Note on the binomial partial difference equation

Abstract

Some formulas for the “general solution” to the binomial partial difference equation cm,n = cm-1,n + cm-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 (Formula Presented) By using a connection with the scalar linear first order difference equation we show that the equation on the domain D\{(0, 0)}, can be solved in closed form by presenting a formula for the solution in terms of the “side” values (Formula Presented).