A small trove of functional equations

Resumen

A new proof is presented for an old algebraic identity which is then used to produce the general functional relation 

\(\ sum_{k=0}^{n-1} \frac{(m)_k}{k!} g(m,k) + \sum_{k=0}^{m-1} \frac{(n)_k}{k!} g(k,n) = g(0,0)\)

where \(\ g\) is an Euler transform, and a related integral identity. Several examples are given.