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Vincent's theorem and sin(sexp) ?
While thinking about the sign changes in the truncated Taylor series for functions such as sin(sinh(x)), sin(e^x) or the more exotic ones such as sin(sexp(x)) , sin(exp^[1/2](x)) i was fascinated by apparant trends.

For instance the signs of the truncated taylors ( ignoring the zero's ) of sin(sinh(x)) and sin(e^x) tend to never have more than 9+'s or 9-'s in a row.
Hence the max amount of sign changes appears to be bounded by n/9 where n is the degree (of the polynomial) of the truncation.

Now I know - as most here - that there are formula's for the Taylor coefficients of f(g(x)) when they are known for f and g.

However that does not make those apparant trends seem trivial.

There is alot written about Taylor series and related series.
There is also written alot about polynomials.
SO I might have forgotten something trivial but I seem to be missing the BIG PICTURE here.

Whenever f(x) grows faster than polynomial , sin(f(x)) is somewhat mysterious to me when it comes to sign changes.

Since sin(exp(x)) or sin(sexp(x)) must have an infinite amount of real zero's it is clear that the zero's of that function relate to the sign changes.

So I was thinking about using the error term theorems about Taylor series together with Vincent's theorem to understand this better.

But Im still a bit confused. And I do not know how to generalize things.

Probably there are some uniqueness conjectures to be made about the sign changes of sin(sexp(x)) too. ( uniqueness for sexp )

Im not sure how " standard " these questions are - like I said I might have forgotten stuff - but if they are classical they seem to be well hidden.



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