I have obtained the following solution to a recursive equation $r(m,n)=\frac{ab^{n-1}c^{m-1}}{(2a-1)^{m+n}}\left[\frac{(2a)^n}{b^{n-1}}-\sum_{i=1}^{n}(2a-b)^i(1-b)^{m-i}\binom{m-1}{i-1}\sum_{j=1}^{n-i+1}\binom{j+i-2}{i-1}(\frac{2a}{b})^{n-j-i+1}\right]$, where a, b, c are combines of some parameters to the recursive equation $r(m,n)=\frac{c}{2a-1}[r(m,n-1)+c\frac{2a-b}{2a-1} \sum_{j=1}^{m-1} \left((1-b)\frac{c}{2a-1}\right)^{m-j-1} r(j,n-1) ]$ and the initial condition $r(m,0)=a\frac{c^{m-1}}{(2a-1)^m}$. I need the above solution to be more compact. It would be better to get a simplified and a single term solution. I guess there may be a simple sum to the series involved there in.
2026-04-08 05:37:16.1775626636
Simplification of a solution of a recursive equation
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