Post #2060191
2026-05-03 19:53 UTC
Just learned that the theory of generating functions can be seen as harmonic analysis on Z?? Then the usual way we use generating functions to solve recurrence relations is a special case of Fourier transforms turning translation into multiplication, which turns the recurrence relation into an algebraic equation!
If you haven't seen this before, it's really fun to work out for yourself ^_^.
Whether or not you've seen this before, post your thoughts here! If you already knew this, I'd love to know where you learned it, and if there's any other applications of this observation! And if this is new to you I'd love to see your thoughts as you try to prove this yourself!
Replies (2)
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@counting_is_hard@mathstodon.xyz 2026-05-03 21:40
@hallasurvivor the laplace transform link has always been there, in e.g. feller, but I always felt it was treated as a kind of magic trick by most undergrad probability classes (when they bothered to include it)
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@MaloTarpin@mathstodon.xyz 2026-05-03 22:29
@hallasurvivor if I understand correctly what you mean, this is pretty well-known in signal processing under the name Z-transform. Although it's not exactly the same as a Fourier transform because you're not limited to the unit circle so there is fun to have with the Hardy space and such.