# Cox-Ross-Rubinstein Model Closed Formula for Call Option

I am I quite new to the topic and at the moment I am self studying the CRR model. My notations are: $$N$$ number of periods, $$\delta T$$ length of one period, $$S_0$$ stock price at time $$t=0$$, $$f_0$$ price of the call option at time $$t=0$$, $$r$$ risk free interest rate (I assume continuously compounding) and $$\widetilde{p}$$ risk neutral probability.

I want to find a closed formula for the "fair" price of the call option at time $$t=i after $$j$$ up movements in the stock price. I already derived the following formula via backward induction for the "fair" price of a call option at time $$t=i$$ and implemented it in R and I get correct results.
$$\begin{equation} f_{i,j}=e^{-r \delta T}\left(\widetilde{p}f_{i+1,j+1}+(1+\widetilde{p})f_{i+1,j} \right), \quad j=0,1, \dots,i \end{equation}$$ $$j$$ denotes the number of up movements in the stock price. I wonder if I can represent this formula in a closed form. For example for $$t=0$$ I know the formula $$\begin{equation} f_0=e^{-rN\delta T}\sum_{j=0}^N {N \choose j}\widetilde{p}^j(1-\widetilde{p})^{N-j}\max\left\{S_0u^jd^{N-j};0\right\} \end{equation}$$ which can be rewritten as $$\begin{equation} f_0=S_0\sum_{j=k}^N {N \choose j} \hat{p}^j(1-\hat{p})^{N-j}-Ke^{-r \cdot N \cdot \delta T}\sum_{j=k}^N {N \choose j} \widetilde{p}^j(1-\widetilde{p})^{N-j} \end{equation}$$ where $$k$$ is the smallest integer such that $$S_0u^kd^{N-k}\geq K$$ and $$\hat{p}=e^{-r \delta T} \widetilde{p} u$$.