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In response to the dead poster: An ordinary generating function converges only when the coefficients of the sequence grow no faster than polynomial growth. On
by goldenkey 5y ago
In response to the dead poster:
An ordinary generating function converges only when the coefficients of the sequence grow no faster than polynomial growth. On the other hand, exponential generating functions converge for sequences that grow faster than polynomials, including some exponential growth. Therefore, calculus of exponential generating functions is wider in scope than that of ordinary generating functions but still not large enough to represent superexponential sequences.
- ogogmad 5y agoGenerating functions are often treated as purely formal objects. You only have to worry about convergence if you need to evaluate them at some complex number: https://en.wikipedia.org/wiki/Formal_power_series https://en.wikipedia.org/wiki/Formal_power_series Are you trying to use complex analysis to study fast-growing sequences?
- goldenkey 5y agoIf a generating function does not condense to some small symbolic expression, it is next to useless. Yes, the radius of convergence is not very relevant for G.F., however, without any convergence, there will be no shorthand symbolic expression. And that's really what we are after with G.F. For example 1/(1-x) for the OGF of the sequence 1,1,1,1.....etc. We don't care that the interval of convergence is between (-1,1). However if there was no convergence at all, the expression 1+x+x^2+x^3..... would not simplify.