Important properties of z-transforms
The proofs can be easily obtained by using basic z-transform definition and transformations in summation.
(1) Linearity If ?[x(n)] = X(z) with the ROC rx1 <|z|< rx2 and ?[y(n)] z2 + 2z , ROC: the whole- = Y(z) with ROC ry1 <|z|< ry2 then ?[a x(n) + b y(n)] = a X(z) + b Y(z) with ROC at least the overlap of the ROC's of Y(z) and X(z) and if there is any pole-zero cancellation because of linear combination, then the ROC can be larger.
(2) Translation (Time-shifting)
If ?[x(n)] = X(z) with ROC r1<|z|< r2 then ?[x(n-k)] = z-k X(z) with same ROC except for the possible deletion or addition of z = 0 or z = ∞ because of z-k.
(3) Multiplication by the complex exponential sequence
If ?[x(n)] = 
X(z) with ROC r1 <|z|< r2 then ?[an x(n)] = X ( z) z→( z / a ) with ROC |a| r1 <|z|< |a| r2.
Example
Given x(n) = {1, 2} and x2(n) = 0.5n x(n-2) find X(z) and X2(z) and their ROCs respectively.
(4) Multiplication by a ramp If ?[x(n)] = X(z) with ROC r1 <|z|< r2 then
?[n x(n)] = - z dX ( z)/dz with ROC r1 <|z|< r2.
Example Given x(n) = {1, 2} and x2(n) = (1+ n + n 2 ) x(n) find X(z) and X2(z) and their ROCs respectively. ?[x2(n)] = ?[1] + ?[n x(n)] + ?[n [n x(n)]] = ...
(5) Time reversal If ?[x(n)] = X(z) with ROC r1 <|z|< r2 then ?[ x(-n)] = X (z -1 ) having ROC (1/ r2 )<|z|< (1/ r1 )
Example Given x(n) = 2- n u(n) and X2 ( z) = z X ( z -1 ) evaluate x2(n).
x(n) = (0.5)n u(n) , X(z) = z/z - 0.5 , ROC: 0.5 <|z|
?-1{ X ( z -1 ) } = x(-n); x2(n) = ?-1{ z X ( z -1 ) } = x(-(n+1)).= 2-( -( n+1)) u(-(n + 1))
(6) Convolution in the time domain leads to multiplication in frequency domain
Given ?[x(n)] =
(7) Multiplication in the time domain leads to convolution in frequency domain
If ?[x(n)] = X(z)
having ROC rx1 <|z|< rx2 and ?[y(n)] = Y(z) with ROC ry1 <|z|< ry2 then
is a complex contour integral and C2 is a closed contour in intersection of the ROCs of X(v) and Y(z/v).
(8) Initial Value Theorem If x(n) is causal sequence with z transform X(z), then
(9) Final Value Theorem If ?[x(n)] = X(z) and poles of X(z) are all in the unit circle then value of x(n) as n→∞ can be given by
some also give this as 
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