar problems are discussed...