Note on the binomial partial difference equation

摘要

Some formulas for the "general solution" to the binomial partial difference equation $$c_{m,n}=c_{m-1,n}+c_{m-1,n-1},$$ are known in the literature. However, it seems that there is no such a formula on the most natural domain connected to the equation, that is, on the set $D=ig{(m,n)inmathbb{N}^2_0 : 0le nle mig}.$ By using a connection with the scalar linear first order difference equation we show that the equation on the domain $Dsetminus{(0,0)}$, can be solved in closed form by presenting a formula for the solution in terms of the "side" values $c_{k,0}$, $c_{k,k}$, $kinmathbb{N}$.