# Apply Formal Operators in Discrete Calculus

Formal operations can be applied with ease for discrete calculus operations such as differencing, summation, and Z transforms.

Verify that is the inverse of Sum.

 In[1]:= XInactive[DifferenceDelta][f[k], k]
 Out[1]=
 In[2]:= XInactive[DifferenceDelta][f[k], k]; Sum[%, k]
 Out[2]=

Verify that DiscreteRatio is the inverse of Product.

 In[3]:= XInactive[DiscreteRatio][f[k], k]
 Out[3]=
 In[4]:= XInactive[DiscreteRatio][f[k], k]; Product[%, k]
 Out[4]=

Apply DifferenceDelta to an indefinite sum.

 In[5]:= XTiming[DifferenceDelta[Inactive[Sum][(1 + x)^200, x], x]]
 Out[5]=

This is significantly faster than the evaluated version.

 In[6]:= XTiming[DifferenceDelta[Sum[(1 + x)^200, x], x]]
 Out[6]=

Apply standard properties of ZTransform.

 In[7]:= XZTransform[Inactive[InverseZTransform][F[z], z, k], k, z]
 Out[7]=
 In[8]:= XD[Inactive[ZTransform][f[k], k, z], z]
 Out[8]=
 In[9]:= XZTransform[k Inactive[InverseZTransform][F[z], z, k], k, z]
 Out[9]=

## Mathematica

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