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Unexpected behavior of sum differentiation and KroneckerDelta 

Sum[D[Sum[f[p[b + c]], {b, 0, Infinity}, {c, 0, Infinity}], 
  p[1 + a]], {a, 0, Infinity}]

results in

Sum[Sum[KroneckerDelta[1, b]*KroneckerDelta[a, c]*Derivative[1][f][p[b + c]], {b, 0, Infinity}, {c, 0, Infinity}], {a, 0, Infinity}]

where one can expect

Sum[Sum[KroneckerDelta[1 + a, b + c]*Derivative[1][f][p[b + c]], {b, 0, Infinity}, {c, 0, Infinity}], {a, 0, Infinity}]

What can one do to obtain correct behaviour?
POSTED BY: Yaroslav Drachov
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Of course, we will fix this issue in D itself. I was just suggesting a workaround that can be used right now.
POSTED BY: Jose Martin-Garcia

Okay, thanks!
POSTED BY: Yaroslav Drachov

Thank you very much for reporting this issue. Indeed, the result is currently incorrect and should be as you say.

A possible workaround it is to shift the index a as follows, obtaining a single KroneckerDelta factor instead of two:

In[]:= Sum[D[Sum[f[p[b + c]], {b, 0, Infinity}, {c, 0, Infinity}], p[a]], {a, 1, Infinity}]
Out[]= Sum[Sum[KroneckerDelta[a, b + c]*Derivative[1][f][p[b + c]], {b, 0, Infinity}, {c, 0, Infinity}], {a, 1, Infinity}]
POSTED BY: Jose Martin-Garcia

Thank you for response! Your solution is great for this particular example, but maybe it is possible to modify the definition of D to obtain correct behaviour without shifting index? When it comes to differentiating dozens of similar sums, it seems not the easiest solution to shift indices everywhere.
POSTED BY: Yaroslav Drachov
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