Strange result with an indefinite double sum

Question

Bug introduced in V10.0.2 or earlier and persists through V11.3

(Simplified version - courtesy Lukas Lang - reported [CASE:4208776])

I try to compute

Sum[2^Max[0, 2 h - b - c - 2] x^b y^c z^(2 h), {h, \[Infinity]}, {b, h}, {c, h}]

and get

(-y z^2 - x y z^2 + x^4 y z^2 + 4 y z^4 + 6 x y z^4 + 3 x^2 y z^4 + x^3 y z^4
 - 4 x^4 y z^4 - 2 x^5 y z^4 + 2 y^2 z^4 + 3 x y^2 z^4 + 2 x^2 y^2 z^4 + x^3 y^2 z^4
 - 2 x^4 y^2 z^4 - x^5 y^2 z^4 - 8 x y z^6 - 8 x^2 y z^6 - 4 x^3 y z^6 - x^4 y z^6
 + 8 x^5 y z^6 - 8 y^2 z^6 - 16 x y^2 z^6 - 14 x^2 y^2 z^6 - 10 x^3 y^2 z^6 + 6 x^4 y^2 z^6
 + 8 x^5 y^2 z^6 + 2 x^6 y^2 z^6 - 2 x y^3 z^6 - 3 x^2 y^3 z^6 - 2 x^3 y^3 z^6
 + 2 x^5 y^3 z^6 + 4 x^4 y z^8 + 16 x y^2 z^8 + 24 x^2 y^2 z^8 + 20 x^3 y^2 z^8
 + 10 x^4 y^2 z^8 - 16 x^5 y^2 z^8 - 8 x^6 y^2 z^8 + 8 x y^3 z^8 + 16 x^2 y^3 z^8
 + 14 x^3 y^3 z^8 + 4 x^4 y^3 z^8 - 8 x^5 y^3 z^8 - 4 x^6 y^3 z^8 - 8 x^4 y^2 z^10
 - 16 x^2 y^3 z^10 - 24 x^3 y^3 z^10 - 16 x^4 y^3 z^10 + 16 x^6 y^3 z^10)
 /(x (-1 + 4 z^2) (-1 + 2 x z^2) (-1 + 2 y z^2) (-1 + x y z^2))

This cannot be correct since the input is symmetric in x and y while the output is not.

At the same time

Sum[2^Max[0, 2 h - b - c - 2] x^b y^c z^(2 h), {h, 2, \[Infinity]}, {b, h}, {c, h}]

seems to sum up correctly.

Is there some known bug or I am doing something wrong?

Answer 1

There is definitely something broken here. Consider only the inner two sums:

s = Sum[2^Max[0, 2 h - b - c - 2] x^b y^c z^(2 h), {b, h}, {c, h}];

Inserting h->1 and simplifying:

s /. h -> 1 // FullSimplify
(* ((-1 - x + x^4) y z^2)/x *)

which is clearly wrong. We get the correct result by specifying h==1 before evaluating the sum:

 Sum[2^Max[0, 2 h - b - c - 2] x^b y^c z^(2 h), {b, h}, {c, h}, Assumptions -> h == 1]
(* x y z^2 *)