Sum of series (x^z-m^x)(y^z-m^y)

              //   y                 \      //   x                 \   //    x*y                   \
              || -----    for |y| < 1|      || -----    for |x| < 1|   ||  -------    for |x*y| < 1|
              || 1 - y               |      || 1 - x               |   ||  1 - x*y                 |
              ||                     |      ||                     |   ||                          |
    x  y    x ||  oo                 |    y ||  oo                 |   ||  oo                      |
oo*m *m  - m *|< ___                 | - m *|< ___                 | + |< ___                      |
              || \  `                |      || \  `                |   || \  `                     |
              ||  \    z             |      ||  \    z             |   ||  \    z  z               |
              ||  /   y    otherwise |      ||  /   x    otherwise |   ||  /   x *y     otherwise  |
              || /__,                |      || /__,                |   || /__,                     |
              \\z = 1                /      \\z = 1                /   \\z = 1                     /

$$\infty m^{x} m^{y} - m^{x} \left(\begin{cases} \frac{y}{1 - y} & \text{for}\: \left|{y}\right| < 1 \\\sum_{z=1}^{\infty} y^{z} & \text{otherwise} \end{cases}\right) - m^{y} \left(\begin{cases} \frac{x}{1 - x} & \text{for}\: \left|{x}\right| < 1 \\\sum_{z=1}^{\infty} x^{z} & \text{otherwise} \end{cases}\right) + \begin{cases} \frac{x y}{- x y + 1} & \text{for}\: \left|{x y}\right| < 1 \\\sum_{z=1}^{\infty} x^{z} y^{z} & \text{otherwise} \end{cases}$$

oo*m^x*m^y - m^x*Piecewise((y/(1 - y), |y| < 1), (Sum(y^z, (z, 1, oo)), True)) - m^y*Piecewise((x/(1 - x), |x| < 1), (Sum(x^z, (z, 1, oo)), True)) + Piecewise((x*y/(1 - x*y), |x*y| < 1), (Sum(x^z*y^z, (z, 1, oo)), True))