Question: how to calculate this example

http://uknowledge.uky.edu/cgi/viewcontent.cgi?article=1001&context=math_etds

Example 3.23 in page 24

degree(2*z^2 + z + 2); #2

how to convert rational function to into (something) - (something) + (something) like in the example

the example 's rational function is 2*z^2+z+2

if rational function is has denominator how to calculate?

2z^2 + z + 2 =

z+2
3


−

z−2
3


+

z+1
2


−

z−4
2


+

z
1


−

z−8
1

when calculate delta p1, what is bi? how to calculate like the example's one?

(z+i-1 i+1-j) - (z+i-bi-1 i+1-1)

sum(d,i=j) (z+i-1 i+1-j) - (z+i-bi-1 i+1-1)

3.23
(z 1) - (z-bi 1)

how to do the expansion in step 3, as i use right shift that can not calculate the same result as in the example.

x0*x1/x0 = x1; # not same as in the example for I

x1^4*x2/x1 = x2*x1^3;  # not same as in the example for I

finally would like to apply to hilbert series 1/((1+z)*(1+z^2))