I have following equation to solve $$p_{k,k+1}v_k=p_{k+N,k}\left(\sum_{j=0}^{k}v_{j+N}\right) \quad \text{if }0\leq k\leq N-1$$ and $$p_{k,k+1}v_k=p_{k,k-N}\left(\sum_{j=1}^{N}v_{j+k}\right) \quad \text{if } k\geq N$$ where $N>0$. Further information is as follows $$\sum_{i=0}^{\infty}v_i=1$$ and (I am not sure if it helps or not) $$p_{k,k}+p_{k,k+1}=1 \quad \text{if } k<N$$ $$p_{k,k-N}+p_{k,k}+p_{k,k+1}=1 \quad \text{if } k\geq N$$ I want to find the formula for $v_k$. Any help in this regard will be appreciated. Thanks in advance.
Edit: My attempt:
I read this problem in one paper in which the author mentions that we can get the value of $v_k$ if we sum $p_{k,k+1}v_k$ (first two equations of this post) on both sides and use the fact that $\sum_{i=0}^{\infty}v_i=1$. In my attempt I assume $N=2$ and I get the following equation $$p_{0,1}v_0+p_{1,2}v_1+p_{2,3}v_2+p_{3,4}v_3+p_{4,5}v_4+p_{5,6}v_5+....=p_{2,0}v_2+p_{3,1}(v_2+v_3)+p_{2,0}(v_3+v_4)+p_{3,1}(v_4+v_5)+p_{4,2}(v_5+v_6)+...$$ But after that I do not know how to get something useful from this infinite looking sequence. Your help in this regard is required. Thanks in advance.
Edit: Related to solution
Fortunately, I have been able to find a proof for this problem. If somebody is interested then I can tell the reference to find the proof. Thank you.