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One thing I've been working on lately is to derive a series that can be used for constructing generating functions for superexponential sequences. Did you know

One thing I've been working on lately is to derive a series that can be used for constructing generating functions for superexponential sequences. Did you know that any GF, ie. OGF, EGF, DGF, etc.. don't exist for these sequences? Because these GF don't converge. The growth of the series needs to be balanced with the growth of the coefficients in order to provide convergence. We need to find a new series if we want to allow for Combinatorial species and Analytic Combinatorics of fast growing sequences.

zmgsabst 5y ago
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 ago
  Generating 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 ago
    If 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.

zmgsabst 5y ago

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 ago
Generating 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 ago
  If 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.