Equation solution needed

I have following equation to solve pk,k+1vk=pk+N,k(k∑j=0vj+N)if 0≤k≤N−1p_{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 pk,k+1vk=pk,k−N(N∑j=1vj+k)if k≥Np_{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>0N>0. Further information is as follows ∞∑i=0vi=1\sum_{i=0}^{\infty}v_i=1 and (I am not sure if it helps or not) pk,k+pk,k+1=1if k