Sequence and Series Reconstruction

[Liquid7800]

After studying Generating Functions some, I wondered this question:

1). Take an arbitrary sequence of numbers for example:

{p1,p2,p3,p4...pN} and get/find a Generating Function for that sequence.

2). Now 'remove' an arbitrary sequence members from the above set :{p1,p2,p3,p4...p100} which might look like this after removal of arbitrary sequence members:

{p1,p2,,p8,,,,p20,p21,p22,,,,,,p45,,,...p100}

Now the question becomes:

Can you produce a Function (another Generating Function?) to make the sequence 'recognizable' again...how would you re-generate or re-facilitate the 'missing' sequence members or 'holes'...maybe using some combination of techniques to define a frame of 'reference' from an initial encoding from the original (full) sequence.

I have no idea if this problem(s) has been researched or even solved before...

but I was just thinking of how you could 'create' or 'fill' missing information mathematically...and the problem of sequence reconstruction came to mind.

If I am unclear as to what I am asking please let me know.

...I guess what I am looking for is some starting points, pre-requisite mathematical knowledge, theory, subject areas, refernce materials etc., from people who know what they are doing--- (anyone worked on this kind of thing before?)...

This is going to be a personal thing to mathematically immerse myself in as a project to explore, since I love series and Generating Functions.

Thanks for listening!

 

[Gerenuk]

Isn't it that you can have a generating function for just every sequence? I mean I write out the series and the generating function is whatever comes you in the end.
In that case of your you cannot regenerate.

Here is a question back to you:
I wrote a program to generate random numbers 0-9. Then I deleted every other number. My result is "3751961764396515591391651592". Now what were the missing numbers in between?