Differing behaviour of GeneratingFunction with different numbers of parameters

Question

When given a single variable sequence, GeneratingFunction is automatically expressed in the most "direct form", for example,

GeneratingFunction[(i + 1) v[i - 1], i, x]

returns something like

2 x GeneratingFunction[v[i], i, x] + v[-1] + x^2 D[GeneratingFunction[v[i], i, x],x]

But, if I add an extra variable that does not otherwise affect the result

GeneratingFunction[(i + 1) v[i - 1, j], {i, j}, {x, y}]

then the function is not evaluated in the same way, although, it would seem that it would be correct to return

2 x GeneratingFunction[(i + 1) v[i - 1, j], {i, j}, {x, y}] + v[-1,j]
 + x^2 D[GeneratingFunction[(i + 1) v[i - 1, j], {i, j}, {x, y}],x]

what is happening here, how can I make it behave in a way that would fit this intuition?

Answer 1

If you don't include j or y in GeneratingFunction[], then you get something like what you want:

GeneratingFunction[(i + 1) v[i - 1, j], i, x]
   2 x GeneratingFunction[v[i, j], i, x] + v[-1, j] +
   x^2 Derivative[0, 0, 1][GeneratingFunction][v[i, j], i, x]