Hypertranscendence of the functional equation 𝑔(𝑥²)=[𝑔(𝑥)]²+𝑐𝑥

Borwein, Peter

doi:10.1090/S0002-9939-1989-0979226-X

Hypertranscendence of the functional equation $g(x^ 2)=[g(x)]^ 2+cx$
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Proc. Amer. Math. Soc. 107 (1989), 215-221 Request permission

Abstract:

The functional equation $g({x^2}) = {[g(x)]^2} + cx$ has a unique nontrivial solution that is analytic at zero. We show, for $c > 0$, that the solution of this equation satisfies no algebraic differential equation.

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Pi and the AGM—a study in analytic number theory and computational complexity

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