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How should I find an explicit solution for the following function: $$f(n,m)=a \, f(n-1,m+1)+b \, f(n-1,m)+c \, f(n-1,m-1)$$ where $f(1,0)=a+b$ and $f(1,1)=c$ for $n\geq 1$, $m\geq 0$. Also $f(n,-1)=0$ for all $n$ and $f(n,m)=0$ for $m\geq n$.

It looks like Pascal triangle with three terms. This recursive function is cumulative number of customers who are waiting in an appointment system with probability of no-show.

Babak Hoseini
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  • It seems $f(2,0)$ is undefined. Also, some context would help: where does it come from?, is it a model of some physical phenomenon?, and over all, what have you tried? – Miguel Aug 20 '15 at 14:52
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    This needs clarification: How is $f(n,0)$ defined? What about $f(1,m)$? – lulu Aug 20 '15 at 14:54

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