How Can I Derive Sumations with Multiple Indices and Powers?

[jetoso]

I have a hard time trying to figure it out how to derive the following sumations:

We know that:
P(z) = Sumation from j=0 to Infinity of [(Pj * (z^j)]
and
Q(z) = Sumation from j=0 to Infinity of [(Kj * ((z^j)]

Where j is a subindex for P and K, and a power for z.


Sumation from j=0 to Infinity of [ Sumation from i=0 to j+1 of [(z^j) * Pi * Kj-i+1] ]

Where i and j are a subindices for P and K, and a power for z.



I know that I have to play with the indices, but I have no clue. Any advice??

 

[mathman]

I may be missing something, but I can't see what your question is?

 

[jetoso]

Reply

Yes, you are right.
From the information above, it has to be simplified to:
[ (1/z) * P(z) * Q(z) ] - [ (1/z) * Po * Q(z) ]

The author of the book does not give much more information about how he derived this equation, but he says is trivial.

 

[mathman]

The basic trick is to switch the order of summation. Then you will have two parts. The main part has i=1,inf with j=i-1,inf. The other part has i=0 with j=0,inf. For the main part let n=j-i+1 (replacing j), then n=0,inf. Put it all together and you should get the required answer.

 

[jetoso]

Sorry

I am sorry, but I still do not get it.

 

[mathman]

Do you understand switching the order of summation?
Do you see the result of switching?
Do you understand the change of variables (n=)?