# Differing behaviour of GeneratingFunction with different numbers of parameters

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?

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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]